such that $F({\mathbf{P}})\subset{\mathbf{Q}}$. Let
${\mathbb{B}}=(X,\mu_{X},\eta_{X},\Delta_{X},\varepsilon_{X})$ be a bimonoid
in ${\mathbf{C}}$ and
$\widetilde{\mathbb{B}}=\mathop{\mathbf{Bimon}}\nolimits(F){\mathbb{B}}$ be
the corresponding bimonoid in ${\mathbf{D}}$. Then the restriction of the
functors (2.105) are the functors (3.67) and (3.73), where we have
${\mathbb{M}}=(X,\mu_{X},\eta_{X})\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$,
$\widetilde{\mathbb{M}}=\mathop{\mathbf{Mon}}\nolimits(F){\mathbb{M}}\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{D}})$
and
${\mathbb{O}}=(X,\Delta_{X},\varepsilon_{X})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{C}})$,
$\widetilde{\mathbb{O}}=\mathop{\mathbf{Comon}}\nolimits(F){\mathbb{O}}\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{D}})$.
Since the functors (2.105) are symmetric strong monoidal, the restriction
gives the symmetric strong monoidal functors

$\displaystyle\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})\to\mathop{\mathbf{Rep}}\nolimits_{\mathbf{Q}}(\widetilde{\mathbb{B}}),$
$\displaystyle\mathop{\mathbf{Corep}}\nolimits_{\mathbf{P}}({\mathbb{B}})\to\mathop{\mathbf{Corep}}\nolimits_{\mathbf{Q}}(\widetilde{\mathbb{B}}).$
(3.75)

For example, let $\rho$ and $\pi$ be representations of the monoid
${\mathbb{M}}$ on $W\in{\mathbf{P}}$ and $Z\in{\mathbf{P}}$. Their monoidal
product in $\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$ is a
representation $\tau$ on $W\otimes Z$. The functor (3.67) maps them to
representations $\widetilde{\rho}$, $\widetilde{\pi}$ and $\widetilde{\tau}$
on $FW$, $FZ$ and $F(W\otimes Z)$ respectively. The monoidal product of
$\widetilde{\rho}$ and $\widetilde{\pi}$ in
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{Q}}(\widetilde{\mathbb{B}})$ is a
representation $\lambda$ on $FW\odot FZ$. The isomorphism $\phi_{W,Z}\colon
FW\odot FZ\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}F(W\otimes Z)$
gives the isomorphism $\big{(}FW\odot
FZ,\lambda\big{)}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\big{(}F(W\otimes
Z),\widetilde{\tau}\big{)}$ in
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{Q}}(\widetilde{\mathbb{B}})$. In the
same way one can describe the isomorphism of the corresponding
corepresentations.

######  Remark 3.14.

The strong monoidality of the functors (3.75) follows from the strong
monoidality of the functors (2.105). Alternatively, one can prove this in a
direct way by using the formula (3.39) and Prop. 2.1.

##  4 Quantum linear spaces

Idea. Consider the sets ${\mathbb{K}}^{n}$ and ${\mathbb{K}}^{m}$ as vector
(linear) spaces, that is as objects of ${\mathbf{Vect}}$ or
${\mathbf{FVect}}$. The morphisms between these vector spaces are linear maps
${\mathbb{K}}^{n}\to{\mathbb{K}}^{m}$ (matrices $m\times n$ over
${\mathbb{K}}$). If we (partially) forget linear structure of these spaces we
obtain the affine spaces ${\mathbb{A}}^{n}$ and ${\mathbb{A}}^{m}$, which are
the objects of ${\mathbf{AlgSet}}\subset{\mathbf{AffSch}}$. Since
${\mathbf{AffSch}}={\mathbf{CommAlg}}^{\mathrm{op}}$ and
$A({\mathbb{A}}^{n})={\mathbb{K}}[x^{1},\ldots,x^{n}]$, the morphisms
${\mathbb{A}}^{n}\to{\mathbb{A}}^{m}$ is in one-to-one correspondence with the
algebra homomorphisms
$f\colon{\mathbb{K}}[y^{1},\ldots,y^{m}]\to{\mathbb{K}}[x^{1},\ldots,x^{n}]$.
Namely, if a homomorphism $f$ is given by the images
$f(y^{i})=P_{i}(x^{1},\ldots,x^{n})\in{\mathbb{K}}[x^{1},\ldots,x^{n}]$ for
some polynomials $P_{i}$, then the corresponding morphism
$\Phi\colon{\mathbb{A}}^{n}\to{\mathbb{A}}^{m}$ is the map
$\Phi(x^{1},\ldots,x^{n})=\big{(}P_{1}(x^{1},\ldots,x^{n}),\ldots,P_{m}(x^{1},\ldots,x^{n})\big{)}$.
Now let us recall back the linear structure of
${\mathbb{K}}^{n}={\mathbb{A}}^{n}$ and ${\mathbb{K}}^{m}={\mathbb{A}}^{m}$.
The map $\Phi$ preserves the linear structure iff all $P_{i}$ are homogeneous
polynomials of order $1$, that is
$P_{i}(x^{1},\ldots,x^{n})=\sum_{j=1}^{n}a_{ij}x^{j}$ for some
$a_{ij}\in{\mathbb{K}}$. This means exactly that the homomorphism $f$
preserves the grading of the algebras ${\mathbb{K}}[y^{1},\ldots,y^{m}]$ and
${\mathbb{K}}[x^{1},\ldots,x^{n}]$ (the $k$-th component of
${\mathbb{K}}[x^{1},\ldots,x^{n}]$ is the space of all the homogeneous
polynomials of order $k$). Any linear map
${\mathbb{K}}^{n}\to{\mathbb{K}}^{m}$ is arisen in this way: it is given by
the $m\times n$ matrix $(a_{ij})$.

Hence a natural candidates to a quantum analogue of a vector spaces are graded
algebras or at least some of them, considered as objects of (a subcategory of)
the category ${\mathbf{GrAlg}}^{\mathrm{op}}$. In [Man87, Man88] Yuri Manin
proposed a subcategory ${\mathbf{FQA}}^{\mathrm{op}}$ to generalise the
finite-dimensional vector spaces. He introduced the notion ‘quantum linear
space’ for this case.

###  4.1 Operations with quadratic algebras

4.1.1. Manin’s binary operations. Let
${\mathcal{A}},{\mathcal{B}}\in{\mathbf{QA}}$ be arbitrary quadratic algebras
(here we do not suppose that the quadratic algebras are finitely generated).
They can be presented in the form

$\displaystyle{\mathcal{A}}=TV/(R),$ $\displaystyle{\mathcal{B}}=TW/(S),$
(4.1)

where $V,W\in{\mathbf{Vect}}$ and $(R)\subset TV$, $(S)\subset TW$ are ideals
generated by subspaces $R\subset V^{\otimes 2}$, $S\subset W^{\otimes 2}$
respectively. Define the following operations [Man87, Man88].

  * •

Manin white product: ${\mathcal{A}}\circ{\mathcal{B}}=T(V\otimes
W)/(R_{\mathrm{w}})$, where

$\displaystyle
R_{\mathrm{w}}=(\mathop{\mathrm{id}}\nolimits_{V}\otimes\sigma_{V,W}\otimes\mathop{\mathrm{id}}\nolimits_{W})(R\otimes
W\otimes W+V\otimes V\otimes S)\subset V\otimes W\otimes V\otimes W.$

  * •

Manin black product: ${\mathcal{A}}\bullet{\mathcal{B}}=T(V\otimes
W)/(R_{\mathrm{b}})$, where

$\displaystyle
R_{\mathrm{b}}=(\mathop{\mathrm{id}}\nolimits_{V}\otimes\sigma_{V,W}\otimes\mathop{\mathrm{id}}\nolimits_{W})(R\otimes
S)\subset V\otimes W\otimes V\otimes W.$

  * •

(Even) tensor product: ${\mathcal{A}}\otimes{\mathcal{B}}=T(V\oplus
W)/\big{(}R\oplus[V,W]\oplus S\big{)}$, where

$[V,W]=\\{v\otimes w-w\otimes v\mid v\in V,w\in W\\}\subset(V\otimes
W)\oplus(W\otimes V)\subset(V\oplus W)^{\otimes 2}.$

  * •

Odd tensor product:
${\mathcal{A}}\mathbin{\underline{\otimes}}{\mathcal{B}}=T(V\oplus
W)/\big{(}R\oplus[V,W]_{+}\oplus S\big{)}$, where

$[V,W]_{+}=\\{v\otimes w+w\otimes v\mid v\in V,w\in W\\}\subset(V\otimes
W)\oplus(W\otimes V)\subset(V\oplus W)^{\otimes 2}.$

4.1.2. Properties of Manin’s binary operations [Man88]. The Manin white
product and the even tensor product coincide with the Manin product
${\mathcal{A}}\circ{\mathcal{B}}$ and the usual tensor product
${\mathcal{A}}\otimes{\mathcal{B}}$ defined in p. 2.3 for general graded
algebras. All four operations are bifunctors
${\mathbf{QA}}\times{\mathbf{QA}}\to{\mathbf{QA}}$ equipping ${\mathbf{QA}}$
with monoidal structures. The unit objects of $({\mathbf{QA}},\circ)$ and
$({\mathbf{QA}},\bullet)$ are ${\mathbb{K}}[u]$ and
${\mathbb{K}}[\epsilon]/(\epsilon^{2})$ respectively. The unit object of
$({\mathbf{QA}},\otimes)$ and $({\mathbf{QA}},\mathbin{\underline{\otimes}})$
is ${\mathbb{K}}$. The subcategory ${\mathbf{FQA}}\subset{\mathbf{QA}}$
inherits all four monoidal structures. The inclusion $R_{\mathrm{b}}\subset
R_{\mathrm{w}}$ defines an epimorphism
${\mathcal{A}}\bullet{\mathcal{B}}\to{\mathcal{A}}\circ{\mathcal{B}}$.

4.1.3. The functor $(-)_{1}$. If we have a graded algebra
${\mathcal{A}}=\bigoplus\limits_{k\in{\mathbb{N}}_{0}}{\mathcal{A}}_{1}$ we
can take its first graded component ${\mathcal{A}}_{1}$. In this way we obtain
the functor $(-)_{1}\colon{\mathbf{GrAlg}}\to{\mathbf{Vect}}$, which maps a
graded homomorphism
$f\in\mathop{\mathrm{Hom}}\nolimits_{{\mathbf{GrAlg}}}({\mathcal{A}},{\mathcal{B}})$
to its first order component
$f_{1}\colon{\mathcal{A}}_{1}\to{\mathcal{B}}_{1}$. Restriction gives functors
$(-)_{1}\colon{\mathbf{QA}}\to{\mathbf{Vect}}$ and
$(-)_{1}\colon{\mathbf{FQA}}\to{\mathbf{FVect}}$, which translate a quadratic
algebra ${\mathcal{A}}=TV/(R)$ to the vector space $V$. If
${\mathcal{A}}\in{\mathbf{QA}}$, then a graded homomorphism
$f\colon{\mathcal{A}}\to{\mathcal{B}}$ can be uniquely restored by the
component $f_{1}$. Hence the functors
$(-)_{1}\colon{\mathbf{QA}}\to{\mathbf{Vect}}$ and
$(-)_{1}\colon{\mathbf{FQA}}\to{\mathbf{FVect}}$ are faithful. By applying
these functors to the result of the Manin’s binary operations we obtain the
following natural isomorphisms:

$\displaystyle({\mathcal{A}}\circ{\mathcal{B}})_{1}=({\mathcal{A}}\bullet{\mathcal{B}})_{1}={\mathcal{A}}_{1}\otimes{\mathcal{B}}_{1},$
$\displaystyle({\mathcal{A}}\otimes{\mathcal{B}})_{1}=({\mathcal{A}}\mathbin{\underline{\otimes}}{\mathcal{B}})_{1}={\mathcal{A}}_{1}\oplus{\mathcal{B}}_{1}.$
(4.2)

This means that the functors $(-)_{1}\colon{\mathbf{QA}}\to{\mathbf{Vect}}$
and $(-)_{1}\colon{\mathbf{FQA}}\to{\mathbf{FVect}}$ are equipped with the
following strong monoidal structures

$\displaystyle({\mathbf{QA}},\circ)\to({\mathbf{Vect}},\otimes),$
$\displaystyle({\mathbf{FQA}},\circ)\to({\mathbf{FVect}},\otimes),$ (4.3)
$\displaystyle({\mathbf{QA}},\bullet)\to({\mathbf{Vect}},\otimes),$
$\displaystyle({\mathbf{FQA}},\bullet)\to({\mathbf{FVect}},\otimes),$ (4.4)
$\displaystyle({\mathbf{QA}},\otimes)\to({\mathbf{Vect}},\oplus),$
$\displaystyle({\mathbf{FQA}},\otimes)\to({\mathbf{FVect}},\oplus),$ (4.5)
$\displaystyle({\mathbf{QA}},\mathbin{\underline{\otimes}})\to({\mathbf{Vect}},\oplus),$
$\displaystyle({\mathbf{FQA}},\mathbin{\underline{\otimes}})\to({\mathbf{FVect}},\oplus).$
(4.6)

The fullness of the functor $(-)_{1}\colon{\mathbf{QA}}\to{\mathbf{Vect}}$ and
the left isomorphism (4.2) imply that the epimorphism
${\mathcal{A}}\bullet{\mathcal{B}}\to{\mathcal{A}}\circ{\mathcal{B}}$ is
natural in ${\mathcal{A}},{\mathcal{B}}\in{\mathbf{QA}}$.

Note also that the functors $T\colon{\mathbf{Vect}}\to{\mathbf{QA}}$ and
$T\colon{\mathbf{FVect}}\to{\mathbf{FQA}}$ considered in p. 2.3 are left
adjoints of $(-)_{1}\colon{\mathbf{QA}}\to{\mathbf{Vect}}$ and
$(-)_{1}\colon{\mathbf{FQA}}\to{\mathbf{FVect}}$ respectively.

We see that the binary operations $\circ$ and $\bullet$ are related with the
tensor product of vector spaces, while $\otimes$ and
$\mathbin{\underline{\otimes}}$ are related with the direct sum $\oplus$.
Below we introduce two embedding of ${\mathbf{FVect}}$ into
${\mathbf{FQA}}^{{\mathrm{op}}}$: ‘even’ and ‘odd’. The former one will relate
the monoidal products $\otimes$ and $\oplus$ on ${\mathbf{FVect}}$ to $\circ$
and $\otimes$ on ${\mathbf{FQA}}$, while the odd embedding will relate
$\otimes$ and $\oplus$ to $\bullet$ and $\mathbin{\underline{\otimes}}$
respectively.

4.1.4. Purely even/odd quadratic super-algebras. Let
${\mathbb{Z}}_{2}={\mathbb{Z}}/2{\mathbb{Z}}=\\{\bar{0},\bar{1}\\}$. A
quadratic super-algebra over ${\mathbb{K}}$ is a quadratic algebra
${\mathcal{A}}=TV/(R)$ with the structure of ${\mathbb{Z}}_{2}$-grading
compatible with multiplication and with ${\mathbb{N}}_{0}$-grading in the
sense that each component ${\mathcal{A}}_{k}$ is a direct sum of its subspaces
of even and odd elements:
${\mathcal{A}}=\bigoplus\limits_{k\in{\mathbb{N}}_{0}}{\mathcal{A}}_{k}=\bigoplus\limits_{k\in{\mathbb{N}}_{0}}({\mathcal{A}}_{k})_{\bar{0}}\oplus({\mathcal{A}}_{k})_{\bar{1}}$.
Such algebra is generated by (basis) elements of the subspaces
$V_{\bar{0}}=(A_{1})_{\bar{0}}$ and $V_{\bar{1}}=(A_{1})_{\bar{1}}$. We call
it commutative if $ab=-ba$ for $a,b\in V_{\bar{1}}$ and $ab=ba$ for $a\in
V_{\bar{0}}$, $b\in V$ (this is exactly the commutativity defined in p. 2.1
for the monoidal category of super-vector spaces [Del, § 1.1, 1.2]). We call a
quadratic super-algebra ${\mathcal{A}}=TV/(R)$ purely even or purely odd if
$V=V_{\bar{0}}$ or $V=V_{\bar{1}}$ respectively. The commutative purely even
and odd quadratic super-algebras over ${\mathbb{K}}$ form categories, which we
denote by ${\mathbf{CommQSA}}_{\mathrm{e}ven}$ and
${\mathbf{CommQSA}}_{\mathrm{o}dd}$. They can be identified with subcategories
of ${\mathbf{QA}}$ consisting of commutative ${\mathcal{A}}\in{\mathbf{QA}}$
(i.e. $ab=ba$ $\;\forall\,a,b\in{\mathcal{A}}$) and of
${\mathcal{A}}\in{\mathbf{QA}}$ satisfying $a_{1}b_{1}=-b_{1}a_{1}$
$\;\forall\,a_{1},b_{1}\in{\mathcal{A}}_{1}$ respectively. The additional
condition that they are finitely generated gives the subcategories

$\displaystyle{\mathbf{FCommQSA}}_{\mathrm{e}ven}={\mathbf{CommQSA}}_{\mathrm{e}ven}\cap{\mathbf{FQA}},$
$\displaystyle{\mathbf{FCommQSA}}_{\mathrm{o}dd}={\mathbf{CommQSA}}_{\mathrm{o}dd}\cap{\mathbf{FQA}}.$

4.1.5. The functors $S$ and $\Lambda$. For a vector space
$V\in{\mathbf{Vect}}$ symmetric algebra and exterior algebra are the quadratic
algebras

$\displaystyle SV=TV/\big{(}\\{v_{1}\otimes v_{2}-v_{2}\otimes v_{1}\mid
v_{1},v_{2}\in V\\}\big{)},$ (4.7) $\displaystyle\Lambda
V=TV/\big{(}\\{v_{1}\otimes v_{2}+v_{2}\otimes v_{1}\mid v_{1},v_{2}\in
V\\}\big{)}.$ (4.8)

We obtain the functors $S\colon{\mathbf{Vect}}\to{\mathbf{QA}}$ and
$\Lambda\colon{\mathbf{Vect}}\to{\mathbf{QA}}$ and their ‘finite-dimensional’
versions $S\colon{\mathbf{FVect}}\to{\mathbf{FQA}}$ and
$\Lambda\colon{\mathbf{FVect}}\to{\mathbf{FQA}}$. The functors $S$ and
$\Lambda$ are left adjoint functors to the restrictions of the functor
$(-)_{1}\colon{\mathbf{QA}}\to{\mathbf{Vect}}$ to the subcategories
${\mathbf{CommQSA}}_{\mathrm{e}ven}$ and ${\mathbf{CommQSA}}_{\mathrm{o}dd}$
respectively. Analogously, the finite-dimensional versions are left adjoints
to the restrictions of $(-)_{1}\colon{\mathbf{FQA}}\to{\mathbf{FVect}}$ to the
subcategories ${\mathbf{FCommQSA}}_{\mathrm{e}ven}$ and
${\mathbf{FCommQSA}}_{\mathrm{o}dd}$.

Denote by $S^{k}V$ and $\Lambda^{k}V$ the $k$-th graded components of $SV$ and
$\Lambda V$. In these notations we can write the following natural
isomorphisms

$\displaystyle S^{k}(V\oplus
W)\cong\bigoplus_{l=0}^{k}(S^{l}V)\otimes(S^{k-l}W),$
$\displaystyle\Lambda^{k}(V\oplus
W)\cong\bigoplus_{l=0}^{k}(\Lambda^{l}V)\otimes(\Lambda^{k-l}W)$ (4.9)

in ${\mathbf{Vect}}$. Summation over $k\in{\mathbb{N}}_{0}$ gives the natural
isomorphisms

$\displaystyle S(V\oplus W)\cong(SV)\otimes(SW),$
$\displaystyle\Lambda(V\oplus W)\cong(\Lambda V)\otimes(\Lambda W)$ (4.10)

in ${\mathbf{QA}}$. They are the structure morphisms of the strong monoidal
functors

$\displaystyle S\colon({\mathbf{Vect}},\oplus)\to({\mathbf{QA}},\otimes),$
$\displaystyle S\colon({\mathbf{FVect}},\oplus)\to({\mathbf{FQA}},\otimes),$
(4.11)
$\displaystyle\Lambda\colon({\mathbf{Vect}},\oplus)\to({\mathbf{QA}},\mathbin{\underline{\otimes}}),$
$\displaystyle\Lambda\colon({\mathbf{FVect}},\oplus)\to({\mathbf{FQA}},\mathbin{\underline{\otimes}}).$
(4.12)

On the other hand the natural transformations

$\displaystyle S(V\otimes W)\to(SV)\circ(SW),$ $\displaystyle(\Lambda
V)\bullet(\Lambda W)\to\Lambda(V\otimes W)$ (4.13)

are not isomorphisms even for finite-dimensional $V$ and $W$. This does not
give strong monoidal functors. Nevertheless, we obtain colax monoidal functors

$\displaystyle S\colon({\mathbf{Vect}},\otimes)\to({\mathbf{QA}},\circ),$
$\displaystyle S\colon({\mathbf{FVect}},\otimes)\to({\mathbf{FQA}},\circ)$
(4.14)

and lax monoidal functors

$\displaystyle\Lambda\colon({\mathbf{Vect}},\otimes)\to({\mathbf{QA}},\bullet),$
$\displaystyle\Lambda\colon({\mathbf{FVect}},\otimes)\to({\mathbf{FQA}},\bullet).$
(4.15)

The functors $S$ and $\Lambda$ are faithful since their compositions with the
functor $(-)_{1}$ coincide with the identical functor: $(SV)_{1}=V$, $(\Lambda
V)_{1}=V$. Moreover, any morphism $SV\to SW$ or $\Lambda V\to\Lambda W$ is
uniquely given by its first component and hence have the form $Sf$ or,
respectively, $\Lambda f$ for some linear map $f\colon V\to W$, so that the
functors $S$ and $\Lambda$ are fully faithful.

