=2.3em

# Affine monoids of corank one

Yulia Zaitseva  HSE University, Faculty of Computer Science, Pokrovsky
Boulvard 11, Moscow, 109028 Russia<EMAIL_ADDRESS>
###### Abstract.

We give a classification of noncommutative algebraic monoid structures of
corank one on affine normal varieties and study their algebraic properties.

###### Key words and phrases:

Algebraic variety, algebraic group, algebraic monoid, toric variety, group
embedding, locally nilpotent derivation, semidirect product

###### 2010 Mathematics Subject Classification:

Primary 14L30, 14R10; Secondary 13E10, 14M25, 20M32

Supported by the RSF-DST grant 22-41-02019

##  1\. Introduction

An (affine) algebraic monoid is an irreducible (affine) algebraic variety $X$
with an associative multiplication

$\mu\colon X\times X\to X,\quad(x,y)\mapsto x*y,$

which is a morphism of algebraic varieties and admits a unity $e\in X$ such
that ${e*x=x*e=x}$ for all $x\in X$. The group of invertible elements $G(X)$
of an algebraic monoid $X$ is an algebraic group, Zariski open in $X$.
According to [16, Theorem 3], every algebraic monoid $X$ with an affine group
of invertible elements $G(X)$ is an affine monoid.

The theory of affine algebraic monoids is a rich area of mathematics lying at
the intersection of algebra, algebraic geometry, combinatorics and
representation theory; see [13, 14, 15, 18]. An affine algebraic monoid $X$ is
called reductive (solvable, commutative) if the group $G(X)$ is a reductive
(solvable, commutative) affine algebraic group. The most developed is the
theory of reductive monoids, see e.g. the combinatorial classification of
reductive monoids in [18, 15].

By the (co)rank of a monoid we mean the (co)dimension of a maximal torus in
$G(X)$. Let the ground field $\mathbb{K}$ be algebraically closed and of
characteristic zero. In the commutative case, the group $G(X)$ splits into the
direct product of an algebraic torus $(\mathbb{K}^{\times})^{r}$ and a
commutative unipotent group $\mathbb{G}_{a}^{s}$, where
$\mathbb{G}_{a}=(\mathbb{K},+)$ is the additive group of the ground field.

In [1], commutative monoids on affine spaces are studied. In particular, [1,
Proposition 1] gives the classification of commutative monoid structures of
ranks $0$, $n-1$ and $n$ on $\mathbb{A}^{n}$. This implies the classification
of monoids on $\mathbb{A}^{1}$ and $\mathbb{A}^{2}$, and [1, Theorem 1]
provides the classification on $\mathbb{A}^{3}$. In [17], these
classifications are extended to not algebraically closed fields of
characteristic zero.

In [8], a classification of commutative monoid structures of rank $0$, $n-1$
and $n$ on normal affine varieties is obtained. It turns out that such affine
algebraic variety is toric, and structures of rank $n-1$ are described by
Demazure roots of the variety. The classification is given in terms of
comultiplications
$\mu^{*}\colon\mathbb{K}[X]\to\mathbb{K}[X]\otimes\mathbb{K}[X]$:

$\chi^{u}\mapsto\chi^{u}\otimes\chi^{u}(1\otimes\chi^{e}+\chi^{e}\otimes
1)^{\langle p,u\rangle},$

where $p$ is the primitive vector on a ray of the cone of $X$ and $e$ is a
Demazure root corresponding to this ray. In particular, it implies the
classification of commutative monoid structures of any rank for surfaces.

In [4] the classification of noncommutative monoids on affine surfaces is
given. In this case $G(X)$ is the semidirect product of $\mathbb{K}^{\times}$
and $\mathbb{G}_{a}$. It is proved that the surface is toric and the
comultiplication has the form

$\chi^{u}\mapsto\chi^{u}\otimes\chi^{u}(1\otimes\chi^{e_{1}}+\chi^{e_{2}}\otimes
1)^{\langle p,u\rangle},$

where $e_{1},e_{2}$ are two Demazure roots corresponding to the same ray.

In this work we generalize results of [8] and [4] and obtain a classification
of noncommutative monoids of an arbitrary dimension and corank one, see
Proposition 1 and Theorem 1. Theorem 2 is the specialization of the
classification to affine spaces. In Theorem 3 we obtain a description of
idempotents in such monoids.

The author is grateful to Ivan Arzhantsev for valuable suggestions, attention
to this work and permanent support.

##  2\. Semidirect products and the toric structure

######  Definition 1.

An (affine) irreducible normal algebraic variety $X$ with a morphism ${X\times
X\to X,\,(x,y)\mapsto x*y}$, is called an _(affine) algebraic monoid_ if
$x*(y*z)=(x*y)*z$ for all $x,y,z\in X$ and there exists a point $1\in X$
called _unity_ such that $1*x=x*1=x$.

Let $X$ be an affine monoid of dimension $n$. Then the group $G(X)$ of
invertible elements of $X$ is a connected affine algebraic group, which is
Zariski open in $X$, see [15, Theorem 1] and [16, Theorem 5]. Recall that the
_rank_ and the _corank_ of an affine algebraic group $G$ are the dimension and
the codimension of a maximal torus in $G$, respectively. By the rank and the
corank of a monoid $X$ we mean the rank and the corank of the group $G(X)$,
respectively.

The aim of the present work is to study affine monoids of corank one. In this
case $G(X)$ has no semisimple part since any root subgroup in a semisimple
group occurs together with an opposite one. So $G(X)$ has a one-dimensional
unipotent radical and $G(X)=\mathbb{G}_{a}\leftthreetimes T$, where
$\mathbb{G}_{a}=(\mathbb{K},+)$ and $T$ is a torus of dimension $n-1$.

By definition of a semidirect product, the group operation in $G(X)$ is
defined by a homomorphism $\chi\colon
T\to\operatorname{Aut}(\mathbb{G}_{a})=\mathbb{K}^{\times}$:

$(\alpha,t]\cdot(\alpha^{\prime},t^{\prime}]=(\alpha+\chi(t)\alpha^{\prime},\,tt^{\prime}],$
(1)

where $\alpha,\alpha^{\prime}\in\mathbb{G}_{a}$, $t,t^{\prime}\in T$. We
denote this group $G_{\chi}$. Thus,

$G_{\chi}=\mathbb{G}_{a}\leftthreetimes T,$

where $T$ is an algebraic torus of dimension $n-1$ and the multiplication is
defined by (1).

Notice that according to (1) we have $(\alpha,t]=(\alpha,1](0,t]=\alpha\cdot
t\in\mathbb{G}_{a}T=G_{\chi}$. There is also a dual way to write the elements
of $G_{\chi}$. Namely, for an element $(\alpha,t]\in G_{\chi}$ consider
$\beta=\chi^{-1}(t)\alpha\in\mathbb{G}_{a}$. Then
$t\cdot\beta=(0,t](\beta,1]=(\chi(t)\beta,t]=(\alpha,t]$. Thus, for any
element $g\in G_{\chi}$ we have two decompositions

$g=(\alpha,t]=\alpha\cdot t=[t,\beta)=t\cdot\beta.$

The dual multiplication rule is

$[t,\beta)\cdot[t^{\prime},\beta^{\prime})=[tt^{\prime},\chi(t^{\prime})^{-1}\beta+\beta^{\prime}).$

The group $G_{\chi}$ is commutative if and only if $\chi=0$.

######  Lemma 1.

If $\chi\neq 0$, then the center
$Z(G_{\chi})=\\{0\\}\times\operatorname{Ker}\chi$.

###### Proof.

Let $(\alpha,t]\in Z(G)$, i.e.

$(\alpha+\chi(t)\alpha^{\prime},\,tt^{\prime}]=(\chi(t^{\prime})\alpha+\alpha^{\prime},\,t^{\prime}t]$
(2)

for any $\alpha^{\prime}\in\mathbb{G}_{a}$, $t\in T$. Since there exists
$t^{\prime}\in T$ with $\chi(t^{\prime})\neq 1$, condition (2) for
$\alpha^{\prime}=0,1$ implies $\alpha=0$, $\chi(t)=1$. Conversely, for
$(\alpha,t]\in\\{0\\}\times\operatorname{Ker}\chi$ equation (2) turns into
$(\alpha^{\prime},\,tt^{\prime}]=(\alpha^{\prime},\,t^{\prime}t]$. ∎

######  Definition 2.

A _group embedding_ of an algebraic group $G$ is an irreducible algebraic
variety $X$ with an open embedding $G\hookrightarrow X$ such that the action
of $G\times G$ on $G$ by left and right multiplications can be extended to an
action of $G\times G$ on $X$.