4.1.6. Koszul duality for quadratic algebras [Man87, Man88]. Consider a
finitely generated quadratic algebra ${\mathcal{A}}=TV/(R)$, where
$V\in{\mathbf{FVect}}$ and $R\subset V^{\otimes 2}$. Define its Koszul dual
quadratic algebra as

$\displaystyle{\mathcal{A}}^{!}=T(V^{*})/(R^{\bot}),$ $\displaystyle
R^{\bot}=\\{\xi\in V^{*}\otimes V^{*}\mid\xi(r)=0\;\forall\,r\in R\\}.$ (4.16)

The operation (4.16) is a contravariant functor
$(-)^{!}\colon{\mathbf{FQA}}\to{\mathbf{FQA}}$. We have the following natural
isomorphisms

$\displaystyle({\mathcal{A}}^{!})^{!}\cong{\mathcal{A}},$
$\displaystyle({\mathcal{A}}\circ{\mathcal{B}})^{!}\cong{\mathcal{A}}^{!}\bullet{\mathcal{B}}^{!},$
$\displaystyle({\mathcal{A}}\bullet{\mathcal{B}})^{!}\cong{\mathcal{A}}^{!}\circ{\mathcal{B}}^{!},$
(4.17) $\displaystyle({\mathcal{A}}^{!})_{1}=({\mathcal{A}}_{1})^{*},$
$\displaystyle({\mathcal{A}}\otimes{\mathcal{B}})^{!}\cong{\mathcal{A}}^{!}\mathbin{\underline{\otimes}}{\mathcal{B}}^{!},$
$\displaystyle({\mathcal{A}}\mathbin{\underline{\otimes}}{\mathcal{B}})^{!}\cong{\mathcal{A}}^{!}\otimes{\mathcal{B}}^{!}.$
(4.18)

We see that this functor is an involutive anti-autoequivalence, which
interchanges the Manin’s operations, and that the diagrams

(4.27)

are commutative.

4.1.7. The functors $S^{*}$ and $\Lambda^{*}$. Define contravariant functors
$S^{*}\colon{\mathbf{FVect}}\to{\mathbf{FQA}}$ and
$\Lambda^{*}\colon{\mathbf{FVect}}\to{\mathbf{FQA}}$ by the formulae
$S^{*}(V)=S(V^{*})$, $\Lambda^{*}(V)=\Lambda(V^{*})$, $V\in{\mathbf{FVect}}$.
As functors ${\mathbf{FVect}}\to{\mathbf{FQA}}^{\mathrm{op}}$ they are the
compositions of
$(-)^{*}\colon{\mathbf{FVect}}\to{\mathbf{FVect}}^{\mathrm{op}}$ with the
functors
$S^{\mathrm{op}}\colon{\mathbf{FVect}}^{\mathrm{op}}\to{\mathbf{FQA}}^{\mathrm{op}}$
and
$\Lambda^{\mathrm{op}}\colon{\mathbf{FVect}}^{\mathrm{op}}\to{\mathbf{FQA}}^{\mathrm{op}}$
respectively, which are the opposite functors to ones defined in p. 4.1.

The algebra $S^{*}(V)$ is the algebra of polynomial functions on
$V\in{\mathbf{FVect}}$. By choosing a basis $(x_{i})_{i=1}^{n}$ in $V$ we
obtain an isomorphism $V\cong{\mathbb{K}}^{n}$ and hence $S^{*}(V)\cong
S^{*}({\mathbb{K}}^{n})={\mathbb{K}}[x^{1},\ldots,x^{n}]$, where
$(x^{i})_{i=1}^{n}$ is the dual basis of $V^{*}\cong({\mathbb{K}}^{n})^{*}$.

As a fully faithful functor the functor
$S^{*}\colon{\mathbf{FVect}}\to{\mathbf{FQA}}^{\mathrm{op}}$ embeds the
finite-dimensional vector spaces into the category
${\mathbf{FQA}}^{\mathrm{op}}$. This justifies the term ‘quantum linear
spaces’ for the objects of ${\mathbf{FQA}}^{\mathrm{op}}$, but this is only
finite-dimensional quantum analogue.

The functor $\Lambda^{*}$ also embeds ${\mathbf{FVect}}$ into
${\mathbf{FQA}}^{\mathrm{op}}$, but in a different way. For
$V={\mathbb{K}}^{n}$ the algebra $\Lambda^{*}({\mathbb{K}}^{n})$ is isomorphic
to the Grassmann algebra with generators $\psi^{1},\ldots,\psi^{n}$ and
relations $\psi^{i}\psi^{j}+\psi^{j}\psi^{i}=0$.

Thus, the functors $S^{*}$ and $\Lambda^{*}$ are two different embeddings of
the category ${\mathbf{FVect}}$ into ${\mathbf{FQA}}^{\mathrm{op}}$. They
identify the category ${\mathbf{FVect}}$ with some full subcategories of the
categories
${\mathbf{FCommQSA}}_{\mathrm{e}ven}^{\mathrm{op}}\subset{\mathbf{FQA}}^{\mathrm{op}}$
and
${\mathbf{FCommQSA}}_{\mathrm{o}dd}^{\mathrm{op}}\subset{\mathbf{FQA}}^{\mathrm{op}}$
respectively. Hence the embeddings $S^{*}$ and $\Lambda^{*}$ can be
interpreted as follows. Any vector space $V$ can be considered as a purely
even or as a purely odd super-vector space. In the former case we apply
$S^{*}$ to consider the vector space as quantum linear vector space
$S^{*}V=S(V^{*})$, in the latter case we apply $\Lambda^{*}$.

Note that $S^{*}(V)=S(V^{*})=(\Lambda V)^{!}$ and
$\Lambda^{*}(V)=\Lambda(V^{*})=(SV)^{!}$. This means that the functor
$(-)^{!}$ maps a vector space to its dual with change of its parity. The
operations ${\mathcal{A}}\circ{\mathcal{B}}$ and
${\mathcal{A}}\otimes{\mathcal{B}}$ are analogues of tensor product and direct
sum of vector spaces for the case of the quantum linear spaces
${\mathcal{A}},{\mathcal{B}}\in{\mathbf{FQA}}^{\mathrm{op}}$ considered as
purely even. The operations ${\mathcal{A}}\bullet{\mathcal{B}}$ and
${\mathcal{A}}\mathbin{\underline{\otimes}}{\mathcal{B}}$ are also analogues
of the tensor product and direct sum, but in this case we need to consider
${\mathcal{A}}$ and ${\mathcal{B}}$ as purely odd.

4.1.8. Monoidal structures of $S^{*}$. Let us chose the ‘even’ interpretation:
consider ${\mathbf{FVect}}$ as the subcategory of
${\mathbf{CommQSA}}_{\mathrm{e}ven}^{\mathrm{op}}\subset{\mathbf{FQA}}^{\mathrm{op}}$
by means of $S^{*}$. Since the monoidal functor (4.11) can be regarded as
strict monoidal, the operation $\oplus$ in ${\mathbf{FVect}}$ coincides with
the operation $\otimes$ in ${\mathbf{FQA}}$ (see the formula (4.10)). On the
other hand, the functor (4.14) is colax monoidal, so its opposite
$S^{\mathrm{op}}\colon({\mathbf{FVect}}^{\mathrm{op}},\otimes)\to({\mathbf{FQA}}^{\mathrm{op}},\circ)$
is lax monoidal. By composing the latter one with the strong monoidal functor
$(-)^{*}\colon({\mathbf{FVect}},\otimes)\to({\mathbf{FVect}}^{\mathrm{op}},\otimes)$
in a proper order we obtain a lax monoidal functor. Thus we have

strong monoidal functor $\displaystyle
S^{*}\colon({\mathbf{FVect}},\oplus)\to({\mathbf{FQA}}^{\mathrm{op}},\otimes),$
(4.28) lax monoidal functor $\displaystyle
S^{*}\colon({\mathbf{FVect}},\otimes)\to({\mathbf{FQA}}^{\mathrm{op}},\circ).$
(4.29)

In particular, the tensor product of vector spaces does not coincide with the
result of the Manin white product after the embedding $S^{*}$, because the
monoidal functor (4.29) is not strong monoidal.

######  Remark 4.1.

One can define a functor
$S^{*}\colon{\mathbf{Vect}}\to{\mathbf{QA}}^{\mathrm{op}}$ as the composition
of the functors $(-)^{*}\colon{\mathbf{Vect}}\to{\mathbf{Vect}}^{\mathrm{op}}$
and
$S^{\mathrm{op}}\colon{\mathbf{Vect}}^{\mathrm{op}}\to{\mathbf{QA}}^{\mathrm{op}}$.
It has a structure of strong monoidal functor
$({\mathbf{Vect}},\oplus)\to({\mathbf{QA}}^{\mathrm{op}},\otimes)$, but it
does not get any monoidal structure
$({\mathbf{Vect}},\otimes)\to({\mathbf{QA}}^{\mathrm{op}},\circ)$, since
$(-)^{*}\colon({\mathbf{Vect}},\otimes)\to({\mathbf{Vect}}^{\mathrm{op}},\otimes)$
is colax monoidal, while
$S^{\mathrm{op}}\colon({\mathbf{Vect}}^{\mathrm{op}},\otimes)\to({\mathbf{QA}}^{\mathrm{op}},\circ)$
is lax monoidal.

4.1.9. Coproduct of quadratic algebras. The product of quantum linear spaces
is the coproduct in the category ${\mathbf{FQA}}$ or, more generally, in
${\mathbf{QA}}$. For any two quadratic algebras
${\mathcal{A}}=TV/(R)\in{\mathbf{QA}}$ and
${\mathcal{A}}^{\prime}=TV^{\prime}/(R^{\prime})\in{\mathbf{QA}}$ their
coproduct exists and has the form

$\displaystyle{\mathcal{A}}\amalg{\mathcal{A}}^{\prime}:=T(V\oplus
V^{\prime})/(R\oplus R^{\prime}).$ (4.30)

Indeed, let ${\mathcal{B}}=TW/(S)\in{\mathbf{QA}}$ and
$f\colon{\mathcal{A}}\to{\mathcal{B}}$,
$f^{\prime}\colon{\mathcal{A}}^{\prime}\to{\mathcal{B}}$ be two morphisms in
${\mathbf{QA}}$. Their first order components satisfy $(f_{1}\otimes
f_{1})R\subset S$, $(f^{\prime}_{1}\otimes f^{\prime}_{1})R^{\prime}\subset
S$. There exists a unique linear map $h_{1}\colon V\oplus V^{\prime}\to W$
such that the diagram

(4.35)

commutes. Since $(h_{1}\otimes h_{1})(R\oplus R^{\prime})=(f_{1}\otimes
f_{1})(R)+(f^{\prime}_{1}\otimes f^{\prime}_{1})(R^{\prime})\subset S$, there
exists a morphism
$h\colon{\mathcal{A}}\amalg{\mathcal{A}}^{\prime}\to{\mathcal{B}}$ with the
first order component $h_{1}$. Hence there is a unique morphism $h$ making the
diagram

(4.40)

commutative.

Note that the inclusion $R\oplus R^{\prime}\subset R\oplus[V,V^{\prime}]\oplus
R^{\prime}$ implies the natural epimorphism

$\displaystyle{\mathcal{A}}\amalg{\mathcal{A}}^{\prime}\twoheadrightarrow{\mathcal{A}}\otimes{\mathcal{A}}^{\prime}.$
(4.41)

######  Remark 4.2.

The coproduct in the category the connected affinely generated graded algebras
is different. Namely, the coproduct of
${\mathcal{A}}=\bigoplus\limits_{k\in{\mathbb{N}}_{0}}{\mathcal{A}}_{k}$ and
${\mathcal{A}}^{\prime}=\bigoplus\limits_{k\in{\mathbb{N}}_{0}}{\mathcal{A}}^{\prime}_{k}$
in this category is the graded algebra
${\mathcal{A}}\coprod\limits_{c.a.g.}{\mathcal{A}}^{\prime}$ with the
components

$\displaystyle\Big{(}{\mathcal{A}}\coprod\limits_{c.a.g.}{\mathcal{A}}^{\prime}\Big{)}_{k}=\bigoplus_{p=1}^{k+1}\bigoplus_{l_{1}=0}^{k}\bigoplus_{l_{2},\ldots,l_{p}=1\atop
l_{1}+l_{2}+\ldots+l_{p}=k}^{k}({\mathcal{A}}_{l_{1}}\otimes{\mathcal{A}}^{\prime}_{l_{2}}\otimes{\mathcal{A}}_{l_{3}}\otimes\cdots).$
(4.42)

4.1.10. Functor $T^{*}$. To extend the tensor product of vector spaces in an
exact way one can consider another embedding such as the fully faithful
functor $T^{*}\colon{\mathbf{FVect}}\to{\mathbf{FQA}}^{\mathrm{op}}$ defined
as the composition of
$(-)^{*}\colon{\mathbf{FVect}}\to{\mathbf{FVect}}^{\mathrm{op}}$ and
$T^{\mathrm{op}}\colon{\mathbf{FVect}}^{\mathrm{op}}\to{\mathbf{FQA}}^{\mathrm{op}}$,
that is $T^{*}(V)=T(V^{*})$. It is a strong monoidal functor
$T^{*}\colon({\mathbf{FVect}},\otimes)\to({\mathbf{FQA}}^{\mathrm{op}},\circ)$,
but in this case the direct sum $\oplus$ is not strong-monoidally related with
the Manin’s operation $\otimes$ with quadratic algebras. Indeed, $T(V)\otimes
T(W)$ is a quotient algebra of $T(V\oplus W)$ over the ideal generated by the
subset $\\{v\otimes w-w\otimes v\mid v\in V,\,w\in W\\}$, so
$T\colon({\mathbf{FVect}},\oplus)\to({\mathbf{FQA}},\otimes)$ is a colax (not
strong) monoidal functor.

The functor $T$ preserves the finite coproducts, hence it has a structure of
strong monoidal functor $({\mathbf{Vect}},\oplus)\to({\mathbf{QA}},\amalg)$.
If ${\mathcal{A}},{\mathcal{A}}^{\prime}\in{\mathbf{FQA}}$ then
${\mathcal{A}}\amalg{\mathcal{A}}^{\prime}\in{\mathbf{FQA}}$, so we also
obtain the strong monoidal functor
$T\colon({\mathbf{FVect}},\oplus)\to({\mathbf{FQA}},\amalg)$. Thus we have:

lax monoidal functor $\displaystyle
T^{*}\colon({\mathbf{FVect}},\oplus)\to({\mathbf{FQA}}^{\mathrm{op}},\otimes),$
(4.43) strong monoidal functor $\displaystyle
T^{*}\colon({\mathbf{FVect}},\oplus)\to({\mathbf{FQA}}^{\mathrm{op}},\amalg),$
(4.44) strong monoidal functor $\displaystyle
T^{*}\colon({\mathbf{FVect}},\otimes)\to({\mathbf{FQA}}^{\mathrm{op}},\circ),$
(4.45) strong monoidal functor $\displaystyle
T^{*}\colon({\mathbf{Vect}},\oplus)\to({\mathbf{QA}}^{\mathrm{op}},\amalg),$
(4.46) colax monoidal functor $\displaystyle
T^{*}\colon({\mathbf{Vect}},\otimes)\to({\mathbf{QA}}^{\mathrm{op}},\circ),$
(4.47)

where $T^{*}\colon{\mathbf{Vect}}\to{\mathbf{QA}}^{\mathrm{op}}$ is the
composition of the functors
$(-)^{*}\colon{\mathbf{Vect}}\to{\mathbf{Vect}}^{\mathrm{op}}$ and
$T^{\mathrm{op}}\colon{\mathbf{Vect}}^{\mathrm{op}}\to{\mathbf{QA}}^{\mathrm{op}}$.

4.1.11. Dequantisation functor. Composition of the functors $(-)^{*}$ and
$(-)_{1}$ gives the functor
$(-)^{*}_{1}\colon{\mathbf{FQA}}^{\mathrm{op}}\to{\mathbf{FVect}}$,
${\mathcal{A}}\mapsto({\mathcal{A}})_{1}^{*}=({\mathcal{A}}_{1})^{*}$. By
composing it with any of the functors $S^{*}$, $\Lambda^{*}$ and $T^{*}$,
which embed ${\mathbf{FVect}}$ into ${\mathbf{FQA}}^{\mathrm{op}}$, we obtain
the identical functor:

$\displaystyle\big{(}S^{*}(V)\big{)}_{1}^{*}=V,$
$\displaystyle\big{(}\Lambda^{*}(V)\big{)}_{1}^{*}=V,$
$\displaystyle\big{(}T^{*}(V)\big{)}_{1}^{*}=V.$ (4.48)

The functor $(-)_{1}^{*}\colon{\mathbf{FQA}}^{\mathrm{op}}\to{\mathbf{FVect}}$
is faithful, but not full. It has the following structures of strong monoidal
functor:

$\displaystyle({\mathbf{FQA}}^{\mathrm{op}},\otimes)\to({\mathbf{FVect}},\oplus),$
$\displaystyle({\mathbf{FQA}}^{\mathrm{op}},\circ)\to({\mathbf{FVect}},\otimes),$
(4.49)
$\displaystyle({\mathbf{FQA}}^{\mathrm{op}},\mathbin{\underline{\otimes}})\to({\mathbf{FVect}},\oplus),$
$\displaystyle({\mathbf{FQA}}^{\mathrm{op}},\bullet)\to({\mathbf{FVect}},\otimes),$
(4.50)
$\displaystyle({\mathbf{FQA}}^{\mathrm{op}},\amalg)\to({\mathbf{FVect}},\oplus).$
(4.51)

This functor extracts the classical part from a quadratic algebra. In other
words, this is a classical limit for quantum linear spaces.

It can be extended to the functor
$(-)^{*}_{1}\colon{\mathbf{QA}}^{\mathrm{op}}\to{\mathbf{Vect}}$, which has a
lax monoidal structure
$(-)_{1}^{*}\colon({\mathbf{QA}}^{\mathrm{op}},\circ)\to({\mathbf{Vect}},\otimes)$,
but the identities (4.48) do not hold for infinite-dimensional vector spaces
$V$.

###  4.2 Manin matrices

To define Manin matrices in terms of matrix idempotents we first describe the
relationship between quadratic algebras and general idempotent operators.

4.2.1. Quadratic algebras for idempotents. Remind that an idempotent is an
element $A$ of a ring ${\mathfrak{R}}$ satisfying $A^{2}=A$. Let $W$ be a
vector space and $R,R^{\prime}\subset W$ be its subspaces such that $W=R\oplus
R^{\prime}$. The composition $W\twoheadrightarrow W/R^{\prime}\cong
R\hookrightarrow W$ is an idempotent in the algebra $\mathop{\mathrm{\bf
end}}\nolimits(W)$. Moreover, the idempotents $A\in\mathop{\mathrm{\bf
end}}\nolimits(W)$ are in one-to-one correspondence with the decompositions
$W=R\oplus R^{\prime}$ into subspaces $R,R^{\prime}\subset W$ such that
$R=\mathop{\mathrm{Im}}\nolimits A$ and
$R^{\prime}=\mathop{\mathrm{Im}}\nolimits(1-A)$, where
$1=\mathop{\mathrm{id}}\nolimits_{V\otimes V}$. The operator
$1-A\in\mathop{\mathrm{\bf end}}\nolimits(W)$ is the dual idempotent, it
corresponds to the decomposition $W=R^{\prime}\oplus R$.

Let ${\mathcal{A}}\in{\mathbf{QA}}$ have the form ${\mathcal{A}}=TV/(R)$ for a
vector space $V\in{\mathbf{Vect}}$ and a subspace $R\subset V^{\otimes 2}$. As
any subspace the subspace $R$ can be given as an image of an idempotent
$A\in\mathop{\mathrm{\bf end}}\nolimits(V^{\otimes 2})$, since $R$ always has
a complement $R^{\prime}\subset V^{\otimes 2}$. Thus any connected quadratic
algebra is isomorphic to $TV/(\mathop{\mathrm{Im}}\nolimits A)$ for some
idempotent $A\in\mathop{\mathrm{\bf end}}\nolimits(V^{\otimes 2})$.

Denote $\Xi_{A}({\mathbb{K}})=TV/(R^{\prime})$ where
$R^{\prime}=\mathop{\mathrm{Im}}\nolimits(1-A)$. Any
${\mathcal{A}}\in{\mathbf{QA}}$ is isomorphic to $\Xi_{A}({\mathbb{K}})$ for
some idempotent operator $A$. More generally, any semi-connected quadratic
algebra is isomorphic to
$\Xi_{A}({\mathfrak{R}}):={\mathfrak{R}}\otimes\Xi_{A}({\mathbb{K}})$ for some
$A$ and ${\mathfrak{R}}\in{\mathbf{Alg}}$.

If $V\in{\mathbf{FVect}}$ then $(V^{\otimes 2})^{*}=(V^{*})^{\otimes 2}$, so
any (idempotent) operator $A\colon V^{\otimes 2}\to V^{\otimes 2}$ gives the
(idempotent) transpose operator $A^{*}\colon(V^{*})^{\otimes
2}\to(V^{*})^{\otimes 2}$. Any quadratic algebra
${\mathcal{A}}\in{\mathbf{FQA}}$ or
${\mathcal{A}}\in{\mathbf{FQA}}_{\mathrm{s}c}$ isomorphic to the algebra
${\mathfrak{X}}_{A}({\mathbb{K}}):=TV^{*}/(\mathop{\mathrm{Im}}\nolimits
A^{*})=\Xi_{1-A^{*}}({\mathbb{K}})$ or
${\mathfrak{X}}_{A}({\mathfrak{R}}):={\mathfrak{R}}\otimes{\mathfrak{X}}_{A}({\mathbb{K}})=\Xi_{1-A^{*}}({\mathfrak{R}})$
respectively for some idempotent $A\in\mathop{\mathrm{\bf
end}}\nolimits(V^{\otimes 2})$. The Koszul duality in these notations has the
form:

$\displaystyle{\mathfrak{X}}_{A}({\mathbb{K}})^{!}=\Xi_{A}({\mathbb{K}}),$
$\displaystyle\Xi_{A}({\mathbb{K}})^{!}={\mathfrak{X}}_{A}({\mathbb{K}}).$
(4.52)

The semi-connected quadratic algebras ${\mathfrak{X}}_{A}({\mathfrak{R}})$ and
$\Xi_{A}({\mathfrak{R}})$ can be considered as values of functors
${\mathfrak{X}}_{A},\Xi_{A}\colon{\mathbf{Alg}}\to{\mathbf{FQA}}_{\mathrm{s}c}$
coinciding with $-\otimes{\mathcal{A}}$ and $-\otimes{\mathcal{A}}^{!}$, where
${\mathcal{A}}={\mathfrak{X}}_{A}({\mathbb{K}})$.

For the idempotent $A_{V}=\dfrac{1-\sigma_{V,V}}{2}$ (anti-symmetrizer of
$V\otimes V$) we obtain

$\displaystyle{\mathfrak{X}}_{A_{V}}({\mathbb{K}})=S^{*}(V),$
$\displaystyle\Xi_{A_{V}}({\mathbb{K}})=\Lambda V,$ (4.53)

The functors $T^{*}$ and $T$ give the quadratic algebras for the trivial
idempotents $0,1\in\mathop{\mathrm{\bf end}}\nolimits(V^{\otimes 2})$:

$\displaystyle{\mathfrak{X}}_{0}({\mathbb{K}})=T^{*}(V),$
$\displaystyle\Xi_{1}({\mathbb{K}})=TV.$ (4.54)

4.2.2. The case of matrix idempotents. A basis $(v_{i})_{i=1}^{n}$ of a vector
space $V\in{\mathbf{FVect}}$ gives the isomorphism $V\cong{\mathbb{K}}^{n}$,
$v\mapsto(x^{1},\ldots,x^{n})$, where $x^{i}$ are components of $v\in V$
defined by the formula $v=\sum_{i=1}^{n}x^{i}v_{i}$. It induces the algebra
isomorphism $\mathop{\mathrm{\bf end}}\nolimits(V)\cong\mathop{\mathrm{\bf
end}}\nolimits({\mathbb{K}}^{n})$, which maps operator on $V$ to its $n\times
n$ matrix in the basis $(v_{i})$. The space $V\otimes V$ has the basis
$(v_{i}\otimes v_{j})_{i,j=1}^{n}$, so the operators on $V\otimes V$
correspond to the $n^{2}\times n^{2}$ matrices, whose entries have two pairs
of indices. Namely, the operator $A\in\mathop{\mathrm{\bf
end}}\nolimits(V^{\otimes 2})$ corresponds to the matrix $A=(A^{ij}_{kl})$
defined by the formula $A(v_{k}\otimes
v_{l})=\sum_{i,j=1}^{n}A^{ij}_{kl}v_{i}\otimes v_{j}$. The transpose operator
$A^{*}\in\mathop{\mathrm{\bf end}}\nolimits(V^{*}\otimes V^{*})$ has the form
$A^{*}(v^{i}\otimes v^{j})=\sum_{k,l=1}^{n}A^{ij}_{kl}v^{k}\otimes v^{l}$,
where $(v^{i})_{i=1}^{n}$ is the dual basis.

Since any $V\in{\mathbf{FVect}}$ has a finite basis $(v_{i})_{i=1}^{n}$, any
algebra ${\mathcal{A}}=TV/(R)\in{\mathbf{FQA}}$ is isomorphic to
$\Xi_{A}({\mathbb{K}})$ for some matrix idempotent $A\in\mathop{\mathrm{\bf
end}}\nolimits({\mathbb{K}}^{n}\otimes{\mathbb{K}}^{n})$. The latter is a
matrix $A=(A^{ij}_{kl})$ such that
$\sum_{k,l=1}^{n}A^{ij}_{kl}A^{kl}_{ab}=A^{ij}_{ab}$. The algebra
${\mathcal{A}}=\Xi_{A}({\mathbb{K}})$ is generated by $v_{1},\ldots,v_{n}$
with the quadratic commutation relations
$\sum_{i,j=1}^{n}A^{ij}_{kl}v_{j}v_{j}=v_{k}v_{l}$, where we identified the
basis $(v_{i})$ with the standard basis of ${\mathbb{K}}^{n}$.

The same algebra is isomorphic to ${\mathfrak{X}}_{A}({\mathbb{K}})=TW/(R)$
for another matrix idempotent $A$ such that $W=({\mathbb{K}}^{n})^{*}$ and
$R=\mathop{\mathrm{Im}}\nolimits A^{*}$. Let $(v^{i})$ be the basis of
$({\mathbb{K}}^{n})^{*}$ dual to the standard basis of ${\mathbb{K}}^{n}$,
then ${\mathfrak{X}}_{A}({\mathbb{K}})$ is generated by $v^{1},\ldots,v^{n}$
with the relations $\sum_{k,l=1}^{n}A^{ij}_{kl}v^{k}v^{l}=0$.

Consider, for instance, the anti-symmetrizer
$A_{n}=A_{{\mathbb{K}}^{n}}\colon{\mathbb{K}}^{n}\otimes{\mathbb{K}}^{n}\to{\mathbb{K}}^{n}\otimes{\mathbb{K}}^{n}$.
It has the entries
$(A_{n})^{ij}_{kl}=\frac{1}{2}(\delta^{i}_{k}\delta^{j}_{l}-\delta^{i}_{l}\delta^{j}_{k})$.
The corresponding quadratic algebras ${\mathfrak{X}}_{A}({\mathbb{K}})$ and
$\Xi_{A}({\mathbb{K}})$ are generated by $v^{i}$ and $v_{i}$ with the
relations $v^{i}v^{j}=v^{j}v^{i}$ and $v_{i}v_{j}=-v_{j}v_{i}$. The former one
is the polynomial algebra ${\mathbb{K}}[x^{1},\ldots,x^{n}]$ with commutative
variables $x^{i}=v^{i}$, while the latter one is the Grassmann algebra with
the anti-commutative variables $v_{i}$.

4.2.3. Manin matrix for a pair of idempotents. Any $n\times m$ matrix $M$ with
entries $M_{ij}=M^{i}_{j}\in{\mathfrak{R}}$ can be considered as an operator
${\mathbb{K}}^{m}\to{\mathfrak{R}}\otimes{\mathbb{K}}^{n}$. In terms of
standard bases $(v_{i})_{i=1}^{n}$ and $(w_{j})_{j=1}^{m}$ of the spaces
${\mathbb{K}}^{n}$ and ${\mathbb{K}}^{m}$ it acts as
$Mw_{j}=\sum_{i=1}^{n}M^{i}_{j}\otimes v_{i}$. Denote by $M^{(a)}$ the
operator $M$ acting on the $a$-th matrix tensor factor ${\mathbb{K}}^{m}$ of
${\mathfrak{R}}\otimes{\mathbb{K}}^{m}\otimes\cdots\otimes{\mathbb{K}}^{m}$.
In the case of two such factors we have

$\displaystyle M^{(1)}(r\otimes w_{j}\otimes
w_{l})=\sum_{i=1}^{n}M^{i}_{j}r\otimes v_{i}\otimes w_{l},$ $\displaystyle
r\in{\mathfrak{R}},$ (4.55) $\displaystyle M^{(2)}(r\otimes w_{j}\otimes
w_{l})=\sum_{k=1}^{n}M^{k}_{l}r\otimes w_{j}\otimes v_{k},$ $\displaystyle
r\in{\mathfrak{R}}.$ (4.56)

######  Definition 4.3.

[S] Let $A\in\mathop{\mathrm{\bf
end}}\nolimits({\mathbb{K}}^{n}\otimes{\mathbb{K}}^{n})$ and
$B\in\mathop{\mathrm{\bf
end}}\nolimits({\mathbb{K}}^{m}\otimes{\mathbb{K}}^{m})$ be matrix
idempotents. The matrix $M$ satisfying

$\displaystyle AM^{(1)}M^{(2)}(1-B)=0$ (4.57)

is called Manin matrix for the pair of idempotents $(A,B)$ or simply
$(A,B)$-Manin matrix. In the case $A=B$ we will say $B$-Manin matrix instead
of $(B,B)$-Manin matrix.

The matrix $M$ gives the operator
$({\mathbb{K}}^{n})^{*}\to{\mathfrak{R}}\otimes({\mathbb{K}}^{m})^{*}$ acting
as $v^{i}\mapsto\sum_{j=1}^{m}M^{i}_{j}\otimes w^{j}$, where $(v^{i})$ and
$(w^{j})$ are bases dual to $(v_{i})$ and $(w_{j})$. Any linear map
${\mathfrak{X}}_{A}({\mathbb{K}})_{1}\to{\mathfrak{X}}_{B}({\mathfrak{R}})_{1}$
has such form for some matrix $M$. In fact, this map gives a homomorphism of
graded algebras
${\mathfrak{X}}_{A}({\mathbb{K}})\to{\mathfrak{X}}_{B}({\mathfrak{R}})$ iff
$M$ is an $(A,B)$-Manin matrix (see [S]), we denote this homomorphism of
graded algebras by
$f_{M}\colon{\mathfrak{X}}_{A}({\mathbb{K}})\to{\mathfrak{X}}_{B}({\mathfrak{R}})$.

In the same way any linear map
$\Xi_{B}({\mathbb{K}})_{1}\to\Xi_{A}({\mathfrak{R}})_{1}$ has the form
${\mathbb{K}}^{m}\to{\mathfrak{R}}\otimes{\mathbb{K}}^{n}$,
$w_{j}\mapsto\sum_{i=1}^{n}M^{i}_{j}v_{i}$. It gives a graded homomorphism
$\Xi_{B}({\mathbb{K}})\to\Xi_{A}({\mathfrak{R}})$ iff $M=(M^{i}_{j})$ is an
$(A,B)$-Manin matrix, we denote this homomorphism by
$f^{M}\colon\Xi_{B}({\mathbb{K}})\to\Xi_{A}({\mathfrak{R}})$.

Thus we have bijections between the sets
$\mathop{\mathrm{Hom}}\nolimits\big{(}{\mathfrak{X}}_{A}({\mathbb{K}}),{\mathfrak{X}}_{B}({\mathfrak{R}})\big{)}$,
$\mathop{\mathrm{Hom}}\nolimits\big{(}\Xi_{B}({\mathbb{K}}),\Xi_{A}({\mathfrak{R}})\big{)}$
and the set of $(A,B)$-Manin matrices.

4.2.4. Usual Manin matrices. The Manin matrices introduced in the papers [CF,
CFR] are $n\times m$ Manin matrices for the pair of idempotents
$(A_{n},A_{m})$. They correspond to the graded homomorphisms of polynomial
algebras
${\mathbb{K}}[x^{1},\ldots,x^{n}]\to{\mathfrak{R}}[y^{1},\ldots,y^{m}]$ or,
equivalently, to the graded homomorphisms $S(V^{*})\to{\mathfrak{R}}\otimes
S(W^{*})$, where $V$ and $W$ are $n$\- and $m$-dimensional vector spaces.

###  4.3 Semi-linear spaces and their quantum analogue

To interpret representations of algebraic monoids and groups on (finite-
dimensional) linear spaces in terms of Subsection 3.2 one needs to include
both algebraic sets and linear spaces to a bigger monoidal category. More
generally, one needs to do the same for the affine schemes. In this way we
will get categories of spaces which are partially linear in some sense. We
call them semi-linear spaces.

4.3.1. Semi-linear algebraic sets. Describe the first case of semi-linear
spaces. For an algebraic set $X\in{\mathbf{AlgSet}}$ and a linear space
$V\in{\mathbf{FVect}}$ consider their set-theoretic product $X\times V$ as an
algebraic set. Define morphisms between two such products $X\times V$ and
$Y\times W$ as morphisms $F\colon X\times V\to Y\times W$ in
${\mathbf{AlgSet}}$ that have the form

$\displaystyle F(x,v)$ $\displaystyle=(\varphi(x),f(x,v)),\qquad\quad x\in
X,\quad v\in V,\qquad\qquad\varphi\colon X\to Y,\qquad f\colon X\times V\to
W,$ $\displaystyle f(x,\alpha v+v^{\prime})=\alpha
f(x,v)+f(x,v^{\prime}),\quad\quad x\in X,\quad v,v^{\prime}\in V.$ (4.58)

In other words, the morphism $F$ is given by arbitrary morphisms
$\varphi\in\mathop{\mathrm{Hom}}\nolimits_{{\mathbf{AlgSet}}}(X,Y)$ and
$f\in\mathop{\mathrm{Hom}}\nolimits_{{\mathbf{AlgSet}}}(X\times V,W)$ such
that $f(x,-)\colon V\to W$ is linear for any $x\in X$. In this way we obtain a
category ${\mathbf{SLAlgSet}}$ with morphisms that partially satisfy the
linearity condition, we call them semi-linear maps. We call the objects
$X\times V\in{\mathbf{SLAlgSet}}$ semi-linear algebraic sets.

4.3.2. Monoidal product of semi-linear algebraic sets. Let
$X,X^{\prime}\in{\mathbf{AlgSet}}$ and $V,V^{\prime}\in{\mathbf{FVect}}$.
Define monoidal product of $X\times V,X^{\prime}\times
V^{\prime}\in{\mathbf{SLAlgSet}}$ as the object $(X\times
X^{\prime})\times(V\otimes V^{\prime})$. For morphisms $F\colon X\times V\to
Y\times W$ and $F^{\prime}\colon X^{\prime}\times V^{\prime}\to
Y^{\prime}\times W^{\prime}$ in ${\mathbf{SLAlgSet}}$, given by maps
$\varphi\colon X\to Y$, $f\colon X\times V\to W$ and $\varphi^{\prime}\colon
X^{\prime}\to Y^{\prime}$, $f^{\prime}\colon X^{\prime}\times V^{\prime}\to
W^{\prime}$, their monoidal product is $F^{\prime\prime}=F\otimes
F^{\prime}\colon(X\times X^{\prime})\times(V\otimes V^{\prime})\to(Y\times
Y^{\prime})\times(W\otimes W^{\prime})$ given by
$\varphi^{\prime\prime}=\varphi\times\varphi^{\prime}\colon X\times X\to
Y\times Y$ and $f^{\prime\prime}\colon(X\times X^{\prime})\times(V\otimes
V^{\prime})\to W\otimes W^{\prime}$, $f^{\prime\prime}(x,x^{\prime},v\otimes
v^{\prime})=f(x,v)\otimes f^{\prime}(x^{\prime},v^{\prime})$. In this way we
obtain a symmetric monoidal category $({\mathbf{SLAlgSet}},\otimes)$ with the
unit object $0\times{\mathbb{K}}$.

The categories $({\mathbf{AlgSet}},\times)$ and $({\mathbf{FVect}},\otimes)$
can be regarded as monoidal subcategories of $({\mathbf{SLAlgSet}},\otimes)$
embedded via the faithful symmetric strict monoidal functors $X\mapsto
X\times{\mathbb{K}}$ and $V\mapsto\\{0\\}\times V$. The latter functor is
full, the former is not, i.e. ${\mathbf{FVect}}$ is a full subcategory of
${\mathbf{SLAlgSet}}$, while the subcategory
${\mathbf{AlgSet}}\subset{\mathbf{SLAlgSet}}$ is not full.

4.3.3. Trivial vector bundles. The category ${\mathbf{SLAlgSet}}$ is
equivalent to the category of trivial vector bundles over algebraic sets. The
semi-linear algebraic set $X\times V$ corresponds to the trivial bundle over
$X$ with the fibre $V$. A semi-linear map $F\colon X\times V\to Y\times W$
given by the formula (4.3) is the vector bundle morphism $(\varphi,F)$ in the
sense that the diagram

(4.63)

is commutative (the vertical arrows mean the projections) and the morphism $F$
induces a linear map on each fibre (see [Lang, III, § 1]). Category embeddings
${\mathbf{AlgSet}}\hookrightarrow{\mathbf{SLAlgSet}}$ and
${\mathbf{FVect}}\hookrightarrow{\mathbf{SLAlgSet}}$ identify an algebraic set
$X$ with the $1$-dimensional trivial bundle over it, and a vector space $V$ –
with the trivial bundle over
$\\{0\\}=\mathop{\mathrm{Spec}}\nolimits{\mathbb{K}}$ with the fibre $V$.

The main difference between the notions of semi-linear algebraic set and
trivial vector bundle is that we consider the former ones not over the
structure of algebraic set, but together with this structure. In particular,
the monoidal product of two semi-linear algebraic sets $X\times V$ and
$X\times W$ is not the same as the tensor product of the corresponding bundles
over $X$. One can consider more general vector bundles in the same manner, but
the category ${\mathbf{SLAffSch}}$ is enough for out purposes, this is a
minimal monoidal category that includes $({\mathbf{AlgSet}},\times)$ and
$({\mathbf{Vect}},\otimes)$.

4.3.4. Algebras of functions on semi-linear algebraic sets. Let
$X\in{\mathbf{AlgSet}}$ and $V\in{\mathbf{FVect}}$. The algebra of regular
functions on $X\times V$ as on algebraic set is $A(X)\otimes SV^{*}$. It
inherits the grading from $SV^{*}$ as a tensor product of an algebra with a
graded algebra (see p. 2.1). Let us denote $S^{*}(X\times V)=A(X)\otimes
SV^{*}\in{\mathbf{GrAlg}}$. For a morphism of the form (4.3) we obtain the
morphism of graded algebras with the reverse direction: $S^{*}(F)\colon
A(Y)\otimes SW^{*}\to A(X)\otimes SV^{*}$, it is defined as

$\displaystyle S^{*}(F)\colon s\mapsto\varphi^{*}(s)=s\cdot\varphi\in A(X),$
$\displaystyle\varphi^{*}(s)(x)=s\big{(}\varphi(x)\big{)},$ (4.64)
$\displaystyle S^{*}(F)\colon\xi\mapsto f^{*}(\xi)=\xi\cdot f\in A(X)\otimes
V^{*},$ $\displaystyle f^{*}(\xi)(x,v)=\xi\big{(}f(x,v)\big{)},$ (4.65)

where $s\in A(Y)$, $\xi\in W^{*}$, $x\in X$, $v\in V$. The formula (4.64)
gives the usual embedding
${\mathbf{AlgSet}}\hookrightarrow{\mathbf{CommAlg}}^{\mathrm{op}}$. Let
$(v_{j})$, $(v^{j})$ and $(w_{i})$, $(w^{i})$ be dual bases of $V$, $V^{*}$
and $W$, $W^{*}$ respectively. Then $f(x,v_{j})=\sum_{i}f^{i}_{j}(x)w_{i}$ for
some $f^{i}_{j}\in A(X)$, so the map (4.65) has the form
$f^{*}(w^{i})=\sum_{j}f^{i}_{j}\otimes v^{j}\in A(X)\otimes V^{*}$.

We obtain a lax monoidal functor
$S^{*}\colon({\mathbf{SLAlgSet}},\otimes)\to({\mathbf{FQA}}_{\mathrm{s}c}^{\mathrm{op}},\circ)$.
In particular, any algebraic set $X$ as the object $X\times{\mathbb{K}}$
corresponds to the graded algebra $S^{*}(X)=A(X)\otimes{\mathbb{K}}[u]$ and we
have the natural isomorphism $S^{*}(X\times V)\cong S^{*}(X)\circ S^{*}(V)$.

4.3.5. Semi-linear affine schemes. Let us generalise the semi-linear spaces to
the case of affine schemes. Among the commutative semi-connected quadratic
algebras consider the algebras of the form ${\mathfrak{R}}\otimes SV^{*}$
where ${\mathfrak{R}}\in{\mathbf{CommAlg}}$, $V\in{\mathbf{FVect}}$. Such
algebra can be interpreted as the ‘function algebra’ on the affine scheme
$X\times V$, where $X=\mathop{\mathrm{Spec}}\nolimits{\mathfrak{R}}$. This is
an affine scheme which corresponds to the algebra ${\mathfrak{R}}\otimes
SV^{*}$ and has an additional structure given by the grading of this algebra.
We call such schemes semi-linear affine schemes. By definition they are
objects of the category ${\mathbf{FQA}}_{\mathrm{s}c}^{\mathrm{op}}$
corresponding to the graded algebras ${\mathfrak{R}}\otimes SV^{*}$, where
${\mathfrak{R}}\in{\mathbf{CommAlg}}$, $V\in{\mathbf{FVect}}$. The embedding
${\mathbf{SLAffSch}}\hookrightarrow{\mathbf{FQA}}_{\mathrm{s}c}^{\mathrm{op}}$
extends the functor
$S^{*}\colon{\mathbf{SLAlgSet}}\to{\mathbf{FQA}}_{\mathrm{s}c}^{\mathrm{op}}$,
$S^{*}(X\times V)={\mathfrak{R}}\otimes SV^{*}$, since ${\mathbf{SLAlgSet}}$
is a subcategory of ${\mathbf{SLAffSch}}$, so we also denote it by $S^{*}$.

Let ${\mathfrak{S}}\in{\mathbf{CommAlg}}$,
$Y=\mathop{\mathrm{Spec}}\nolimits{\mathfrak{S}}$ and $W\in{\mathbf{FVect}}$.
A morphism $X\times V\to Y\times W$ in ${\mathbf{SLAffSch}}$ is given by a
graded homomorphism ${\mathfrak{S}}\otimes SW^{*}\to{\mathfrak{R}}\otimes
SV^{*}$ generated by an algebra homomorphism
$\alpha\colon{\mathfrak{S}}\to{\mathfrak{R}}$ and a linear map $t\colon
W^{*}\to{\mathfrak{R}}\otimes V^{*}$. In the order $1$ it has the form

$\displaystyle(\alpha,t)\colon{\mathfrak{S}}\otimes
W^{*}\to{\mathfrak{R}}\otimes V^{*},$
$\displaystyle(\alpha,t)(s\otimes\xi)=\alpha(s)t(\xi).$ (4.66)

For a morphism in ${\mathbf{SLAlgSet}}$ of the form (4.3) we have
$\alpha=\varphi^{*}$ and $t=f^{*}$, $(\alpha,t)=F^{*}$.

######  Remark 4.4.

The semi-linear affine scheme $X\times V$ can be identified with a quasi-
coherent sheaf corresponding to the free ${\mathfrak{R}}$-module
${\mathfrak{R}}\otimes V$ (or rather to its dual ${\mathfrak{R}}\otimes
V^{*}$). The category ${\mathbf{SLAffSch}}$ is equivalent to the category of
such free sheaves (${\mathfrak{R}}$-modules) with appropriate morphisms. This
generalises the equivalence described in p. 4.3.