If $X$ is a monoid, then the embedding $G(X)\hookrightarrow X$ is a group
embedding. For affine monoids the converse is true, i.e. if $X$ is an affine
group embedding of $G$, then the multiplication on $G$ extends to a
multiplication $X\times X\to X$ in such a way that $G$ is the group $G(X)$ of
invertible elements of $X$, see [18, Theorem 1] for characteristic zero and
[15, Proposition 1] for the general case.

Recall that a normal irreducible algebraic variety is called _toric_ if it
admits an action of an algebraic torus with an open orbit, see [6, 10, 12]. In
other words, a toric variety is a group embedding of an algebraic torus. The
main result of this section is the following statement.

######  Proposition 1.

Any affine monoid $X$ of corank one is a toric variety. Moreover, the group of
invertible elements $G(X)$ is invariant with respect to the acting torus
$\mathbb{T}$.

###### Proof.

If $\chi=0$, then $G_{\chi}$ is commutative and the result follows from [2,
Theorem 2 and Lemma 2].

Let $\chi\neq 0$. First consider the actions of $G_{\chi}$ on itself by left
and right multiplications. Let a group homomorphism $\theta\colon T\times
T\to\operatorname{Aut}(G_{\chi})$ be defined by

$\theta(t_{1},t_{2})(g)=t_{1}gt_{2}^{-1}.$

An element $(t_{1},t_{2})\in T\times T$ belongs to the kernel of $\theta$ if
and only if $(t_{1},t_{2})$ belongs to the diagonal of the center
$\operatorname{diag}Z(G_{\chi})=\\{(g,g)\mid g\in Z(G_{\chi})\\}\subseteq
G_{\chi}\times G_{\chi}$. By Lemma 1, the center
$Z(G_{\chi})=\\{0\\}\times\operatorname{Ker}\chi\subseteq\mathbb{G}_{a}\leftthreetimes
T$, so $\operatorname{Ker}\theta=\operatorname{diag}\operatorname{Ker}\chi$.
Since $T\times T$ is a torus, the image

$\mathbb{T}=\mathop{\text{Im}}\theta\subseteq\operatorname{Aut}(G_{\chi})$

is a torus as well. It is isomorphic to $(T\times
T)/\operatorname{diag}\operatorname{Ker}\chi$, so the torus $\mathbb{T}$ has
dimension $\dim\mathbb{T}=(n-1)+(n-1)-(n-2)=n$.

Thus, we have an effective action of $\mathbb{T}$ on $G_{\chi}$, coming from
left and right multiplications of $G_{\chi}$ on itself. By definition of a
group embedding, this action can be extended to the action on $X$, so there
exists an effective action of $\mathbb{T}$ on $X$. It is known that any
effective action of a torus of dimension coinciding with the dimension of a
variety has an open orbit, see [7, Corollary 1]. We have $\dim\mathbb{T}=\dim
G_{\chi}=\dim X=n$, so both actions of $\mathbb{T}$ on $G_{\chi}$ and $X$ have
an open orbit. ∎

The idea of the following generalization of Proposition 1 was proposed by
Sergey Gorchinskiy and Constantin Shramov. Consider a solvable monoid $X$,
i.e. a monoid with a solvable group of invertible elements $G(X)$. Then
$G(X)=U\leftthreetimes T$, where $T$ is a torus and $U$ is the unipotent
radical of $G(X)$. In some cases it can be proved that $X$ is toric as well.