Let ${\mathfrak{R}},{\mathfrak{R}}^{\prime}\in{\mathbf{CommAlg}}$,
$X=\mathop{\mathrm{Spec}}\nolimits{\mathfrak{R}}$,
$X^{\prime}=\mathop{\mathrm{Spec}}\nolimits{\mathfrak{R}}^{\prime}$,
$V,V^{\prime}\in{\mathbf{FVect}}$. Define the monoidal product of the semi-
linear affine schemes $X\times V$ and $X^{\prime}\times V^{\prime}$ as the
semi-linear affine scheme $(X\times X^{\prime})\times(V\otimes V^{\prime})$
with the function algebra ${\mathfrak{R}}\otimes{\mathfrak{R}}^{\prime}\otimes
S^{*}(V\otimes V^{\prime})$. Note that the embedding
${\mathbf{SLAffSch}}\hookrightarrow{\mathbf{FQA}}_{\mathrm{s}c}^{\mathrm{op}}$
is a fully faithful symmetric lax monoidal functor
$S^{*}\colon({\mathbf{SLAffSch}},\otimes)\to({\mathbf{FQA}}_{\mathrm{s}c}^{\mathrm{op}},\circ)$.

4.3.6. Semi-linear algebraic monoids and semi-linear affine monoid schemes. A
monoid in $({\mathbf{SLAlgSet}},\otimes)$ is a semi-linear algebraic set
$X\times V\in{\mathbf{SLAlgSet}}$ with the semi-linear maps $\mu_{X\times
V}\colon(X\times X)\times(V\otimes V)\to X\times V$, $\eta_{X\times
V}\colon{\mathbb{K}}\to X\times V$ which have the form

$\displaystyle\mu_{X\times V}(x,y,u\otimes v)=\big{(}xy,f(x,y,u\otimes
v)\big{)},$ $\displaystyle\eta_{X\times V}(1)=\big{(}e,1_{V}\big{)},$ (4.67)

where $xy=\mu_{X}(x,y)$, $e=\eta_{X}(0)$ are defined by a monoid
$(X,\mu_{X},\eta_{X})\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{AlgSet}},\times)$
and $f\colon X\times X\times(V\otimes V)\to V$ is a morphism in
${\mathbf{AlgSet}}$ linear in the last argument such that $f(e,x,1_{V}\otimes
v)=v=f(x,e,v\otimes 1_{V})$, $f\big{(}x,yz,u\otimes f(y,z,v\otimes
w)\big{)}=f\big{(}xy,z,f(x,y,u\otimes v)\otimes w\big{)}$, $x,y,z\in X$,
$u,v,w\in V$. Denote by $V_{x}$ the copy of $V$ considered as a fibre over
$x\in X$, then for any $x,y\in X$ we obtain the linear map $f_{x,y}\colon
V_{x}\otimes V_{y}\to V_{xy}$, $f_{x,y}(u\otimes v)=f(x,y,u\otimes v)$, and
the last condition is equivalent to the commutativity of the diagram

(4.72)

while $1_{V}$ belongs to $V_{e}$. If $X=\\{e\\}$ is a one-point set, then to
equip $V=X\times V$ with a monoid structure is the same as to choose the
algebra
$(V,f_{e,e},1)\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{Vect}},\otimes)$. If
$V={\mathbb{K}}$, then a monoid structure on $X\times V=X\times{\mathbb{K}}$
is given by an algebraic monoid
$(X,\mu_{X},e)\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{AlgSet}},\times)$
with collection of coefficients $p_{x,y}\in{\mathbb{K}}$, while
$q\in{\mathbb{K}}\backslash\\{0\\}$ satisfying
$p_{x,yz}p_{y,z}=p_{xy,z}p_{x,y}$ and $p_{e,x}=p_{x,e}=q^{-1}$ (the linear map
$f_{x,y}\colon{\mathbb{K}}\otimes{\mathbb{K}}\to{\mathbb{K}}$ is a
multiplication by the coefficient $p_{x,y}$, while $1_{V}=q$). The case of a
usual algebraic monoid corresponds to the values $p_{x,y}=q=1$. In this way we
see that the categorical embeddings
${\mathbf{FVect}}\hookrightarrow{\mathbf{SLAlgSet}}$ and
${\mathbf{AlgSet}}\hookrightarrow{\mathbf{SLAlgSet}}$ induce the fully
faithful functor
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{FVect}},\otimes)\to\mathop{\mathbf{Mon}}\nolimits({\mathbf{SLAlgSet}},\otimes)$
and the faithful functor
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{AlgSet}},\times)\to\mathop{\mathbf{Mon}}\nolimits({\mathbf{SLAlgSet}},\otimes)$,
which are symmetric strong monoidal due to p. 2.2.

Let ${\mathfrak{R}}=A(X)$. The homomorphisms
$\alpha=\mu_{X}^{*}\colon{\mathfrak{R}}\to{\mathfrak{R}}\otimes{\mathfrak{R}}$
and $\beta=\eta_{X}^{*}\colon{\mathfrak{R}}\to{\mathbb{K}}$ gives the
structure of coalgebra on ${\mathfrak{R}}$. The linear maps $t=f^{*}\colon
V^{*}\to{\mathfrak{R}}\otimes{\mathfrak{R}}\otimes V^{*}\otimes V^{*}$ and
$\varepsilon_{V^{*}}=1_{V}\colon V^{*}\to{\mathbb{K}}$ make the diagrams

$\textstyle{{\mathfrak{R}}\otimes
V^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\alpha,t)}$$\scriptstyle{(\alpha,t)}$$\textstyle{{\mathfrak{R}}\otimes{\mathfrak{R}}\otimes
V^{*}\otimes
V^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{(23)}}$$\textstyle{{\mathfrak{R}}\otimes{\mathfrak{R}}\otimes
V^{*}\otimes
V^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{(23)}}$$\textstyle{{\mathfrak{R}}\otimes
V^{*}\otimes{\mathfrak{R}}\otimes
V^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{id}}\nolimits\otimes(\alpha,t)}$$\textstyle{{\mathfrak{R}}\otimes
V^{*}\otimes{\mathfrak{R}}\otimes
V^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\alpha,t)\otimes\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{id}}\nolimits}$$\textstyle{{\mathfrak{R}}\otimes
V^{*}\otimes{\mathfrak{R}}\otimes{\mathfrak{R}}\otimes V^{*}\otimes
V^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{(34)}\sigma^{(23)}}$$\textstyle{{\mathfrak{R}}\otimes{\mathfrak{R}}\otimes
V^{*}\otimes V^{*}\otimes{\mathfrak{R}}\otimes
V^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma^{(34)}\sigma^{(45)}}$$\textstyle{{\mathfrak{R}}\otimes{\mathfrak{R}}\otimes{\mathfrak{R}}\otimes
V^{*}\otimes V^{*}\otimes V^{*}}$ (4.81) $\textstyle{{\mathfrak{R}}\otimes
V^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\alpha,t)}$$\textstyle{{\mathbb{K}}\otimes{\mathfrak{R}}\otimes{\mathbb{K}}\otimes
V^{*}}$$\textstyle{{\mathfrak{R}}\otimes{\mathfrak{R}}\otimes V^{*}\otimes
V^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta\otimes\mathop{\mathrm{id}}\nolimits\otimes\varepsilon_{V^{*}}\otimes\mathop{\mathrm{id}}\nolimits}$$\scriptstyle{\mathop{\mathrm{id}}\nolimits\otimes\beta\otimes\mathop{\mathrm{id}}\nolimits\otimes\varepsilon_{V^{*}}}$$\textstyle{{\mathfrak{R}}\otimes{\mathbb{K}}\otimes
V^{*}\otimes{\mathbb{K}}}$ (4.86)

to be commutative, where
$\sigma^{(i,i+1)}=\mathop{\mathrm{id}}\nolimits^{\otimes(i-1)}\otimes\sigma\otimes\mathop{\mathrm{id}}\nolimits\otimes\cdots\otimes\mathop{\mathrm{id}}\nolimits$.
Since $({\mathfrak{R}},\alpha,\beta)$ is a comonoid in
$({\mathbf{CommAlg}},\otimes)$, it is enough to check the commutativity of the
diagrams (4.81) by substituting the elements of the form
$1\otimes\xi\in{\mathfrak{R}}\otimes V^{*}$.

Now consider monoids in $({\mathbf{SLAffSch}},\otimes)$, which we call semi-
linear affine monoid schemes. Let ${\mathfrak{R}}\in{\mathbf{CommAlg}}$,
$V\in{\mathbf{FVect}}$ and $X=\mathop{\mathrm{Spec}}\nolimits{\mathfrak{R}}$.
A structure of monoid on $X\times V$ is defined by the morphisms $\mu_{X\times
V}\colon(X\times V)\otimes(X\times V)\to X\times V$ and $\eta_{X\times
V}\colon{\mathbb{K}}\to X\times V$, which are given by graded homomorphisms
${\mathfrak{R}}\otimes S(V^{*})\to{\mathfrak{R}}\otimes{\mathfrak{R}}\otimes
S(V^{*})\otimes S(V^{*})$ and ${\mathfrak{R}}\otimes
S(V^{*})\to{\mathbb{K}}[u]$. The zero and first order components of these
homomorphisms are
$\alpha\colon{\mathfrak{R}}\to{\mathfrak{R}}\otimes{\mathfrak{R}}$,
$(\alpha,t)\colon{\mathfrak{R}}\otimes
V^{*}\to{\mathfrak{R}}\otimes{\mathfrak{R}}\otimes V^{*}\otimes V^{*}$ and
$\beta\colon{\mathfrak{R}}\to{\mathbb{K}}$,
$(\beta,\varepsilon_{V^{*}})\colon{\mathfrak{R}}\otimes V^{*}\to{\mathbb{K}}$,
where $t\colon V^{*}\to{\mathfrak{R}}\otimes{\mathfrak{R}}\otimes V^{*}\otimes
V^{*}$ and $\varepsilon_{V^{*}}\colon V^{*}\to{\mathbb{K}}$ are linear maps.
They give a structure of a monoid on $X\times V$ iff
$({\mathfrak{R}},\alpha,\beta)\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{CommAlg}},\otimes)$
and the diagrams (4.81) commute. The embedding
${\mathbf{AffSch}}\hookrightarrow{\mathbf{SLAffSch}}$ induces the faithful
symmetric strong monoidal functor
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{AffSch}},\times)\to\mathop{\mathbf{Mon}}\nolimits({\mathbf{SLAffSch}},\otimes)$.

4.3.7. Quantum semi-linear spaces. The ‘algebra of functions’ on a semi-linear
space $X\times V$ is the tensor product of a quadratic algebra $S^{*}V$ with a
commutative algebra that is the ‘algebra of functions’ on $X$. By following
the Manin’s idea described in the beginning of Section 3 we call quantum semi-
linear space any semi-connected quadratic algebra
${\mathfrak{R}}\otimes{\mathcal{A}}$ (where ${\mathfrak{R}}\in{\mathbf{Alg}}$,
${\mathcal{A}}\in{\mathbf{QA}}$) considered as an object of the dual category
${\mathbf{QA}}_{\mathrm{s}c}^{\mathrm{op}}$. The finite-dimensional case
corresponds to the objects of ${\mathbf{FQA}}_{\mathrm{s}c}^{\mathrm{op}}$.
The quantum linear spaces correspond to the connected case
${\mathfrak{R}}={\mathbb{K}}$. In Section 5 we will construct Quantum
Representation Theory as the theory of representations of monoids in
${\mathbf{C}}=({\mathbf{QA}}_{\mathrm{s}c}^{\mathrm{op}},\circ)$ on the
objects of a monoidal subcategory ${\mathbf{P}}={\mathbf{FQA}}^{\mathrm{op}}$.

The composition of the embeddings
${\mathbf{AffSch}}\hookrightarrow{\mathbf{SLAffSch}}\hookrightarrow{\mathbf{FQA}}_{\mathrm{s}c}^{\mathrm{op}}$
is a faithful symmetric strong monoidal functor
$({\mathbf{AffSch}},\times)\to({\mathbf{FQA}}_{\mathrm{s}c}^{\mathrm{op}},\circ)$,
its dual is the functor
$({\mathbf{CommAlg}},\otimes)\to({\mathbf{FQA}}_{\mathrm{s}c},\circ)$, which
maps ${\mathfrak{R}}$ to ${\mathfrak{R}}\otimes{\mathbb{K}}[u]$. Thus, we
consider an affine non-commutative space
${\mathfrak{R}}\in{\mathbf{Alg}}^{\mathrm{op}}$ as the quantum semi-linear
space ${\mathfrak{R}}\otimes{\mathbb{K}}[u]\in{\mathbf{FQA}}^{\mathrm{op}}$.

Any morphism $f\colon{\mathcal{B}}\to{\mathcal{A}}$ in
${\mathbf{QA}}_{\mathrm{s}c}$ between arbitrary semi-connected quadratic
algebras ${\mathcal{B}}={\mathfrak{S}}\otimes\big{(}TW/(S)\big{)}$ and
${\mathcal{A}}={\mathfrak{R}}\otimes\big{(}TV/(R)\big{)}$ is uniquely defined
by its zero and first order components
$\alpha=f_{0}\colon{\mathfrak{S}}\to{\mathfrak{R}}$ and
$f_{1}\colon{\mathfrak{S}}\otimes W\to{\mathfrak{R}}\otimes V$. We have
$f_{1}=(\alpha,t)$ in the sense that $f_{1}(s\otimes
w)=\alpha(s)t(w)=t(w)\alpha(s)$, where $t\colon W\to{\mathfrak{R}}\otimes V$
is defined by the formula $t(w)=f_{1}(1\otimes w)$. Since ${\mathcal{A}}_{2}$
is the quotient of $({\mathfrak{R}}\otimes
V)\otimes_{\mathfrak{R}}({\mathfrak{R}}\otimes V)={\mathfrak{R}}\otimes
V\otimes V$ over ${\mathfrak{R}}\otimes R$, the linear map
$f_{2}\colon{\mathcal{B}}_{2}\to{\mathcal{A}}_{2}$ is induced by

$\displaystyle{\mathfrak{S}}\otimes W\otimes W=({\mathfrak{S}}\otimes
W)\otimes_{\mathfrak{S}}({\mathfrak{S}}\otimes
W)\xrightarrow{f_{1}\otimes_{\mathfrak{S}}f_{1}}({\mathfrak{R}}\otimes
V)\otimes_{\mathfrak{R}}({\mathfrak{R}}\otimes V)={\mathfrak{R}}\otimes
V\otimes V,$ $\displaystyle s\otimes w\otimes w^{\prime}=(s\otimes
w)\otimes_{{\mathfrak{S}}}(1\otimes
w^{\prime})\mapsto\big{(}\alpha(s)t(w)\big{)}\otimes_{{\mathfrak{R}}}t(w^{\prime})=\alpha(s)\big{(}t(w)\otimes_{\mathfrak{R}}t(w^{\prime})\big{)},$
$\displaystyle 1\otimes w\otimes
w^{\prime}\mapsto\big{(}t(w)\otimes_{\mathfrak{R}}t(w^{\prime})\big{)}=(t\mathbin{\dot{\otimes}}t)(w\otimes
w^{\prime}),$

where $t\mathbin{\dot{\otimes}}t\colon W\otimes W\to{\mathfrak{R}}\otimes
V\otimes V$ is the composition

$\displaystyle W\otimes W\xrightarrow{t\otimes t}{\mathfrak{R}}\otimes
V\otimes{\mathfrak{R}}\otimes
V\xrightarrow{\mathop{\mathrm{id}}\nolimits_{\mathfrak{R}}\otimes\sigma_{V,{\mathfrak{R}}}\otimes\mathop{\mathrm{id}}\nolimits_{V}}{\mathfrak{R}}\otimes{\mathfrak{R}}\otimes
V\otimes
V\xrightarrow{\mu_{\mathfrak{R}}\otimes\mathop{\mathrm{id}}\nolimits_{V}\otimes\mathop{\mathrm{id}}\nolimits_{V}}{\mathfrak{R}}\otimes
V\otimes V.$

An algebra homomorphism $\alpha\colon{\mathfrak{S}}\to{\mathfrak{R}}$ and a
linear map $t\colon W\to{\mathfrak{R}}\otimes V$ define a homomorphism
$f\colon{\mathcal{B}}\to{\mathcal{A}}$ iff

$\displaystyle\alpha(s)t(w)=t(w)\alpha(s)\qquad\forall\,s\in{\mathfrak{S}},\,w\in
W$ (4.87)

and $(t\mathbin{\dot{\otimes}}t)S\subset{\mathfrak{R}}\otimes R$. In
particular, $({\mathcal{B}},\Delta,\varepsilon)$ is a comonoid in
$({\mathbf{QA}}_{\mathrm{s}c},\circ)$ iff the zero and first order components
of the graded homomorphisms
$\Delta\colon{\mathcal{B}}\to{\mathcal{B}}\circ{\mathcal{B}}$ and
$\varepsilon\colon{\mathcal{B}}\to{\mathbb{K}}[u]$ satisfy the following
conditions:
$\alpha=\Delta_{0}\colon{\mathfrak{S}}\to{\mathfrak{S}}\otimes{\mathfrak{S}}$
and $\beta=\varepsilon_{0}\colon{\mathfrak{S}}\to{\mathbb{K}}$ are algebra
homomorphisms, $\Delta_{1}(s\otimes w)=\alpha(s)t(w)$ and
$\varepsilon_{1}(s\otimes w)=\varepsilon_{0}(s)\varepsilon_{W}(w)$ for some
linear maps $t\colon W\to{\mathfrak{S}}\otimes{\mathfrak{S}}\otimes W\otimes
W$, $\varepsilon_{W}\colon W\to{\mathbb{K}}$ subjected to the commutativity
condition (4.87), the inclusion conditions
$(t\mathbin{\dot{\otimes}}t)S\subset{\mathfrak{S}}\otimes{\mathfrak{S}}\otimes\sigma^{(23)}(S\otimes
W^{\otimes 2}+W^{\otimes 2}\otimes S)$,
$(\varepsilon_{W}\otimes\varepsilon_{W})S=0$ and making the diagrams (4.81) to
be commutative (where $V^{*}=W$ and ${\mathfrak{R}}={\mathfrak{S}}$).

A coaction of the comonoid
${\mathbb{O}}=({\mathcal{A}},\Delta,\varepsilon)\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{QA}}_{\mathrm{s}c},\circ)$
on an object ${\mathcal{B}}\in{\mathbf{QA}}_{\mathrm{s}c}$ is a graded
homomorphism $\delta\colon{\mathcal{B}}\to{\mathcal{A}}\circ{\mathcal{B}}$.
Its zero component
$\delta_{0}\colon{\mathcal{B}}_{0}\to{\mathcal{A}}_{0}\otimes{\mathcal{B}}_{0}$
should be coaction of
$({\mathcal{A}}_{0},\Delta_{0},\varepsilon_{0})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{Alg}},\otimes)$
on ${\mathcal{B}}_{0}\in{\mathbf{Alg}}$ and the first order component is
$\delta_{1}=(\delta_{0},p)$ for some linear map $p\colon
W\to{\mathcal{A}}_{0}\otimes{\mathcal{B}}_{0}\otimes V\otimes W$. To obtain
the condition for $p$ one needs to take first order components of the
morphisms in the commutative diagrams (2.90), this leads to diagrams similar
to (4.81).

###  4.4 Internal hom for quantum linear spaces

The next ingredient, which we need to construct quantum representations, is a
quantum analogue of the internal hom of vector spaces. This is an internal hom
in $({\mathbf{FQA}}^{\mathrm{op}},\circ)$, i.e. an internal cohom in
$({\mathbf{FQA}},\circ)$. We need also its generalised version in the sense of
p. 3.1.

4.4.1. Internal cohom and universal Manin matrix. It was shown in [Man87,
Man88] that the category ${\mathbf{FQA}}$ is coclosed with respect to the
Manin product, namely, proved the existence of a natural isomorphism
$\mathop{\mathrm{Hom}}\nolimits({\mathcal{A}}\bullet{\mathcal{B}}^{!},{\mathcal{C}})\cong\mathop{\mathrm{Hom}}\nolimits({\mathcal{A}},{\mathcal{B}}\circ{\mathcal{C}})$.
Hence the internal cohom for ${\mathcal{A}},{\mathcal{B}}\in{\mathbf{FQA}}$
can be defined as333Manin denoted the cohom-functor $\mathop{\mathrm{\bf
cohom}}\nolimits\colon{\mathbf{FQA}}^{\mathrm{op}}\times{\mathbf{FQA}}\to{\mathbf{FQA}}$
as $\underline{\hom}$.

$\displaystyle\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{A}})={\mathcal{B}}^{!}\bullet{\mathcal{A}}.$
(4.88)

Let us calculate it for ${\mathcal{A}}={\mathfrak{X}}_{A}({\mathbb{K}})$ and
${\mathcal{B}}={\mathfrak{X}}_{B}({\mathbb{K}})$, where
$A\in\mathop{\mathrm{\bf end}}\nolimits(V^{\otimes 2})$ and
$B\in\mathop{\mathrm{\bf end}}\nolimits(W^{\otimes 2})$ are idempotents and
$V={\mathbb{K}}^{n}$, $W={\mathbb{K}}^{m}$. Since
${\mathcal{B}}^{!}=\Xi_{B}({\mathbb{K}})=TW/\big{(}\mathop{\mathrm{Im}}\nolimits(1-B)\big{)}$
and ${\mathcal{A}}=TV^{*}/(\mathop{\mathrm{Im}}\nolimits A^{*})$ we have

$\displaystyle{\mathcal{B}}^{!}\bullet{\mathcal{A}}=T(W\otimes
V^{*})/\big{(}\sigma^{(23)}(\mathop{\mathrm{Im}}\nolimits(1-B)\otimes\mathop{\mathrm{Im}}\nolimits
A^{*})\big{)},$ (4.89)

where
$\sigma^{(23)}=\mathop{\mathrm{id}}\nolimits\otimes\sigma\otimes\mathop{\mathrm{id}}\nolimits$
(the indices $2,3$ mean that $\sigma$ acts on the second and third tensor
factors). Let $(v_{i})$ and $(w_{j})$ be bases of $V$ and $W$ respectively.
Let $(v^{i})$ be basis of $V^{*}$ dual to $(v_{i})$. Then the basis of
$W\otimes V^{*}=\mathop{\mathrm{Hom}}\nolimits(V,W)$ consists of
${\mathcal{M}}^{i}_{j}=w_{j}\otimes v^{i}$. The subspace
$\sigma^{(23)}(\mathop{\mathrm{Im}}\nolimits(1-B)\otimes\mathop{\mathrm{Im}}\nolimits
A^{*})\subset W\otimes V^{*}\otimes W\otimes V^{*}$ is spanned by the vectors

$\sigma^{(23)}\sum_{k,l,s,t}(1-B)^{kl}_{ij}A^{pq}_{st}w_{k}\otimes
w_{l}\otimes v^{s}\otimes
v^{t}=\sum_{k,l,s,t}A^{pq}_{st}({\mathcal{M}}_{k}^{s}\otimes{\mathcal{M}}_{l}^{t})(1-B)^{kl}_{ij},$

where $(1-B)^{kl}_{ij}=\delta^{k}_{i}\delta^{l}_{j}-B^{kl}_{ij}$. Therefore,
the algebra $\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{A}})$ is generated by
${\mathcal{M}}^{i}_{j}$ with the quadratic relation

$\displaystyle A{\mathcal{M}}^{(1)}{\mathcal{M}}^{(2)}(1-B)=0,$ (4.90)

where ${\mathcal{M}}$ is the matrix with entries ${\mathcal{M}}^{i}_{j}$. The
relation (4.90) means exactly that ${\mathcal{M}}$ is an $(A,B)$-Manin matrix
over ${\mathfrak{R}}=\mathop{\mathrm{\bf
cohom}}\nolimits\big{(}{\mathfrak{X}}_{B}({\mathbb{K}}),{\mathfrak{X}}_{A}({\mathbb{K}})\big{)}$.
We call it universal $(A,B)$-Manin matrix. Note also that

$\mathop{\mathrm{\bf
cohom}}\nolimits\big{(}{\mathfrak{X}}_{B}({\mathbb{K}}),{\mathfrak{X}}_{A}({\mathbb{K}})\big{)}={\mathfrak{X}}_{B}({\mathbb{K}})^{!}\bullet{\mathfrak{X}}_{A}({\mathbb{K}})=\Xi_{A}({\mathbb{K}})^{!}\bullet\Xi_{B}({\mathbb{K}})=\\\
=\mathop{\mathrm{\bf
cohom}}\nolimits\big{(}\Xi_{A}({\mathbb{K}}),\Xi_{B}({\mathbb{K}})\big{)}.$
(4.91)

4.4.2. Connection with the Manin matrices. In [S] we denote by
${\mathfrak{U}}_{A,B}$ the algebra (4.91) considered as an object of
${\mathbf{Alg}}$. This algebra was interpreted there via left adjoints to the
functors
${\mathfrak{X}}_{B}\colon{\mathbf{Alg}}\to{\mathbf{FQA}}_{\mathrm{s}c}$ and
$\Xi_{A}\colon{\mathbf{Alg}}\to{\mathbf{FQA}}_{\mathrm{s}c}$. In particular,
we derived the natural bijections

$\displaystyle\mathop{\mathrm{Hom}}\nolimits_{\mathbf{GrAlg}}\big{(}{\mathfrak{X}}_{A}({\mathbb{K}}),{\mathfrak{X}}_{B}({\mathfrak{R}})\big{)}\cong\mathop{\mathrm{Hom}}\nolimits_{\mathbf{Alg}}({\mathfrak{U}}_{A,B},{\mathfrak{R}})\cong\mathop{\mathrm{Hom}}\nolimits_{\mathbf{GrAlg}}\big{(}\Xi_{B}({\mathbb{K}}),\Xi_{A}({\mathfrak{R}})\big{)},$
(4.92)

which can be described as follows. The left and right sets consists of the
graded homomorphisms $f_{M}$ and $f^{M}$ for $(A,B)$-Manin matrices $M$ over
${\mathfrak{R}}$ (see p. 4.2). A formula ${\mathcal{M}}^{i}_{j}\mapsto
M^{i}_{j}$ gives a homomorphism ${\mathfrak{U}}_{A,B}\to{\mathfrak{R}}$ (in
${\mathbf{Alg}}$) iff $M=(M^{i}_{j})$ is an $(A,B)$-Manin matrix.

Consider the case when ${\mathfrak{R}}$ is also a quadratic algebra:
${\mathfrak{R}}={\mathfrak{X}}_{C}({\mathbb{K}})$ for some idempotent $C$.
Note that
${\mathfrak{X}}_{B}\big{(}{\mathfrak{X}}_{C}({\mathbb{K}})\big{)}_{k}={\mathfrak{X}}_{C}({\mathbb{K}})\otimes{\mathfrak{X}}_{B}({\mathbb{K}})_{k}$,
so the algebra
${\mathfrak{X}}_{B}\big{(}{\mathfrak{X}}_{C}({\mathbb{K}})\big{)}$ does not
coincide with
${\mathfrak{X}}_{C}({\mathbb{K}})\otimes{\mathfrak{X}}_{B}({\mathbb{K}})$ as
graded algebra in the sense of the Manin’s operation $\otimes$ (see p. 4.1).
Consider the subset of the set (4.92) corresponding to the $(A,B)$-Manin
matrices $M$ with entries of the first order:
$M^{i}_{j}\in{\mathfrak{X}}_{C}({\mathbb{K}})_{1}$. As a subset of
$\mathop{\mathrm{Hom}}\nolimits_{\mathbf{GrAlg}}\Big{(}{\mathfrak{X}}_{A}({\mathbb{K}}),{\mathfrak{X}}_{B}\big{(}{\mathfrak{X}}_{C}({\mathbb{K}})\big{)}\Big{)}$
it consists of the graded homomorphisms
${\mathfrak{X}}_{A}({\mathbb{K}})\to{\mathfrak{X}}_{B}\big{(}{\mathfrak{X}}_{C}({\mathbb{K}})\big{)}$
which factors through the morphism
${\mathfrak{X}}_{C}({\mathbb{K}})\circ{\mathfrak{X}}_{B}({\mathbb{K}})\to{\mathfrak{X}}_{B}\big{(}{\mathfrak{X}}_{C}({\mathbb{K}})\big{)}$.
This subset can be identified with the set
$\mathop{\mathrm{Hom}}\nolimits_{\mathbf{GrAlg}}\big{(}{\mathfrak{X}}_{A}({\mathbb{K}}),{\mathfrak{X}}_{C}({\mathbb{K}})\circ{\mathfrak{X}}_{B}({\mathbb{K}})\big{)}$.
As a subset of
$\mathop{\mathrm{Hom}}\nolimits_{\mathbf{Alg}}\Big{(}{\mathfrak{U}}_{A,B},{\mathfrak{X}}_{B}\big{(}{\mathfrak{X}}_{C}({\mathbb{K}})\big{)}\Big{)}$
it consists of homomorphisms
${\mathfrak{U}}_{A,B}\to{\mathfrak{X}}_{C}({\mathbb{K}})$ which preserve the
grading, where the grading of the algebra ${\mathfrak{U}}_{A,B}$ arises, if
one considers it as the cohom-object $\mathop{\mathrm{\bf
cohom}}\nolimits\big{(}{\mathfrak{X}}_{B}({\mathbb{K}}),{\mathfrak{X}}_{A}({\mathbb{K}})\big{)}$.
This leads to the adjunction from the beginning of p. 4.4, in the notations of
this paragraph it takes the form

$\displaystyle\mathop{\mathrm{Hom}}\nolimits\big{(}{\mathfrak{X}}_{A}({\mathbb{K}}),{\mathfrak{X}}_{C}({\mathbb{K}})\circ{\mathfrak{X}}_{B}({\mathbb{K}})\big{)}\cong\mathop{\mathrm{Hom}}\nolimits\Big{(}\mathop{\mathrm{\bf
cohom}}\nolimits\big{(}{\mathfrak{X}}_{B}({\mathbb{K}}),{\mathfrak{X}}_{A}({\mathbb{K}})\big{)},{\mathfrak{X}}_{C}({\mathbb{K}})\Big{)},$
(4.93)

this isomorphism is induced by the isomorphism (4.92).

4.4.3. Internal hom for finite-dimensional vector spaces. Let us regard how
the internal hom of quantum linear spaces agrees with the internal hom of
vector spaces. Remind that the internal hom-object $\mathop{\mathrm{\bf
hom}}\nolimits(W,V)$ in ${\mathbf{Vect}}$ is the set
$\mathop{\mathrm{Hom}}\nolimits(W,V)$ equipped with the usual structure of
vector space (see p. 3.1). In the case $W\in{\mathbf{FVect}}$ we have natural
linear isomorphism $\mathop{\mathrm{\bf hom}}\nolimits(W,V)=W^{*}\otimes V$.
By applying the strong monoidal functor
$(-)^{*}_{1}\colon({\mathbf{FQA}}^{\mathrm{op}},\circ)\to({\mathbf{FVect}},\otimes)$
to the internal hom (4.88) we obtain exactly $\big{(}\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{A}})\big{)}^{*}_{1}=W^{*}\otimes
V=\mathop{\mathrm{\bf hom}}\nolimits(W,V)$, where $W={\mathcal{B}}_{1}^{*}$,
$V={\mathcal{A}}_{1}^{*}$. In the notations of Subsection 3.3 it means that
the morphisms $\Phi_{{\mathcal{B}},{\mathcal{A}}}$ for the strong monoidal
functor
$(-)^{*}_{1}\colon({\mathbf{FQA}}^{\mathrm{op}},\circ)\to({\mathbf{FVect}},\otimes)$
are isomorphisms.

On the other hand the quadratic algebra $S^{*}\mathop{\mathrm{\bf
hom}}\nolimits(W,V)$ does not coincide with the algebra $\mathop{\mathrm{\bf
cohom}}\nolimits(S^{*}W,S^{*}V)=(S^{*}W)^{!}\bullet S^{*}V=\Lambda(W)\bullet
S^{*}(V)$, which means that $\Phi_{W,V}$ for the lax monoidal functor
$S^{*}\colon({\mathbf{FVect}},\otimes)\to({\mathbf{FQA}}^{\mathrm{op}},\circ)$
are not isomorphisms.

4.4.4. Coevaluation and cocomposition morphisms. Let $(v_{i})$, $(w_{j})$ be
the bases of $V,W\in{\mathbf{FVect}}$ and let $(v^{i})$, $(w^{j})$ be the dual
bases of $V^{*}$, $W^{*}$. The elements $v^{i}$, $w^{j}$ and
${\mathcal{M}}^{i}_{j}=w_{j}\otimes v^{i}$ are generators of the algebras
${\mathcal{A}}=TV^{*}/(R)$, ${\mathcal{B}}=TW^{*}/(S)$ and
$\mathop{\mathrm{\bf cohom}}\nolimits({\mathcal{B}},{\mathcal{A}})$. In these
terms the coevaluation (3.15) (for $X={\mathcal{A}}$, $Y={\mathcal{B}}$) has
the form (see [Man88])

$\displaystyle\mathop{\mathrm{coev}}\nolimits\colon{\mathcal{A}}\to\mathop{\mathrm{\bf
cohom}}\nolimits\big{(}{\mathcal{B}},{\mathcal{A}}\big{)}\circ{\mathcal{B}},$
$\displaystyle v^{i}\mapsto\sum_{j}{\mathcal{M}}^{i}_{j}\otimes w^{j}.$ (4.94)

Let ${\mathcal{C}}\in{\mathbf{FQA}}$ and $(z_{l})$ be a basis of
$Z={\mathcal{C}}_{1}^{*}$. Then ${\mathcal{N}}^{j}_{l}=z_{l}\otimes w^{j}$ and
${\mathcal{K}}^{i}_{l}=z_{l}\otimes v^{i}$ are the generators of the algebras
$\mathop{\mathrm{\bf cohom}}\nolimits({\mathcal{C}},{\mathcal{B}})$ and
$\mathop{\mathrm{\bf cohom}}\nolimits({\mathcal{C}},{\mathcal{A}})$. The
cocomposition morphism (3.17) (for $X={\mathcal{A}}$, $Y={\mathcal{B}}$,
$Z={\mathcal{C}}$) reads (see [S])

$\displaystyle
d=d_{{\mathcal{A}},{\mathcal{B}},{\mathcal{C}}}\colon\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{C}},{\mathcal{A}})\to\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{A}})\circ\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{C}},{\mathcal{B}}),$
$\displaystyle{\mathcal{K}}^{i}_{l}\mapsto\sum_{j=1}^{m}{\mathcal{M}}^{i}_{j}\otimes{\mathcal{N}}^{j}_{l}.$
(4.95)

Let $B=(B^{ij}_{kl})_{i,j,k,l=1}^{m}$ be a matrix idempotent such that the
elements $\sum\limits_{i,j=1}^{m}B^{ij}_{kl}w_{i}\otimes w_{j}$ span the
subspace $S\subset W\otimes W$, then
${\mathcal{B}}={\mathfrak{X}}_{B}({\mathbb{K}})$ and $\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{B}})$ is the quadratic algebra
generated by ${\mathcal{M}}^{i}_{j}=w_{j}\otimes w^{j}$ with the relations
$B{\mathcal{M}}^{(1)}{\mathcal{M}}^{(2)}(1-B)=0$, where
${\mathcal{M}}=({\mathcal{M}}^{i}_{j})_{i,j=1}^{m}$ is the universal $B$-Manin
matrix. The comonoid $\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{FQA}},\circ)$
is the quadratic algebra $\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{B}})$ with the structure morphisms
$d_{\mathcal{B}}\colon\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})\to\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})\circ\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})$ and $v_{\mathcal{B}}\colon\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})\to{\mathbb{K}}[u]$ defined on generators as

$\displaystyle
d_{\mathcal{B}}({\mathcal{M}}^{i}_{j})=\sum_{k=1}^{m}{\mathcal{M}}^{i}_{k}\otimes{\mathcal{M}}^{k}_{j},$
$\displaystyle v_{\mathcal{B}}({\mathcal{M}}^{i}_{j})=\delta^{i}_{j}u.$ (4.96)

Since
${\mathcal{B}}^{!}=\Xi_{B}({\mathbb{K}})=TW/\big{(}\mathop{\mathrm{Im}}\nolimits(1-B)\big{)}={\mathfrak{X}}_{1-B^{\top}}({\mathbb{K}})$
is generated by $w_{1},\ldots,w_{m}\in W$, the algebra $\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}}^{!})$ is generated by the elements
$\widetilde{\mathcal{M}}_{ij}=w^{j}\otimes w_{i}\in W^{*}\otimes W$. The
latter ones are entries of the universal $(1-B^{\top})$-Manin matrix
$\widetilde{\mathcal{M}}=(\widetilde{\mathcal{M}}_{ij})$ transposed to the
universal $B$-Manin matrix ${\mathcal{M}}$. Due to the formulae (4.96) this
implies

$\displaystyle d_{{\mathcal{B}}^{!}}=d_{\mathcal{B}}^{\mathrm{cop}},$
$\displaystyle v_{{\mathcal{B}}^{!}}=v_{\mathcal{B}}.$ (4.97)

4.4.5. Infinite-dimensional case. Let
${\mathcal{A}},\widetilde{\mathcal{A}}\in{\mathbf{QA}}$ and
${\mathcal{B}}\in{\mathbf{FQA}}$. Then ${\mathcal{A}}=TV/(R)$,
$\widetilde{\mathcal{A}}=T\widetilde{V}/(\widetilde{R})$ and
${\mathcal{B}}=TW/(S)$ for some $V,\widetilde{V}\in{\mathbf{Vect}}$,
$W\in{\mathbf{FVect}}$, $R\subset V^{\otimes 2}$,
$\widetilde{R}\subset\widetilde{V}^{\otimes 2}$, $S\subset W^{\otimes 2}$. Let
us prove that there exists $\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{A}})$ in the sense of p. 3.1, where
the role of parameter is played by ${\mathcal{B}}\in{\mathbf{FQA}}$.

######  Proposition 4.5.

For any ${\mathcal{B}}\in{\mathbf{FQA}}$ the functor
$-\circ{\mathcal{B}}\colon{\mathbf{QA}}\to{\mathbf{QA}}$ has a left adjoint
$\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},-)\colon{\mathbf{QA}}\to{\mathbf{QA}}$. We have
$\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{A}})=TV^{\prime}/(R^{\prime})$ where
$V^{\prime}=\mathop{\mathrm{\bf hom}}\nolimits(W,V)$ and

$\displaystyle R^{\prime}=\\{U\in\mathop{\mathrm{\bf
hom}}\nolimits(W,V)^{\otimes 2}\mid U(W^{\otimes 2})\subset R,U(S)=0\\}.$
(4.98)

Proof. Let $(w_{i})_{i=1}^{m}$ and $(w^{i})_{i=1}^{m}$ be dual bases of
$W^{*}$ and $(W^{*})^{*}=W$ respectively. These bases give identifications
$W=({\mathbb{K}}^{m})^{*}$ and $W^{*}={\mathbb{K}}^{m}$. Let
$B\in\mathop{\mathrm{\bf
end}}\nolimits({\mathbb{K}}^{m}\otimes{\mathbb{K}}^{m})$ be a matrix
idempotent such that $S=\mathop{\mathrm{Im}}\nolimits B^{*}$, then
${\mathcal{B}}={\mathfrak{X}}_{B}({\mathbb{K}})=TW/(\mathop{\mathrm{Im}}\nolimits
B^{*})$. By identifying $W^{*}\otimes W$ with $\mathop{\mathrm{\bf
end}}\nolimits(W)$ we obtain

$\displaystyle\sum_{i,j}w_{i}\otimes w^{i}\otimes w_{j}\otimes
w^{j}=\sum_{i,j,k,l}\big{(}B_{ij}^{kl}+(1-B)_{ij}^{kl}\big{)}w_{k}\otimes
w^{i}\otimes w_{l}\otimes w^{j},$ (4.99)

where the coefficients $B^{ij}_{kl}\in{\mathbb{K}}$ and
$(1-B)^{ij}_{kl}=\delta^{i}_{k}\delta^{j}_{l}-B^{ij}_{kl}$ are entries of the
operators $B$ and $1-B$ respectively. Consider the linear map $\eta_{V}\colon
V\to\mathop{\mathrm{\bf hom}}\nolimits(W,V)\otimes W$ defined in the proof of
Prop. 3.2, that is $\eta_{V}(v)=\sum_{i}(w_{i}\otimes v)\otimes w^{i}$. It
generates a morphism $\eta\colon{\mathcal{A}}\to\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{A}})\circ{\mathcal{B}}$ with the
first order component $\eta_{1}=\eta_{V}$. Indeed, for any element
$r=\sum_{s}x_{s}\otimes y_{s}\in R$ we have

$\sigma^{(23)}(\eta_{V}\otimes\eta_{V})r=\sum_{s,i,j}(w_{i}\otimes
x_{s})\otimes(w_{j}\otimes y_{s})\otimes w^{i}\otimes w^{j}=\\\
=\sum_{s,i,j,k,l}\big{(}B_{ij}^{kl}+(1-B)_{ij}^{kl}\big{)}w_{k}\otimes
x_{s}\otimes w_{l}\otimes y_{s}\otimes w^{i}\otimes w^{j}.$ (4.100)

Since $B_{ij}^{kl}w^{i}\otimes w^{j}\in S$ and $(1-B)_{ij}^{kl}w_{k}\otimes
x_{s}\otimes w_{l}\otimes y_{s}\in R^{\prime}$, the element (4.100) belongs to
$\mathop{\mathrm{\bf hom}}\nolimits(W,V)^{\otimes 2}\otimes
S+R^{\prime}\otimes W^{\otimes 2}$.

Now we need to prove that for any $\widetilde{\mathcal{A}}\in{\mathbf{QA}}$
and
$f\in\mathop{\mathrm{Hom}}\nolimits({\mathcal{A}},\widetilde{\mathcal{A}}\circ{\mathcal{B}})$
there exists a unique
$h\in\mathop{\mathrm{Hom}}\nolimits\big{(}\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{A}}),\widetilde{\mathcal{A}}\big{)}$
making the diagram

(4.105)

commutative. For purposes of the next paragraph we do it for a slightly more
general case $\widetilde{\mathcal{A}}\in{\mathbf{QA}}_{\mathrm{s}c}$. Let
$\widetilde{\mathfrak{R}}=\widetilde{\mathcal{A}}_{0}$, then
$\widetilde{\mathcal{A}}=\widetilde{\mathfrak{R}}\otimes
T\widetilde{V}/(\widetilde{R})$ for some vector space
$\widetilde{V}\in{\mathbf{Vect}}$ and subspace
$\widetilde{R}\subset\widetilde{V}\otimes\widetilde{V}$. We have
$\widetilde{\mathcal{A}}_{1}=\widetilde{\mathfrak{R}}\otimes\widetilde{V}$ and
$\widetilde{\mathcal{A}}=T_{\widetilde{\mathfrak{R}}}{\mathcal{A}}_{1}/(\widetilde{\mathfrak{R}}\otimes\widetilde{R})$.
Note that any graded homomorphism
$g\colon{\mathcal{A}}\to\widetilde{\mathcal{A}}$ is uniquely determined by its
first order component $g_{1}\colon
V\to\widetilde{\mathfrak{R}}\otimes\widetilde{V}$ and, conversely, this
component gives the whole graded homomorphism
$g\colon{\mathcal{A}}\to\widetilde{\mathcal{A}}$ if it is linear and satisfies
$(g_{1}\mathbin{\dot{\otimes}}g_{1})R\subset\widetilde{\mathfrak{R}}\otimes\widetilde{R}$,
where $g_{1}\mathbin{\dot{\otimes}}g_{1}\colon V\otimes
V\to{\mathfrak{R}}\otimes\widetilde{V}\otimes\widetilde{V}$ is defined as in
p. 4.4, that is
$g_{1}\mathbin{\dot{\otimes}}g_{1}=(\mu_{\widetilde{\mathfrak{R}}}\otimes\mathop{\mathrm{id}}\nolimits_{\widetilde{V}}\otimes\mathop{\mathrm{id}}\nolimits_{\widetilde{V}})\cdot\sigma^{(23)}\cdot(g_{1}\otimes
g_{1})$. By using the natural isomorphism
$({\mathfrak{R}}\otimes{\mathcal{A}})\circ{\mathcal{B}}={\mathfrak{R}}\otimes({\mathcal{A}}\circ{\mathcal{B}})$
we obtain
$\widetilde{\mathcal{A}}\circ{\mathcal{B}}=\widetilde{\mathfrak{R}}\otimes
T(\widetilde{V}\otimes W)/(R_{\mathrm{w}})$, where
$R_{\mathrm{w}}=\sigma^{(23)}(\widetilde{V}^{\otimes 2}\otimes
S+\widetilde{R}\otimes W^{\otimes 2})$. In particular,
$(\widetilde{\mathcal{A}}\circ{\mathcal{B}})_{1}=\widetilde{\mathfrak{R}}\otimes\widetilde{V}\otimes
W=\mathop{\mathrm{\bf
hom}}\nolimits(W^{*},\widetilde{\mathfrak{R}}\otimes\widetilde{V})$.