Namely, let the multiplication in a semidirect product $G=U\leftthreetimes T$
be given by a homomorphism $\psi\colon T\to\operatorname{Aut}(U)$:

$(u,t]\cdot(u^{\prime},t^{\prime}]=(u\cdot\psi(t)(u^{\prime}),tt^{\prime}],\quad\quad
u,u^{\prime}\in U,\;\;t,t^{\prime}\in T.$

######  Definition 3.

Let us call this semidirect product _active_ if $\dim
T+\dim\mathop{\text{Im}}\psi=\dim G$, or, equivalently,
$\dim\mathop{\text{Im}}\psi=\dim U$.

######  Remark 1.

Any affine algebraic group of corank one is solvable, and the semidirect
product $G_{\chi}=\mathbb{G}_{a}\leftthreetimes T$ is active if and only if
$\chi\neq 0$.

Since $U$ is a unipotent group, it is isomorphic to a vector space as an
algebraic variety. Let us prove that we can choose coordinates in $U$ in such
a way that $T$ acts on $U$ linearly.

######  Lemma 2.

There are coordinates in $U$ such that $\psi(T)\subseteq\operatorname{GL}(U)$.

###### Proof.

For $G=U\leftthreetimes T$ and its Lie algebra
${\mathfrak{g}}={\mathfrak{u}}\oplus{\mathfrak{t}}$, consider the conjugation

$\psi\colon
G\to\operatorname{Aut}(G),\quad\psi(g)(g^{\prime})=gg^{\prime}g^{-1},$

and the adjoint representation

$\operatorname{Ad}\colon
G\to\operatorname{Aut}({\mathfrak{g}}),\quad\operatorname{Ad}(g)=d\psi(g).$

Since $U$ is normal in $G$, we have restrictions

$\psi\colon
T\to\operatorname{Aut}(U)\quad\text{and}\quad\operatorname{Ad}\colon
T\to\operatorname{Aut}({\mathfrak{u}})\subseteq\operatorname{GL}({\mathfrak{u}}).$

It is known that the exponential map $\exp\colon{\mathfrak{u}}\to U$ is
bijective and commutes with the differential, i.e.
$\exp\circ\,\psi(t)=\operatorname{Ad}(t)\circ\exp$ for any $t\in T$. So $\exp$
takes coordinates on ${\mathfrak{u}}$ to $U$ in such a way that
$\psi(t)\in\operatorname{GL}(U)$ for any $t\in T$. ∎

So one may assume that
$\psi(t)=\operatorname{diag}(\chi_{1}(t),\ldots,\chi_{k}(t))\in\operatorname{GL}(U)$,
where $\chi_{i},\,1\leqslant i\leqslant k$, are characters of the torus $T$.
In these terms the semidirect product is active if and only if $\chi_{i}$ are
linearly independent.

######  Example 1.

Notice that $\dim\mathop{\text{Im}}\psi\leqslant\dim T$, so for an active
semidirect product there is a necessary condition $\dim
T\geqslant\frac{1}{2}\dim G$. For example, the solvable group $B$ of upper
triangular $n\times n$ matrices is the semidirect product of the group of
unitriangular matrices $U$ and the torus of diagonal matrices $T$. It is not
active for $n>3$ since $\dim T=n<\frac{1}{2}\dim G=\frac{n(n+1)}{4}$. For
$n=3$ we have $\dim T=3=\frac{1}{2}\dim B$, however the semidirect product is
not active as well since $\psi(\operatorname{diag}(t_{1},t_{2},t_{3}))$ acts
on $U$ with linear dependent characters
$t_{1}t_{2}^{-1},\,t_{1}t_{3}^{-1},\,t_{2}t_{3}^{-1}$.

######  Example 2.

The group

$G=\left\\{\begin{pmatrix}t_{1}&0&\ldots&0&u_{1}\\\ 0&t_{2}&\ldots&0&u_{2}\\\
\ldots&\ldots&\ddots&\ldots&\ldots\\\ 0&0&\ldots&t_{n-1}&u_{n-1}\\\
0&0&\ldots&0&t_{n}\end{pmatrix},\,t_{i}\in\mathbb{K}^{\times},u_{j}\in\mathbb{K}\right\\}$

is an active semidirect product of the group $U$ of unitriangular matrices of
$G$ and the torus of diagonal matrices $T$ since
$\psi(T)\subseteq\operatorname{GL}(U)$ is generated by linearly independent
characters $t_{1}t_{n}^{-1},\,\ldots,t_{n-1}t_{n}^{-1}$.