Due to Prop. 3.2 there exists a unique linear map
$h_{1}\colon\mathop{\mathrm{\bf
hom}}\nolimits(W,V)\to\widetilde{\mathfrak{R}}\otimes\widetilde{V}$ such that
the diagram

(4.110)

commute, it has the form $h_{1}(w_{i}\otimes v)=f_{1}(v)(w_{i})$. This implies
the uniqueness of $h$. Let us prove its existence. Let $U\in R^{\prime}$, it
has the form $U=\sum\limits_{s,i,j}U_{s}^{ij}w_{i}\otimes x_{s}\otimes
w_{j}\otimes y_{s}$ for some $U_{s}^{ij}\in{\mathbb{K}}$ and $x_{s},y_{s}\in
V$ such that $x_{s}\otimes y_{s}\in R$. Since $U(S)=0$ we have
$\sum\limits_{s,i,j}U_{s}^{ij}B_{ij}^{kl}x_{s}\otimes y_{s}=0$, which implies
$U=\sum\limits_{s,i,j,k,l}U_{s}^{ij}(1-B)_{ij}^{kl}w_{k}\otimes x_{s}\otimes
w_{l}\otimes y_{s}$. Hence

$\displaystyle(h_{1}\mathbin{\dot{\otimes}}h_{1})U=\sum_{s,i,j,k,l}U_{s}^{ij}(1-B)_{ij}^{kl}(f_{1}\mathbin{\dot{\otimes}}f_{1})(x_{s}\otimes
y_{s})(w_{k}\otimes w_{l}),$ (4.111)

where $(f_{1}\mathbin{\dot{\otimes}}f_{1})(x_{s}\otimes y_{s})$ belongs to
$\widetilde{\mathfrak{R}}\otimes\widetilde{V}\otimes
W\otimes\widetilde{V}\otimes W=\mathop{\mathrm{\bf hom}}\nolimits(W^{*}\otimes
W^{*},\widetilde{\mathfrak{R}}\otimes\widetilde{V}\otimes\widetilde{V})$.
Since $(f_{1}\mathbin{\dot{\otimes}}f_{1})R\subset R_{\mathrm{w}}$, we obtain

$\displaystyle(f_{1}\otimes f_{1})(x_{s}\otimes y_{s})\in(f_{1}\otimes
f_{1})R\subset\widetilde{\mathfrak{R}}\otimes(\mathop{\mathrm{id}}\nolimits_{V}\otimes\sigma_{V,W}\otimes\mathop{\mathrm{id}}\nolimits_{W})(V\otimes
V\otimes S+\widetilde{R}\otimes W\otimes W).$

The formula $(1-B^{*})(\mathop{\mathrm{Im}}\nolimits B^{*})=0$ implies
$\sum_{k,l}(1-B)_{ij}^{kl}(w_{k}\otimes w_{l})(S)=0$, so we derive
$(h_{1}\otimes h_{1})U\in\widetilde{\mathfrak{R}}\otimes\widetilde{R}$.

We proved that $\eta$ is a universal arrow from ${\mathcal{B}}$ to the functor
$-\circ{\mathcal{B}}\colon{\mathbf{QA}}\to{\mathbf{QA}}$, hence this functor
has a left adjoint $\mathop{\mathrm{\bf cohom}}\nolimits({\mathcal{B}},-)$,
which acts on an object ${\mathcal{A}}$ as defined in the proposition (see
[MacLane, § 4.1, Th. 2 (ii)]). ∎

######  Remark 4.6.

The proof of Prop. 4.5 can be directly generalised to the case of arbitrary
connected affinely generated graded algebras
${\mathcal{A}},\widetilde{\mathcal{A}},{\mathcal{B}}$ such that
${\mathcal{B}}$ is finely generated: $\dim{\mathcal{B}}_{1}<\infty$.

4.4.6. Semi-connected case. The generalised internal cohom found in the
previous paragraph can be extended to the case
${\mathbf{C}}=({\mathbf{QA}}_{\mathrm{s}c},\circ)$,
${\mathbf{P}}={\mathbf{FQA}}$. Remind that any
${\mathcal{D}}\in{\mathbf{QA}}_{\mathrm{s}c}$ has the form
${\mathcal{D}}={\mathfrak{R}}\otimes{\mathcal{A}}$ for
${\mathfrak{R}}={\mathcal{D}}_{0}\in{\mathbf{Alg}}$ and
${\mathcal{A}}\in{\mathbf{QA}}$.

######  Proposition 4.7.

For any ${\mathcal{B}}\in{\mathbf{FQA}}$ the functor
$-\circ{\mathcal{B}}\colon{\mathbf{QA}}_{\mathrm{s}c}\to{\mathbf{QA}}_{\mathrm{s}c}$
has a left adjoint $\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},-)\colon{\mathbf{QA}}_{\mathrm{s}c}\to{\mathbf{QA}}_{\mathrm{s}c}$.
Let ${\mathcal{D}}={\mathfrak{R}}\otimes{\mathcal{A}}$ for some
${\mathfrak{R}}\in{\mathbf{Alg}}$ and ${\mathcal{A}}\in{\mathbf{QA}}$. Then
$\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{D}})={\mathfrak{R}}\otimes\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{A}})$, where $\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{A}})\in{\mathbf{QA}}$ is defined as
in Prop. 4.5.

Proof. Let $\widetilde{\mathcal{A}}\in{\mathbf{QA}}_{\mathrm{s}c}$. For an
arbitrary graded homomorphism
$\widehat{f}\colon{\mathfrak{R}}\otimes{\mathcal{A}}\to\widetilde{\mathcal{A}}$
consider its zero component
$\alpha=\widehat{f}_{0}\in\mathop{\mathrm{Hom}}\nolimits_{{\mathbf{Alg}}}({\mathfrak{R}},\widetilde{A}_{0})$
and its restriction
$f\in\mathop{\mathrm{Hom}}\nolimits_{{\mathbf{GrAlg}}}({\mathcal{A}},\widetilde{\mathcal{A}})$
to the subalgebra ${\mathcal{A}}\subset{\mathfrak{R}}\otimes{\mathcal{A}}$.
They satisfy the commutativity condition $\alpha(r)f(a_{1})=f(a_{1})\alpha(r)$
$\;\forall\,r\in{\mathfrak{R}},\,a_{1}\in{\mathcal{A}}_{1}$. Any pair of such
morphisms subjected to the commutativity condition uniquely defines a morphism
$\widehat{f}=(\alpha,f)\in\mathop{\mathrm{Hom}}\nolimits_{{\mathbf{GrAlg}}}({\mathfrak{R}}\otimes{\mathcal{A}},\widetilde{\mathcal{A}})$
(see [S, § 2.5], cf. p. 4.4).

Let $\widehat{f}=(\alpha,f)\colon{\mathcal{D}}\to\widetilde{\mathcal{A}}$ be a
graded homomorphism to an algebra
$\widetilde{\mathcal{A}}\in{\mathbf{QA}}_{\mathrm{s}c}$, where
$\alpha\colon{\mathfrak{R}}\to\widetilde{\mathcal{A}}_{0}$ and
$f\colon{\mathcal{A}}\to\widetilde{\mathcal{A}}$ are morphisms subjected to
the commutativity condition. As it was proved in p. 4.4 there exists a unique
graded homomorphism $h\colon\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{A}})\to\widetilde{\mathcal{A}}$
making the digram (4.105) commutative. By substituting
$f(v)=\sum_{i}h_{1}(w_{i}\otimes v)\otimes w^{i}$, $v\in V$, to the
commutativity condition $\alpha(r)f(v)=f(v)\alpha(r)$ we obtain the
commutativity condition for the pair $(\alpha,h)$, so it gives a graded
homomorphism
$\widehat{h}=(\alpha,h)\colon{\mathfrak{R}}\otimes\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{A}})\to\widetilde{\mathcal{A}}$ and
we obtain the commutative diagram

(4.116)

The uniqueness of $h$ implies the uniqueness of $\widehat{h}$. ∎

4.4.7. Internal cohom on the morphisms. Prop. 4.7 implies the existence of the
generalised internal cohom-functor $\mathop{\mathrm{\bf
cohom}}\nolimits\colon{\mathbf{FQA}}^{\mathrm{op}}\times{\mathbf{QA}}_{\mathrm{s}c}\to{\mathbf{QA}}_{\mathrm{s}c}$
and gives its values on objects. Let us calculate this functor on morphisms.
Let ${\mathcal{B}},{\mathcal{B}}^{\prime}\in{\mathbf{FQA}}$,
${\mathfrak{R}},{\mathfrak{R}}^{\prime}\in{\mathbf{Alg}}$ and
${\mathcal{A}},{\mathcal{A}}^{\prime}\in{\mathbf{QA}}$. Set
$W={\mathcal{B}}_{1}$, $W^{\prime}={\mathcal{B}}^{\prime}_{1}$,
$V={\mathcal{A}}_{1}$, $V^{\prime}={\mathcal{A}}^{\prime}_{1}$. Consider
arbitrary morphisms $f\colon{\mathcal{B}}^{\prime}\to{\mathcal{B}}$ and
$g\colon{\mathfrak{R}}\otimes{\mathcal{A}}\to{\mathfrak{R}}^{\prime}\otimes{\mathcal{A}}^{\prime}$.
They are uniquely determined by the components $f_{1}\colon W^{\prime}\to W$,
$\alpha=g_{0}\colon{\mathfrak{R}}\to{\mathfrak{R}}^{\prime}$ and
$g_{1}=(\alpha,t)\colon{\mathfrak{R}}\otimes
V\to{\mathfrak{R}}^{\prime}\otimes V^{\prime}$, where $t\colon
V\to{\mathfrak{R}}^{\prime}\otimes V^{\prime}$. Then the morphism
$\mathop{\mathrm{\bf
cohom}}\nolimits(f,g)\colon{\mathfrak{R}}\otimes\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}},{\mathcal{A}})\to{\mathfrak{R}}^{\prime}\otimes\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}}^{\prime},{\mathcal{A}}^{\prime})$ is uniquely
determined by the components

$\displaystyle\mathop{\mathrm{\bf
cohom}}\nolimits(f,g)_{0}\colon{\mathfrak{R}}\to{\mathfrak{R}}^{\prime},$
(4.117) $\displaystyle\mathop{\mathrm{\bf
cohom}}\nolimits(f,g)_{1}\colon{\mathfrak{R}}\otimes\mathop{\mathrm{\bf
hom}}\nolimits(W,V)\to{\mathfrak{R}}^{\prime}\otimes\mathop{\mathrm{\bf
hom}}\nolimits(W^{\prime},V^{\prime}).$ (4.118)

######  Proposition 4.8.

The components (4.117), (4.118) equal

$\displaystyle\mathop{\mathrm{\bf cohom}}\nolimits(f,g)_{0}=\alpha,$
$\displaystyle\mathop{\mathrm{\bf
cohom}}\nolimits(f,g)_{1}=(\alpha,t_{*}f_{1}^{*}),$ (4.119)

where $g_{0}=\alpha$, $g_{1}=(\alpha,t)$ and the operator
$t_{*}f_{1}^{*}\colon\mathop{\mathrm{\bf
hom}}\nolimits(W,V)\to{\mathfrak{R}}^{\prime}\otimes\mathop{\mathrm{\bf
hom}}\nolimits(W^{\prime},V^{\prime})$ is the composition

$\displaystyle\mathop{\mathrm{\bf
hom}}\nolimits(W,V)\xrightarrow{f_{1}^{*}}\mathop{\mathrm{\bf
hom}}\nolimits(W^{\prime},V)\xrightarrow{t_{*}}\mathop{\mathrm{\bf
hom}}\nolimits(W^{\prime},{\mathfrak{R}}^{\prime}\otimes
V^{\prime})={\mathfrak{R}}^{\prime}\otimes\mathop{\mathrm{\bf
hom}}\nolimits(W^{\prime},V^{\prime}).$ (4.120)

Proof. Due to the formulae (3.16) and (3.50) we obtain

$\displaystyle\big{(}\mathop{\mathrm{\bf
cohom}}\nolimits(f,g)\circ\mathop{\mathrm{id}}\nolimits\big{)}\cdot\mathop{\mathrm{coev}}\nolimits=\vartheta\big{(}\mathop{\mathrm{\bf
cohom}}\nolimits(f,g)\big{)}=\big{(}\mathop{\mathrm{id}}\nolimits\circ
f\big{)}\cdot\mathop{\mathrm{coev}}\nolimits\cdot g,$ (4.121)

where $\mathop{\mathrm{coev}}\nolimits$ coincides with the horizontal arrow in
(4.116). This gives us the commuting diagram

(4.126)

By taking zero and first order components in this diagram we obtain (4.119). ∎

Consider the case ${\mathcal{A}},{\mathcal{A}}^{\prime}\in{\mathbf{FQA}}$. Up
to isomorphisms we have ${\mathcal{A}}={\mathfrak{X}}_{A}({\mathbb{K}})$,
${\mathcal{A}}^{\prime}={\mathfrak{X}}_{A^{\prime}}({\mathbb{K}})$,
${\mathcal{B}}={\mathfrak{X}}_{B}({\mathbb{K}})$,
${\mathcal{B}}={\mathfrak{X}}_{B^{\prime}}({\mathbb{K}})$ for some matrix
idempotents $A,A^{\prime},B,B^{\prime}$. Then the morphisms
$f\colon{\mathcal{B}}^{\prime}\to{\mathcal{B}}$ and
$g\colon{\mathfrak{R}}\otimes{\mathcal{A}}\to{\mathfrak{R}}^{\prime}\otimes{\mathcal{A}}^{\prime}$
have the form

$\displaystyle
f=f_{K}\colon{\mathfrak{X}}_{B^{\prime}}({\mathbb{K}})\to{\mathfrak{X}}_{B}({\mathbb{K}})$
and $\displaystyle
g=(\alpha,f_{M})\colon{\mathfrak{X}}_{A}({\mathfrak{R}})\to{\mathfrak{X}}_{A^{\prime}}({\mathfrak{R}}^{\prime})$
(4.127)

respectively for a $(B^{\prime},B)$-Manin matrix $K$ over ${\mathbb{K}}$, an
$(A,A^{\prime})$-Manin matrix $M$ over ${\mathfrak{R}}^{\prime}$ and a
homomorphism $\alpha\colon{\mathfrak{R}}\to{\mathfrak{R}}^{\prime}$ such that
$\alpha(r)M^{i}_{j}=M^{i}_{j}\alpha(r)$ $\;\forall\,r\in{\mathfrak{R}}$. Let
${\mathcal{M}}$ and ${\mathcal{N}}$ be the universal $(A,B)$\- and
$(A^{\prime},B^{\prime})$-Manin matrices respectively. Their entries are
generators of $\mathop{\mathrm{\bf
cohom}}\nolimits\big{(}{\mathfrak{X}}_{B}({\mathbb{K}}),{\mathfrak{X}}_{A}({\mathbb{K}})\big{)}$
and $\mathop{\mathrm{\bf
cohom}}\nolimits\big{(}{\mathfrak{X}}_{B^{\prime}}({\mathbb{K}}),{\mathfrak{X}}_{A^{\prime}}({\mathbb{K}})\big{)}$.
The graded homomorphism

$\mathop{\mathrm{\bf
cohom}}\nolimits(f,g)\colon{\mathfrak{R}}\otimes\mathop{\mathrm{\bf
cohom}}\nolimits\big{(}{\mathfrak{X}}_{B}({\mathbb{K}}),{\mathfrak{X}}_{A}({\mathbb{K}})\big{)}\to{\mathfrak{R}}^{\prime}\otimes\mathop{\mathrm{\bf
cohom}}\nolimits\big{(}{\mathfrak{X}}_{B^{\prime}}({\mathbb{K}}),{\mathfrak{X}}_{A^{\prime}}({\mathbb{K}})\big{)}$

acts on the generators as

$\displaystyle\mathop{\mathrm{\bf cohom}}\nolimits(f,g)\colon r\otimes
1\mapsto\alpha(r)\otimes 1,\qquad r\in{\mathfrak{R}},$ (4.128)
$\displaystyle\mathop{\mathrm{\bf cohom}}\nolimits(f,g)\colon
1\otimes{\mathcal{M}}^{i}_{j}\mapsto\sum_{a,b}M^{i}_{a}\otimes{\mathcal{N}}^{a}_{b}K^{b}_{j}.$
(4.129)

Note that the matrix $M{\mathcal{N}}K$ appeared in (4.129) is a $(A,B)$-Manin
matrix, which follows directly from [S, Prop. 2.26].

##  5 Quantum Representation Theory

By taking the category
${\mathbf{C}}={\mathbf{QA}}_{\mathrm{s}c}^{\mathrm{op}}$ with the Manin
product ‘$\circ$’ and parameter category
${\mathbf{P}}={\mathbf{FQA}}^{\mathrm{op}}$ we obtain a class of
representations on quantum linear spaces. Quantum Representation Theory
presented here investigates representations for this case and, more generally,
for ${\mathbf{C}}=({\mathbf{GrAlg}}^{\mathrm{op}},\circ)$. It can be regarded
as a generalisation of Representation Theory on the usual vector spaces
(classical representation theory). The latter one can be embedded into Quantum
Representation Theory in two ways: by the functor $S^{*}$ or $T^{*}$. The
binary and duality operations with representations can be generalised to the
quantum case.

###  5.1 Quantum representations

First we describe objects which we are going to ‘represent’. In the classical
representation theory these were algebraic monoids and algebras. In the
quantum case we need to consider monoids in
$({\mathbf{QA}}_{\mathrm{s}c}^{\mathrm{op}},\circ)$ or, more generally, in
$({\mathbf{GrAlg}}^{\mathrm{op}},\circ)$. We call these monoids quantum
algebras.

5.1.1. Quantum monoids. A quantum analogue of algebraic monoids is a comonoid
in the monoidal category $({\mathbf{Alg}},\otimes)$. This is an algebra
${\mathfrak{R}}\in{\mathbf{Alg}}$ with comultiplication
$\Delta_{\mathfrak{R}}\colon{\mathfrak{R}}\to{\mathfrak{R}}\otimes{\mathfrak{R}}$
and counit $\varepsilon_{\mathfrak{R}}\colon{\mathfrak{R}}\to{\mathbb{K}}$.
These are algebra homomorphisms satisfying the equations
$(\varepsilon_{\mathfrak{R}}\otimes\mathop{\mathrm{id}}\nolimits_{\mathfrak{R}})\cdot\Delta_{\mathfrak{R}}=\mathop{\mathrm{id}}\nolimits_{\mathfrak{R}}=(\mathop{\mathrm{id}}\nolimits_{\mathfrak{R}}\otimes\varepsilon_{\mathfrak{R}})\cdot\Delta_{\mathfrak{R}}$
and the coassociativity condition
$(\Delta_{\mathfrak{R}}\otimes\mathop{\mathrm{id}}\nolimits_{\mathfrak{R}})\cdot\Delta_{\mathfrak{R}}=(\mathop{\mathrm{id}}\nolimits_{\mathfrak{R}}\otimes\Delta_{\mathfrak{R}})\cdot\Delta_{\mathfrak{R}}$.
Such comonoids
$({\mathfrak{R}},\Delta_{\mathfrak{R}},\varepsilon_{\mathfrak{R}})$ are
exactly the bialgebras:
$\mathop{\mathbf{Comon}}\nolimits({\mathbf{Alg}},\otimes)=\mathop{\mathbf{Bimon}}\nolimits({\mathbf{Vect}},\otimes)$.

To define representations of a quantum monoid we need to consider it as a
quantum algebra, so we should translate it by means of the (not full)
embedding ${\mathbf{Alg}}\hookrightarrow{\mathbf{QA}}_{\mathrm{s}c}$,
${\mathfrak{R}}\mapsto{\mathfrak{R}}\otimes{\mathbb{K}}[u]$. This is a strong
monoidal functor
$({\mathbf{Alg}},\otimes)\hookrightarrow({\mathbf{QA}}_{\mathrm{s}c},\circ)$,
it translates the bialgebra
$({\mathfrak{R}},\Delta_{\mathfrak{R}},\varepsilon_{\mathfrak{R}})$ to the
comonoid

$\displaystyle{\mathbb{O}}_{\mathfrak{R}}=({\mathfrak{R}}\otimes{\mathbb{K}}[u],\Delta,\varepsilon)\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{QA}}_{\mathrm{s}c},\circ),$
$\displaystyle\Delta=\Delta_{\mathfrak{R}}\otimes\mathop{\mathrm{id}}\nolimits_{{\mathbb{K}}[u]},$
$\displaystyle\varepsilon=\varepsilon_{\mathfrak{R}}\otimes\mathop{\mathrm{id}}\nolimits_{{\mathbb{K}}[u]}.$
(5.1)

Component-wise we have
$({\mathfrak{R}}\otimes{\mathbb{K}}[u])_{k}={\mathfrak{R}}$,
$\Delta_{k}=\Delta_{\mathfrak{R}}$ and
$\varepsilon_{k}=\varepsilon_{\mathfrak{R}}$. The comonoid
${\mathbb{O}}_{\mathfrak{R}}$ is a quantum monoid considered as a comonoid in
$({\mathbf{QA}}_{\mathrm{s}c},\circ)$. Note, however, that not all the
comonoid structure on the semi-connected graded algebra
${\mathfrak{R}}\otimes{\mathbb{K}}[u]$ has this form (because the embedding
functor is not full).

5.1.2. Comonoids in $({\mathbf{GrAlg}},\circ)$. A linear version of an
algebraic monoid is an algebra. Its quantum analogue is a comonoid in
$({\mathbf{QA}},\circ)$. A quantum semi-linear monoid is a comonoid in
$({\mathbf{QA}}_{\mathrm{s}c},\circ)$. For wider generality we describe
comonoids in $({\mathbf{GrAlg}},\circ)$, which we call quantum algebras. Such
comonoid is a graded algebra
${\mathcal{A}}=\bigoplus\limits_{k\in{\mathbb{N}}_{0}}{\mathcal{A}}_{k}$ with
graded homomorphisms
$\Delta\colon{\mathcal{A}}\to{\mathcal{A}}\circ{\mathcal{A}}$ and
$\varepsilon\colon{\mathcal{A}}\to{\mathbb{K}}[u]$, their components are
linear maps
$\Delta_{k}\colon{\mathcal{A}}_{k}\to{\mathcal{A}}_{k}\otimes{\mathcal{A}}_{k}$
and $\varepsilon_{k}\colon{\mathcal{A}}_{k}\to{\mathbb{K}}$ satisfying the
conditions
$({\mathcal{A}}_{k},\Delta_{k},\varepsilon_{k})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{Vect}},\otimes)$
$\;\forall\,k\in{\mathbb{N}}_{0}$ and
$\Delta_{k+l}(a_{l}b_{l})=\Delta_{k}(a_{k})\Delta_{l}(b_{l})$
$\;\forall\,k,l\in{\mathbb{N}}_{0}$, $a_{k}\in{\mathcal{A}}_{k}$,
$b_{l}\in{\mathcal{A}}_{l}$.

By composing $\Delta$ and $\varepsilon$ with the embedding
${\mathcal{A}}\circ{\mathcal{A}}\hookrightarrow{\mathcal{A}}\otimes{\mathcal{A}}$
and evaluation ${\mathbb{K}}[u]\to{\mathbb{K}}$ at $u=1$ we get a
comultiplication
$\Delta_{\mathcal{A}}\colon{\mathcal{A}}\to{\mathcal{A}}\otimes{\mathcal{A}}$
and a counit $\varepsilon_{\mathcal{A}}\colon{\mathcal{A}}\to{\mathbb{K}}$
respectively (they are not graded). The conditions on the graded homomorphisms
$\Delta$ and $\varepsilon$ imply that
$({\mathcal{A}},\Delta_{\mathcal{A}},\varepsilon_{\mathcal{A}})$ is a
bialgebra. Conversely, let a graded algebra ${\mathcal{A}}\in{\mathbf{GrAlg}}$
has a bialgebra structure:
$({\mathcal{A}},\Delta_{\mathcal{A}},\varepsilon_{\mathcal{A}})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{Alg}},\otimes)$,
then it gives a comonoid $({\mathcal{A}},\Delta,\varepsilon)$ in
$({\mathbf{GrAlg}},\circ)$ iff

$\displaystyle\Delta_{\mathcal{A}}({\mathcal{A}}_{k})\subset{\mathcal{A}}_{k}\otimes{\mathcal{A}}_{k}$
$\displaystyle k\in{\mathbb{N}}_{0}.$ (5.2)

5.1.3. Comonoids in $({\mathbf{QA}}_{\mathrm{s}c},\circ)$ and
$({\mathbf{QA}},\circ)$. If ${\mathcal{A}}\in{\mathbf{QA}}_{\mathrm{s}c}$,
then it is enough to require (5.2) for $k=0$ and $k=1$:

$\displaystyle\Delta_{\mathcal{A}}({\mathcal{A}}_{0})\subset{\mathcal{A}}_{0}\otimes{\mathcal{A}}_{0},$
$\displaystyle\Delta_{\mathcal{A}}({\mathcal{A}}_{1})\subset{\mathcal{A}}_{1}\otimes{\mathcal{A}}_{1}.$
(5.3)

In this case $\Delta$ and $\varepsilon$ are completely determined by their
zero and first order components $\Delta_{0}=\alpha$, $\Delta_{1}=(\alpha,t)$,
$\varepsilon_{0}=\beta$, $\varepsilon_{1}=\beta\otimes\varepsilon_{V}$ (see p.
4.4 for details).

In the connected case ${\mathcal{A}}\in{\mathbf{QA}}$ we have
${\mathcal{A}}_{0}={\mathbb{K}}$, so it is enough to impose the condition
(5.2) for $k=1$, the graded homomorphisms $\Delta$ and $\varepsilon$ are
uniquely determined by their first order components
$\Delta_{1}=\Delta_{V}\colon V\to V\otimes V$ and
$\varepsilon_{1}=\varepsilon_{V}\colon V\to{\mathbb{K}}$, where
$V={\mathcal{A}}_{1}\in{\mathbf{Vect}}$. Then the commutativity of the
diagrams (4.81) reduces to the requirement that these components gives a
structure of coalgebra
$(V,\Delta_{V},\varepsilon_{V})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{Vect}},\otimes)$.
The comonoid structure on a connected quadratic algebra ${\mathcal{A}}=TV/(R)$
is given by a coalgebra
$(V,\Delta_{V},\varepsilon_{V})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{Vect}},\otimes)$
such that $(\Delta_{V}\otimes\Delta_{V})R\subset\sigma^{(23)}(R\otimes
V^{\otimes 2}+V^{\otimes 2}\otimes R)$ and
$(\varepsilon_{V}\otimes\varepsilon_{V})R=0$.

5.1.4. Quantum representations. Let
${\mathbb{O}}=({\mathcal{A}},\Delta,\varepsilon)$ be a comonoid in
$({\mathbf{QA}}_{\mathrm{s}c},\circ)$, that is a semi-connected quadratic
algebra ${\mathcal{A}}$ with a bialgebra structure
$(\Delta_{\mathcal{A}},\varepsilon_{\mathcal{A}})$ satisfying (5.3). Due to
the propositions 4.5, 4.7 we have the category
$\mathop{\mathbf{Corep}}\nolimits_{{\mathbf{FQA}}}({\mathbb{O}})$, which
consists of corepresentations of ${\mathbb{O}}$ on finitely generated
quadratic algebras ${\mathcal{B}}\in{\mathbf{FQA}}$. These corepresentations
can be regarded as representations of the corresponding monoid in
$({\mathbf{QA}}_{\mathrm{s}c}^{\mathrm{op}},\circ)$ on finite-dimensional
quantum linear spaces. Let us define a slightly more general notion.

######  Definition 5.1.

A quantum representation is a corepresentation
$\omega\colon\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})\to{\mathbb{O}}$ of a comonoid
${\mathbb{O}}=({\mathcal{A}},\Delta,\varepsilon)\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{GrAlg}},\circ)$
on a quadratic algebra ${\mathcal{B}}\in{\mathbf{FQA}}$. A morphism between
corepresentations $\omega\colon\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})\to{\mathbb{O}}$ and
$\omega^{\prime}\colon\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}}^{\prime})\to{\mathbb{O}}$ is a morphism
$f\colon{\mathcal{B}}\to{\mathcal{B}}^{\prime}$ in ${\mathbf{FQA}}$ such that
the diagram

(5.8)

commute. The objects $({\mathcal{B}},\omega)$, where
${\mathcal{B}}\in{\mathbf{FQA}}$ and $\omega\colon\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})\to{\mathbb{O}}$ is a corepresentation, form a
category with such defined morphisms. We denote it by
$\mathop{\mathbf{Corep}}\nolimits_{{\mathbf{FQA}}}({\mathbb{O}})$.

Note that in the case
${\mathbb{O}}\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{QA}}_{\mathrm{s}c},\circ)$
the category $\mathop{\mathbf{Corep}}\nolimits_{\mathbf{FQA}}({\mathbb{O}})$
coincides with one defined in p. 3.6, so in this case Theorem 3.8 implies that
it is a full subcategory of $\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{O}})$.

More correctly, we should say that the category of quantum representations is
rather
$\mathop{\mathbf{Rep}}\nolimits_{{\mathbf{FQA}}^{\mathrm{op}}}({\mathbb{M}})$,
where
${\mathbb{M}}\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{GrAlg}}^{\mathrm{op}},\circ)$
is the monoid corresponding to the comonoid ${\mathbb{O}}$. This is the
category opposite to
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{FQA}}({\mathbb{O}})$. The morphisms
of quantum representations are reversed morphisms of corepresentations.
However, it is more convenient for us to work in terms of category
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{FQA}}({\mathbb{O}})$.

5.1.5. Multiplicative Manin matrices. An $m\times m$ matrix $M=(M^{i}_{j})$
with entries in a bialgebra
${\mathfrak{R}}=({\mathfrak{R}},\Delta_{\mathfrak{R}},\varepsilon_{\mathfrak{R}})$
(or, more generally in a coalgebra) is called multiplicative if

$\displaystyle\Delta_{\mathfrak{R}}(M^{i}_{j})=\sum_{k=1}^{m}M^{i}_{k}\otimes
M^{k}_{j},$
$\displaystyle\varepsilon_{\mathfrak{R}}(M^{i}_{j})=\delta^{i}_{j},$
$\displaystyle i,j=1,\ldots,m$ (5.9)

(see [Man88, § 2.6], [Man91, § 4.1.1]). For example, the formulae (4.96) means
exactly that the universal $B$-Manin matrix
${\mathcal{M}}=({\mathcal{M}}^{i}_{j})$ is a multiplicative $m\times m$ matrix
over the algebra $\mathop{\mathrm{\bf
coend}}\nolimits\big{(}{\mathfrak{X}}_{B}({\mathbb{K}})\big{)}$. Recall that
${\mathcal{M}}^{i}_{j}$ are generators of this algebra.

Let us establish a relationship between the notions of multiplicative Manin
matrix and of quantum representation. Consider a comonoid
${\mathbb{O}}=({\mathcal{A}},\Delta,\varepsilon)$ in
$({\mathbf{GrAlg}},\circ)$.

######  Theorem 5.2.

Let
$B\in\mathop{\mathrm{End}}\nolimits({\mathbb{K}}^{m}\otimes{\mathbb{K}}^{m})$
be a matrix idempotent. Any corepresentation $\omega\colon\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})\to{\mathbb{O}}$ of the comonoid ${\mathbb{O}}$
on the quadratic algebra ${\mathcal{B}}={\mathfrak{X}}_{B}({\mathbb{K}})$ acts
on the generators by the formula

$\displaystyle\omega({\mathcal{M}}^{i}_{j})=M^{i}_{j},$ (5.10)

where $M=(M^{i}_{j})$ is a multiplicative first order $B$-Manin matrix over
${\mathcal{A}}$, that is an $m\times m$ matrix with entries
$M^{i}_{j}\in{\mathcal{A}}_{1}$ satisfying (5.9) and

$\displaystyle BM^{(1)}M^{(2)}(1-B)=0.$ (5.11)

Conversely, any such matrix $M$ defines a corepresentation of the comonoid
${\mathbb{O}}$ on ${\mathcal{B}}$ by the formula (5.10).

Proof. Any graded homomorphism $\omega\colon\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})\to{\mathcal{A}}$ is uniquely determined by its
values $M^{i}_{j}\in{\mathcal{A}}_{1}$ on the generators
${\mathcal{M}}^{i}_{j}$. The values (5.10) give a graded homomorphism $\omega$
iff they belong to the component ${\mathcal{A}}_{1}$ and satisfy the same
commutation relations that ${\mathcal{M}}^{i}_{j}$ do, i.e.the relations
(5.11). The commutativity of the corresponding diagrams (2.67) is equivalent
to the condition (5.9). ∎

Warning. The same multiplicative first order matrix $M$ defines different
corepresentations (different objects of the category
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{FQA}}({\mathbb{O}})$) on different
quadratic algebras ${\mathcal{B}}={\mathfrak{X}}_{B}({\mathbb{K}})$ and
${\mathcal{B}}^{\prime}={\mathfrak{X}}_{B^{\prime}}({\mathbb{K}})$ if
${\mathcal{B}}_{1}={\mathcal{B}}^{\prime}_{1}$ and $M$ is simultaneously $B$\-
and $B^{\prime}$-Manin matrix. In other words, a matrix $M$ is not enough to
fix a corepresentation, one also needs to know the quadratic algebra
${\mathcal{B}}$ (and also the comonoid ${\mathbb{O}}$, of cause).

By a quantum representation of a quantum monoid we understand a
corepresentation of the comonoid, obtained by embedding of the bialgebra
$({\mathfrak{R}},\Delta_{\mathfrak{R}},\varepsilon_{\mathfrak{R}})$ via
${\mathbf{Alg}}\hookrightarrow{\mathbf{QA}}_{\mathrm{s}c}$. This is a
corepresentation of the comonoid
${\mathbb{O}}_{\mathfrak{R}}=({\mathfrak{R}}\otimes{\mathbb{K}}[u],\Delta_{\mathfrak{R}}\otimes\mathop{\mathrm{id}}\nolimits_{{\mathbb{K}}[u]},\varepsilon_{\mathfrak{R}}\otimes\mathop{\mathrm{id}}\nolimits_{{\mathbb{K}}[u]})$
defined by (5.1). Theorem 5.2 implies in this case a one-to-one correspondence
between corepresentations of ${\mathbb{O}}_{\mathfrak{R}}$ on
${\mathcal{B}}={\mathfrak{X}}_{B}({\mathbb{K}})$ and multiplicative $B$-Manin
matrices over ${\mathfrak{R}}$.

A trivial but important example of quantum representation is the identical
morphism $\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})\xrightarrow{\mathop{\mathrm{id}}\nolimits}\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})$. It is a corepresentation of the comonoid
${\mathbb{O}}=\mathop{\mathrm{\bf coend}}\nolimits({\mathcal{B}})$ on the
object ${\mathcal{B}}\in{\mathbf{FQA}}$. It is given by the universal
$B$-Manin matrix ${\mathcal{M}}$. The corresponding coaction coincides with
the coevaluation (4.94) for ${\mathcal{A}}={\mathcal{B}}$.

######  Remark 5.3.

In [RTF, Def. 20] the following generalisation of representation of a group
was introduced. Representation of a quantum group
${\mathfrak{R}}\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{Alg}},\otimes)$ on
a vector space $W\in{\mathbf{FVect}}$ was defined as a coalgebra homomorphism
$\rho\colon\mathop{\mathrm{\bf end}}\nolimits(W)^{*}\to{\mathfrak{R}}$. In the
basis $(w_{i})_{i=1}^{m}$ it is given by an arbitrary multiplicative first
order $m\times m$ matrix $M=(M^{i}_{j})$ over ${\mathfrak{R}}$. On one hand,
$\rho$ can be considered as a particular case of quantum representation: the
corresponding corepresentation $\omega\colon\mathop{\mathrm{\bf
coend}}\nolimits(TW^{*})=T\mathop{\mathrm{\bf
end}}\nolimits(W)^{*}\to{\mathbb{O}}_{\mathfrak{R}}$ is given by the matrix
$M$ or, equivalently, by the condition $\omega_{1}=\rho$. In that sense the
representations $\rho$ of the quantum group ${\mathfrak{R}}$ form the
subcategory
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{FTA}}({\mathbb{O}}_{\mathfrak{R}})$
in
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{FQA}}({\mathbb{O}}_{\mathfrak{R}})$,
where ${\mathbf{FTA}}\subset{\mathbf{FQA}}$ is the subcategory of finitely
generated tensor algebras (it is coreflective in the sense [MacLane, § 4.3]
with coreflection ${\mathbf{FQA}}\to{\mathbf{FTA}}$, ${\mathcal{B}}\mapsto
T{\mathcal{B}}_{1}$). On the other hand, any quantum representation of a
comonoid
${\mathbb{O}}\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{GrAlg}},\circ)$ on
${\mathcal{B}}\in{\mathbf{FQA}}$ is described by a multiplicative first order
matrix, which, in particular, gives corepresentation of the comonoid
${\mathbb{O}}$ on $T{\mathcal{B}}_{1}\in{\mathbf{FTA}}$. We get a faithful
functor
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{FQA}}({\mathbb{O}})\to\mathop{\mathbf{Corep}}\nolimits_{\mathbf{FTA}}({\mathbb{O}})$
(it is also a coreflection).

5.1.6. Morphisms of quantum representations. Consider two quantum
representations $\omega\colon\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})\to{\mathbb{O}}$ and
$\omega^{\prime}\colon\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}}^{\prime})\to{\mathbb{O}}$ of the same
${\mathbb{O}}=({\mathcal{A}},\Delta,\varepsilon)\in({\mathbf{GrAlg}},\circ)$
on quadratic algebras ${\mathcal{B}},{\mathcal{B}}^{\prime}\in{\mathbf{FQA}}$.
Suppose that these algebras are defined by matrix idempotents
$B\in\mathop{\mathrm{End}}\nolimits({\mathbb{K}}^{m}\otimes{\mathbb{K}}^{m})$,
$B^{\prime}\in\mathop{\mathrm{End}}\nolimits({\mathbb{K}}^{m^{\prime}}\otimes{\mathbb{K}}^{m^{\prime}})$,
i.e. ${\mathcal{B}}={\mathfrak{X}}_{B}({\mathbb{K}})$,
${\mathcal{B}}^{\prime}={\mathfrak{X}}_{B^{\prime}}({\mathbb{K}})$. Due to
Theorem 5.2 these corepresentations are given by multiplicative first order
$B$\- and $B^{\prime}$-Manin matrices $M$ and $M^{\prime}$ respectively:
$\omega({\mathcal{M}})=M$, $\omega({\mathcal{M}}^{\prime})=M^{\prime}$, where
${\mathcal{M}}$ and ${\mathcal{M}}^{\prime}$ are the universal $B$\- and
$B^{\prime}$-Manin matrices.

######  Proposition 5.4.

Any morphism
$f\colon({\mathcal{B}},\omega)\to({\mathcal{B}}^{\prime},\omega^{\prime})$ in
the category $\mathop{\mathbf{Corep}}\nolimits_{\mathbf{FQA}}({\mathbb{O}})$
has the form
$f=f_{K}\colon{\mathfrak{X}}_{B}({\mathbb{K}})\to{\mathfrak{X}}_{B^{\prime}}({\mathbb{K}})$
for an $m\times m^{\prime}$ matrix
$K\in\mathop{\mathrm{Hom}}\nolimits({\mathbb{K}}^{m^{\prime}},{\mathbb{K}}^{m})$
satisfying $BK^{(1)}K^{(2)}(1-B^{\prime})=0$ and

$\displaystyle KM^{\prime}=MK.$ (5.12)

This gives a bijection between the morphisms
$f\colon({\mathcal{B}},\omega)\to({\mathcal{B}}^{\prime},\omega^{\prime})$ and
$(B,B^{\prime})$-Manin matrices $K$ over ${\mathbb{K}}$ satisfying (5.12).

Proof. The graded homomorphisms
$f\colon{\mathfrak{X}}_{B}({\mathbb{K}})\to{\mathfrak{X}}_{B^{\prime}}({\mathbb{K}})$
is in one-to-one correspondence with the $(B,B^{\prime})$-Manin matrices over
${\mathbb{K}}$ (see p. 4.2). By taking the entries of the universal
$(B,B^{\prime})$-Manin matrix in the left upper corner of the diagram (5.8) we
obtain exactly the condition (5.12). Since these entries generate the algebra
$\mathop{\mathrm{\bf cohom}}\nolimits({\mathcal{B}}^{\prime},{\mathcal{B}})$
the commutativity of the diagram (5.8) is equivalent to (5.12). ∎

######  Corollary 5.5.

Let ${\mathcal{B}}_{1}={\mathcal{B}}^{\prime}_{1}$. Suppose that this
identification induces a morphism
$f\colon{\mathcal{B}}\to{\mathcal{B}}^{\prime}$ in ${\mathbf{FQA}}$, that is
$f_{1}=\mathop{\mathrm{id}}\nolimits_{{\mathcal{B}}_{1}}$ (this is not always
true). Then $f$ is a morphism
$({\mathcal{B}},\omega)\to({\mathcal{B}}^{\prime},\omega^{\prime})$ in
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{FQA}}({\mathbb{O}})$ iff the
corepresentations $\omega$ and $\omega^{\prime}$ are given by the same matrix
$M=M^{\prime}$.

Proof. Since $f_{1}=\mathop{\mathrm{id}}\nolimits_{{\mathcal{B}}_{1}}$ we have
$f=f_{K}$ for the identity matrix $K=1$, so the condition (5.12) reduces to
$M=M^{\prime}$ (the equation $B(1-B^{\prime})=0$ is equivalent to the
requirement that the matrix $K=1$ defines a morphism
$f_{K}\colon{\mathfrak{X}}_{B}({\mathbb{K}})\to{\mathfrak{X}}_{B^{\prime}}({\mathbb{K}})$
in the category ${\mathbf{FQA}}$). ∎

5.1.7. Coactions in $({\mathbf{GrAlg}},\circ)$. Let ${\mathbb{O}}$ be a
comonoid in $({\mathbf{GrAlg}},\circ)$ corresponding to a bialgebra
$({\mathcal{A}},\Delta_{\mathcal{A}},\varepsilon_{\mathcal{A}})$ in the sense
of p. 5.1. Consider a coaction
$\delta\colon{\mathcal{B}}\to{\mathcal{A}}\circ{\mathcal{B}}$ of
${\mathbb{O}}$ on ${\mathcal{B}}\in{\mathbf{GrAlg}}$. By composing it with the
embedding
${\mathcal{A}}\circ{\mathcal{B}}\hookrightarrow{\mathcal{A}}\otimes{\mathcal{B}}$
we obtain a coaction
$\delta_{\mathcal{A}}\colon{\mathcal{B}}\to{\mathcal{A}}\otimes{\mathcal{B}}$
of the bialgebra
$({\mathcal{A}},\Delta_{\mathcal{A}},\varepsilon_{\mathcal{A}})$ in the
monoidal category $({\mathbf{Alg}},\otimes)$. Conversely, a coaction
$\delta_{\mathcal{A}}\colon{\mathcal{B}}\to{\mathcal{A}}\otimes{\mathcal{B}}$
of the bialgebra
$({\mathcal{A}},\Delta_{\mathcal{A}},\varepsilon_{\mathcal{A}})$ gives a
coaction $\delta\colon{\mathcal{B}}\to{\mathcal{A}}\circ{\mathcal{B}}$ of
${\mathbb{O}}$ in $({\mathbf{GrAlg}},\circ)$ iff

$\displaystyle\delta_{\mathcal{A}}({\mathcal{B}}_{k})\subset{\mathcal{A}}_{k}\otimes{\mathcal{B}}_{k},$
$\displaystyle k\in{\mathbb{N}}_{0}.$ (5.13)

If ${\mathcal{B}}\in{\mathbf{QA}}$ or
${\mathcal{B}}\in{\mathbf{QA}}_{\mathrm{s}c}$ then it is enough to require
this condition for $k=1$ only or for $k=0,1$ only respectively.

######  Proposition 5.6.

Each coaction
$\delta\colon{\mathfrak{X}}_{B}({\mathbb{K}})\to{\mathcal{A}}\circ{\mathfrak{X}}_{B}({\mathbb{K}})$
has the form

$\displaystyle\delta(w^{i})=\sum_{j}M^{i}_{j}\otimes w^{j},$ (5.14)

where $M=(M^{i}_{j})$ is a multiplicative first order $B$-Manin matrix and
$(w^{j})$ is the standard basis of $({\mathbb{K}}^{m})^{*}$. Any such matrix
$M$ define a coaction of ${\mathbb{O}}$ on ${\mathfrak{X}}_{B}({\mathbb{K}})$.
If $\omega$ is a corepresentation of ${\mathbb{O}}$ on
${\mathfrak{X}}_{B}({\mathbb{K}})$ given by the matrix $M$, then the diagram

(5.19)

commute. In particular, if ${\mathcal{A}}\in{\mathbf{QA}}_{\mathrm{s}c}$, then
$\vartheta(\omega)=\delta$.

Proof. Any graded homomorphism
$\delta\colon{\mathfrak{X}}_{B}({\mathbb{K}})\to{\mathcal{A}}\circ{\mathfrak{X}}_{B}({\mathbb{K}})$
has the form (5.14) for a $B$-Manin matrix $M$ (see p. 4.2). By taking $w^{i}$
in the left upper corners of the diagrams (2.90) we obtain the conditions
(5.9). The commutativity of the diagram (5.19) is checked on the generators
$w^{i}$ in a straightforward way. ∎

We established the bijection between corepresentations
$\omega\colon\mathop{\mathrm{\bf
cohom}}\nolimits({\mathcal{B}})\to{\mathbb{O}}$ and coactions
$\delta\colon{\mathcal{B}}\to{\mathcal{A}}\circ{\mathcal{B}}$ for an arbitrary
comonoid
${\mathbb{O}}=({\mathcal{A}},\Delta,\varepsilon)\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{GrAlg}},\circ)$.
This generalises the case ${\mathbf{C}}={\mathbf{QA}}_{\mathrm{s}c}$ of the
bijection (3.51), so we denote it by the same letter:
$\vartheta\colon\omega\leftrightarrow\delta$.

######  Proposition 5.7.

Let $\delta$ and $\delta^{\prime}$ be coactions of ${\mathbb{O}}$ on
${\mathfrak{X}}_{B}({\mathbb{K}})$ and
${\mathfrak{X}}_{B^{\prime}}({\mathbb{K}})$. Let $M$ and $M^{\prime}$ be the
corresponding multiplicative first order $B$\- and $B^{\prime}$-Manin
matrices. Then morphisms
$f\colon\big{(}{\mathfrak{X}}_{B}({\mathbb{K}}),\delta\big{)}\to\big{(}{\mathfrak{X}}_{B^{\prime}}({\mathbb{K}}),\delta^{\prime}\big{)}$
in $\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{O}})$ are homomorphisms
$f=f_{K}$ for $(B,B^{\prime})$-Manin matrices $K$ over ${\mathbb{K}}$
satisfying (5.12). Hence
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{FQA}}({\mathbb{O}})$ is a
subcategory of $\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{O}})$.

Proof. By substituting $w^{i}$ to the left upper corner of the left diagram
(2.99) we obtain the condition (5.12). ∎

######  Remark 5.8.

Non-linear versions of the quantum representations are the comodule algebras
[Kass, Def. 3.7.1]. By means of the notion of coaction in
$({\mathbf{Alg}},\otimes)$ one can define a comodule algebra over a bialgebra
${\mathfrak{R}}=({\mathfrak{R}},\Delta_{\mathfrak{R}},\varepsilon_{\mathfrak{R}})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{Alg}},\otimes)$
as an object of the category
$\mathop{\mathbf{Lcoact}}\nolimits({\mathfrak{R}})$. The strict monoidal
embedding $({\mathbf{Alg}},\otimes)\hookrightarrow({\mathbf{GrAlg}},\circ)$,
${\mathfrak{R}}\mapsto{\mathfrak{R}}\otimes{\mathbb{K}}[u]$, induces (not
fully) faithful functor
$\mathop{\mathbf{Lcoact}}\nolimits({\mathfrak{R}})\to\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{O}}_{\mathfrak{R}})$,
which embeds the comodule algebras over the bialgebra ${\mathfrak{R}}$ to the
category of coactions of comonoid ${\mathbb{O}}_{\mathfrak{R}}$ defined by the
formula (5.1). On the other hand, the forgetful functor
$U\colon{\mathbf{GrAlg}}\to{\mathbf{Alg}}$ has a strict monoidal structure
$({\mathbf{GrAlg}},\otimes)\to({\mathbf{Alg}},\otimes)$ and hence induces
functors between comonoids and between coactions. Thus we get the functor
$\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{O}})\to\mathop{\mathbf{Lcoact}}\nolimits({\mathcal{A}},\Delta_{\mathcal{A}},\varepsilon_{\mathcal{A}})\to\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{B}})$,
which translates coactions $\delta$ of a comonoid
${\mathbb{O}}=({\mathcal{A}},\Delta,\varepsilon)\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{GrAlg}},\circ)$
on an object ${\mathcal{B}}\in{\mathbf{GrAlg}}$ to the comodule algebra
$U{\mathcal{B}}\in{\mathbf{Alg}}$ over the bialgebra
${\mathbb{B}}=(U{\mathcal{A}},U\Delta_{\mathcal{A}},U\varepsilon_{\mathcal{A}})$,
where the coaction is given by the homomorphism $U\delta_{\mathcal{A}}$. In
particular, we obtain a forgetful functor from the category of quantum
representations of the comonoid ${\mathbb{O}}$ to the category of comodule
algebras over ${\mathbb{B}}$, which generalises the classical situation where
the linear representations are regarded as actions in
$({\mathbf{AlgSet}},\times)$.

###  5.2 Operations with quantum representations

We have tree operations with vector spaces: direct sum, tensor product and
duality. They induce operations with representations (in the case of tensor
product and duality we need an additional structure on the represented
object). Similarly, the operations with quadratic algebras give operations
with quantum representations. We construct these operations in terms of
coactions and Manin matrices.

5.2.1. Direct sum of Manin matrices. According to the Manins’ ideas [Man88]
the quantum analogue of the direct sum of vector spaces (in the ‘even’
interpretation) is the (even) tensor product of quadratic algebras (see p. 4.1
and p. 4.1). Therefore, we describe first the tensor product
${\mathcal{B}}\otimes{\mathcal{C}}$ for quadratic algebras
${\mathcal{B}},{\mathcal{C}}\in{\mathbf{FQA}}$.

For arbitrary matrix idempotents
$B\in\mathop{\mathrm{End}}\nolimits({\mathbb{K}}^{m}\otimes{\mathbb{K}}^{m})$
and
$C\in\mathop{\mathrm{End}}\nolimits({\mathbb{K}}^{n}\otimes{\mathbb{K}}^{n})$
define the idempotent
$D=\mathop{\mathrm{DiS}}\nolimits(B,C)\in\mathop{\mathrm{End}}\nolimits({\mathbb{K}}^{m+n}\otimes{\mathbb{K}}^{m+n})$
with the only non-zero entries

$\displaystyle D^{ij}_{kl}=B^{ij}_{kl},$ $\displaystyle
D^{\bar{a},\bar{b}}_{\bar{c},\bar{d}}=C^{ab}_{cd}$ $\displaystyle
D^{i,\bar{a}}_{i,\bar{a}}=D^{\bar{a},i}_{\bar{a},i}=\frac{1}{2},$
$\displaystyle
D^{i,\bar{a}}_{\bar{a},i}=D^{i,\bar{a}}_{\bar{a},i}=-\frac{1}{2},$ (5.20)

where $i,j,k,l=1,\ldots,m$, $a,b,c,d=1,\ldots,n$, $\bar{a}=m+a$. Then
${\mathfrak{X}}_{B}({\mathbb{K}})\otimes{\mathfrak{X}}_{C}({\mathbb{K}})={\mathfrak{X}}_{D}({\mathbb{K}})$.

Let
$B^{\prime}\in\mathop{\mathrm{End}}\nolimits({\mathbb{K}}^{m^{\prime}}\otimes{\mathbb{K}}^{m^{\prime}})$
and
$C^{\prime}\in\mathop{\mathrm{End}}\nolimits({\mathbb{K}}^{n^{\prime}}\otimes{\mathbb{K}}^{n^{\prime}})$
be idempotents as well. Denote
$D^{\prime}=\mathop{\mathrm{DiS}}\nolimits(B^{\prime},C^{\prime})\in\mathop{\mathrm{End}}\nolimits({\mathbb{K}}^{m^{\prime}+n^{\prime}}\otimes{\mathbb{K}}^{m^{\prime}+n^{\prime}})$.
Graded operators
$f\colon{\mathfrak{X}}_{B}({\mathbb{K}})\to{\mathfrak{X}}_{B^{\prime}}({\mathfrak{R}})$
and
$g\colon{\mathfrak{X}}_{C}({\mathbb{K}})\to{\mathfrak{X}}_{C^{\prime}}({\mathfrak{R}})$
give one more graded operator by the composition

${\mathfrak{X}}_{D}({\mathbb{K}})={\mathfrak{X}}_{B}({\mathbb{K}})\otimes{\mathfrak{X}}_{C}({\mathbb{K}})\xrightarrow{f\otimes
g}{\mathfrak{R}}\otimes{\mathfrak{X}}_{B^{\prime}}({\mathbb{K}})\otimes{\mathfrak{R}}\otimes{\mathfrak{X}}_{C^{\prime}}({\mathbb{K}})\xrightarrow{\sigma^{(23)}}\\\
{\mathfrak{R}}\otimes{\mathfrak{R}}\otimes{\mathfrak{X}}_{B^{\prime}}({\mathbb{K}})\otimes{\mathfrak{X}}_{C^{\prime}}({\mathbb{K}})\xrightarrow{\mu_{\mathfrak{R}}\otimes\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{id}}\nolimits}{\mathfrak{R}}\otimes{\mathfrak{X}}_{B^{\prime}}({\mathbb{K}})\otimes{\mathfrak{X}}_{C^{\prime}}({\mathbb{K}})={\mathfrak{X}}_{D^{\prime}}({\mathfrak{R}}).$
(5.21)

The condition that this operator is an algebra homomorphism can be written in
terms of Manin matrices.

######  Proposition 5.9.

Let $M$ and $N$ be $m\times m^{\prime}$ and $n\times n^{\prime}$ matrices over
${\mathfrak{R}}$. Consider their direct sum

$\displaystyle L=M\oplus N=\begin{pmatrix}M&0\\\ 0&N\end{pmatrix}.$ (5.22)

This is an $(m+n)\times(m^{\prime}+n^{\prime})$ matrix over ${\mathfrak{R}}$
with the only non-zero entries $L^{i}_{j}=M^{i}_{j}$,
$L^{\bar{a}}_{\bar{b}}=N^{a}_{b}$, where $i,j=1,\ldots,m$, $a,b=1,\ldots,n$.
The matrix $L$ is a $(D,D^{\prime})$-Manin matrix iff $M$ is
$(B,B^{\prime})$-Manin matrix, $N$ is $(C,C^{\prime})$-Manin matrix and

$\displaystyle M^{i}_{j}N^{a}_{b}=N^{a}_{b}M^{i}_{j},$
$\displaystyle\forall\,i,j=1,\ldots,m,\quad a,b=1,\ldots,n.$ (5.23)

In other words, the graded operator (5.21) is a graded homomorphisms iff $f$,
$g$ are algebra homomorphisms and the corresponding Manin matrices entry-wise
commute.

Proof. By writing the condition $DL^{(1)}L^{(2)}(1-D^{\prime})=0$ in entries
one sees that it is equivalent to the conditions
$BM^{(1)}M^{(2)}(1-B^{\prime})=0$, $CN^{(1)}N^{(2)}(1-C^{\prime})=0$ and
(5.23). ∎

5.2.2. Direct sum of quantum representations. Consider coactions
$\delta\colon{\mathcal{B}}\to{\mathcal{A}}\circ{\mathcal{B}}$ and
$\gamma\colon{\mathcal{C}}\to{\mathcal{A}}\circ{\mathcal{C}}$ of the comonoid
${\mathbb{O}}=({\mathcal{A}},\Delta,\varepsilon)\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{GrAlg}},\circ)$
on the objects ${\mathcal{B}},{\mathcal{C}}\in{\mathbf{GrAlg}}$. Let
$({\mathcal{A}},\Delta_{\mathcal{A}},\varepsilon_{\mathcal{A}})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{Alg}},\otimes)$
be the bialgebra corresponding to the comonoid ${\mathbb{O}}$ and
$\delta_{\mathcal{A}}\colon{\mathcal{B}}\to{\mathcal{A}}\otimes{\mathcal{B}}$,
$\gamma_{\mathcal{A}}\colon{\mathcal{C}}\to{\mathcal{A}}\otimes{\mathcal{C}}$
be the corresponding coactions of the bialgebra
$({\mathcal{A}},\Delta_{\mathcal{A}},\varepsilon_{\mathcal{A}})$. The letter
one is the bimonoid
$({\mathcal{A}},\mu_{\mathcal{A}},\varepsilon_{\mathcal{A}},\Delta_{\mathcal{A}},\varepsilon_{\mathcal{A}})$
in $({\mathbf{Vect}},\otimes)$, where
$\mu_{\mathcal{A}}\colon{\mathcal{A}}\otimes{\mathcal{A}}\to{\mathcal{A}}$ and
$\eta_{\mathcal{A}}\colon{\mathbb{K}}\to{\mathcal{A}}$ are the multiplication
and unity map in the algebra ${\mathcal{A}}$, i.e. $\mu_{\mathcal{A}}(a\otimes
b)=ab$, $\eta_{\mathcal{A}}(1)=1_{\mathcal{A}}\in{\mathcal{A}}_{0}$.

We define an analogue of direct sum for the coactions $\delta$ and $\gamma$ as
a coaction of ${\mathbb{O}}$ on ${\mathcal{B}}\otimes{\mathcal{C}}$ by using
the monoidal category $({\mathbf{Alg}},\otimes)$. According to p. 2.2 we need
to introduce the structure of a bimonoid on the comonoid
$({\mathcal{A}},\Delta_{\mathcal{A}},\varepsilon_{\mathcal{A}})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{Alg}},\otimes)$,
this is a pair $(\mu^{\prime}_{\mathcal{A}},\eta^{\prime}_{\mathcal{A}})$ such
that
$({\mathcal{A}},\mu^{\prime}_{\mathcal{A}},\eta^{\prime}_{\mathcal{A}},\Delta_{\mathcal{A}},\varepsilon_{\mathcal{A}})$
is a bimonoid in $({\mathbf{Alg}},\otimes)$, where ${\mathcal{A}}$ is
understood as the algebra
$({\mathcal{A}},\mu_{\mathcal{A}},\eta_{\mathcal{A}})\in{\mathbf{Alg}}$. But
$({\mathbf{Alg}},\otimes)=\mathop{\mathbf{Mon}}\nolimits({\mathbf{Vect}},\otimes)$,
so due to Prop. 2.2 we obtain

$\mathop{\mathbf{Bimon}}\nolimits({\mathbf{Alg}},\otimes)=\mathop{\mathbf{Comon}}\nolimits\big{(}\mathop{\mathbf{Mon}}\nolimits({\mathbf{Alg}},\otimes)\big{)}=\mathop{\mathbf{Comon}}\nolimits\Big{(}\mathop{\mathbf{Mon}}\nolimits\big{(}\mathop{\mathbf{Mon}}\nolimits({\mathbf{Vect}},\otimes)\big{)}\Big{)}=\\\
\mathop{\mathbf{Comon}}\nolimits\big{(}\mathop{\mathbf{\leftidx{{}_{c}}Mon}}\nolimits({\mathbf{Vect}},\otimes)\big{)}=\mathop{\mathbf{Comon}}\nolimits({\mathbf{CommAlg}},\otimes).$

This implies
$({\mathcal{A}},\mu^{\prime}_{\mathcal{A}},\eta^{\prime}_{\mathcal{A}})=({\mathcal{A}},\mu_{\mathcal{A}},\eta_{\mathcal{A}})\in{\mathbf{CommAlg}}$,
so we can define the direct sum of coactions by the formula (2.101) iff the
algebra ${\mathcal{A}}$ is commutative. We need to use in this case the
multiplication $\mu_{\mathcal{A}}$ in (2.101). In the non-commutative case we
define the direct sum in an immediate way, but not for every pair
$(\delta,\gamma)$.

Consider the linear map

$\displaystyle{\mathcal{B}}\otimes{\mathcal{C}}\xrightarrow{\delta_{\mathcal{A}}\otimes\gamma_{\mathcal{A}}}{\mathcal{A}}\otimes{\mathcal{B}}\otimes{\mathcal{A}}\otimes{\mathcal{C}}\xrightarrow{\sigma^{(23)}}{\mathcal{A}}\otimes{\mathcal{A}}\otimes{\mathcal{B}}\otimes{\mathcal{C}}\xrightarrow{\mu_{\mathcal{A}}\otimes\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{id}}\nolimits}{\mathcal{A}}\otimes{\mathcal{B}}\otimes{\mathcal{C}}.$
(5.24)

The conditions (5.13) and
$\mu_{\mathcal{A}}({\mathcal{A}}_{k}\otimes{\mathcal{A}}_{l})\subset{\mathcal{A}}_{k+l}$
imply that the image of ${\mathcal{B}}_{k}\otimes{\mathcal{C}}_{l}$ under the
map (5.24) lies in
${\mathcal{A}}_{k+l}\otimes{\mathcal{B}}_{k}\otimes{\mathcal{C}}_{l}$, so this
map also satisfies (5.13). Hence it gives the graded map
${\mathcal{B}}\otimes{\mathcal{C}}\to{\mathcal{A}}\circ({\mathcal{B}}\otimes{\mathcal{C}})$,
which we denote by $\delta\dotplus\gamma$.

######  Definition 5.10.

We say that the direct sum of the coactions $\delta,\gamma$ exists if
$\delta\dotplus\gamma$ is an algebra homomorphism. In this case
$\delta\dotplus\gamma$ is a coaction of ${\mathbb{O}}$ on
${\mathcal{B}}\otimes{\mathcal{C}}$ called direct sum of the coactions
$\delta$ and $\gamma$. Suppose ${\mathcal{B}},{\mathcal{C}}\in{\mathbf{FQA}}$,
so we can consider the corresponding corepresentations
$\omega=\vartheta^{-1}(\delta)\colon\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}})\to{\mathbb{O}}$ and
$\nu=\vartheta^{-1}(\gamma)\colon\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{C}})\to{\mathbb{O}}$. If the direct sum of
$\delta,\gamma$ exists, then

$\displaystyle\omega\dotplus\nu:=\vartheta^{-1}(\delta\dotplus\gamma)\colon\mathop{\mathrm{\bf
coend}}\nolimits({\mathcal{B}}\otimes{\mathcal{C}})\to{\mathbb{O}}$

is a corepresentation of the comonoid ${\mathbb{O}}$ on the object
${\mathcal{B}}\otimes{\mathcal{C}}$ called direct product of corepresentations
$\omega$ and $\nu$. In this case we say that the direct sum of
corepresentations $\omega,\nu$ exists.

######  Proposition 5.11.

Let ${\mathcal{B}}={\mathfrak{X}}_{B}({\mathbb{K}})$ and
${\mathcal{C}}={\mathfrak{X}}_{C}({\mathbb{K}})$,
$D=\mathop{\mathrm{DiS}}\nolimits(B,C)$. Let $\delta$ and $\gamma$ be
coactions of ${\mathbb{O}}$ given by multiplicative first order $B$\- and
$C$-Manin matrices $M$ and $N$ respectively. The direct sum of the coactions
$\delta,\gamma$ exists iff $M$ and $N$ entry-wise commute in the sense of
(5.23). Thus the direct sum of the corepresentations $\omega,\nu$ exists iff
the corresponding Manin matrices $M$ and $N$ satisfy this condition. In this
case the coaction
$\delta\dotplus\gamma\colon{\mathfrak{X}}_{D}({\mathbb{K}})\to{\mathcal{A}}\circ{\mathfrak{X}}_{D}({\mathbb{K}})$
and corepresentation $\omega\dotplus\nu\colon\mathop{\mathrm{\bf
coend}}\nolimits\big{(}{\mathfrak{X}}_{D}({\mathbb{K}})\big{)}\to{\mathbb{O}}$
are given by the multiplicative first order $D$-Manin matrix (5.22).

Proof. Since ${\mathcal{B}}_{0}={\mathcal{C}}_{0}={\mathbb{K}}$ we have
$({\mathcal{B}}\otimes{\mathcal{C}})_{1}={\mathcal{B}}_{1}\oplus{\mathcal{C}}_{1}$,
so the first order component of $\delta\dotplus\gamma$ coincides with
$\delta_{1}\oplus\gamma_{1}$. Hence the graded map
$\delta\dotplus\gamma\colon{\mathcal{B}}\otimes{\mathcal{C}}\to{\mathcal{A}}\circ({\mathcal{B}}\otimes{\mathcal{C}})$
is given by the matrix $L=M\oplus N$. Since $M$ and $N$ are multiplicative
first order matrices, so is $L$. The rest follows from Prop. 5.6 and 5.9. ∎

In general, the operation of direct sum does not give a monoidal structure on
$\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{O}})$ nor on
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{FQA}}({\mathbb{O}})$. However if
$\delta,\gamma,\delta^{\prime},\gamma^{\prime}$ are coactions of
${\mathbb{O}}$ on ${\mathcal{B}}$, ${\mathcal{C}}$, ${\mathcal{B}}^{\prime}$,
${\mathcal{C}}^{\prime}$ such that the direct sums of the coactions
$\delta,\gamma$ and of $\delta^{\prime},\gamma^{\prime}$ exist, then due to
Prop. 5.11 and 5.7 any morphisms
$f\colon({\mathcal{B}},\delta)\to({\mathcal{B}}^{\prime},\delta^{\prime})$ and
$g\colon({\mathcal{C}},\gamma)\to({\mathcal{C}}^{\prime},\gamma^{\prime})$ in
$\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{O}})$ give a morphism

$f\dotplus
g\colon({\mathcal{B}}\otimes{\mathcal{C}},\delta\dotplus\gamma)\to({\mathcal{B}}^{\prime}\otimes{\mathcal{C}}^{\prime},\delta^{\prime}\dotplus\gamma^{\prime}).$

Thus we have the functor from a full subcategory of