######  Proposition 2.

Any solvable affine monoid $X$ with active group of invertible elements is a
toric variety. Moreover, the group of invertible elements $G(X)$ is invariant
with respect to the acting torus $\mathbb{T}$.

###### Proof.

Following the proof of Proposition 1, consider the map $\theta\colon T\times
T\to\operatorname{Aut}(G)$ and the torus
$\mathbb{T}=\mathop{\text{Im}}\theta$. An element $(t_{1},t_{2})\in T\times T$
belongs to $\operatorname{Ker}\theta$ if and only if $t_{1}=t_{2}\in Z(G)$.
Notice that $t=(0,t]\in Z(G)$ if and only if $t\in\operatorname{Ker}\psi$.
Indeed, if $(\psi(t)(u^{\prime}),\,tt^{\prime}]=(u^{\prime},\,t^{\prime}t]$
for any $u^{\prime}\in U$, $t\in T$, then $\psi(t)=\operatorname{id}$. So
$\operatorname{Ker}\theta=\operatorname{diag}\operatorname{Ker}\psi$.

Since $\dim T+\dim\mathop{\text{Im}}\psi=\dim G$, the torus $\mathbb{T}$ has
dimension

$\dim(T\times T)-\dim\operatorname{Ker}\theta=\dim T+\dim T-(\dim
T-\dim\mathop{\text{Im}}\psi)=\dim G=\dim X.$

As above, it follows that the action of $\mathbb{T}$ has an open orbit on
$G(X)$ and $X$. ∎

It would be interesting to investigate affine algebraic monoids with an
arbitrary active group of invertible elements, and we plan to do it in a
further publication. In this article we concentrate on affine monoids of
corank one.

##  3\. Preliminaries on toric varieties

###  3.1. Polyhedral cones of an affine toric variety

Let us recall basic facts on affine toric varieties, see for example [10,
Section 1.3]. Let $N$ be a lattice of rank $n$,
$N_{\mathbb{Q}}=N\otimes_{\mathbb{Z}}\mathbb{Q}$ be the rational vector space
of dimension $n$, and $\sigma\subseteq N_{\mathbb{Q}}$ be a strongly convex
polyhedral cone. Below we give a construction of an affine toric variety
$X_{\sigma}$ of dimension $n$ corresponding to the cone $\sigma$. It is known
that all affine toric varieties arise in this way.

Let $M=\operatorname{Hom}(N,\mathbb{Z})$ be the dual lattice,
$M_{\mathbb{Q}}=M\otimes_{\mathbb{Z}}\mathbb{Q}$ be the corresponding rational
vector space, and $\langle\,\cdot\,,\,\cdot\,\rangle\colon
N_{\mathbb{Q}}\times M_{\mathbb{Q}}\to\mathbb{Q}$ be the natural pairing. We
can consider $M$ as the lattice of characters of a torus
$\mathbb{T}=\operatorname{Hom}(M,\mathbb{K}^{\times})$ of dimension $n$, and
for $u\in M$ denote the corresponding character by
$\chi^{u}\colon\mathbb{T}\to\mathbb{K}^{\times}$. Then the dual lattice $N$ is
identified with the lattice of one-parameter subgroups of the torus
$\mathbb{T}=N\otimes_{\mathbb{Z}}\mathbb{K}^{\times}$. Consider the polyhedral
cone $\sigma^{\vee}$ dual to the cone $\sigma$:

$\sigma^{\vee}=\\{u\in M_{\mathbb{Q}}\mid\langle v,u\rangle\geqslant 0\;\text{
for all }v\in\sigma\\}.$

Then $\sigma^{\vee}$ is a cone of full dimension,
$S_{\sigma}=M\cap\sigma^{\vee}$ is a finitely generated semigroup with
$S_{\sigma}-S_{\sigma}=M$ and
$\mathbb{K}[S_{\sigma}]=\\!\bigoplus\limits_{u\in
S_{\sigma}}\\!\mathbb{K}\chi^{u}$ is a finitely generated
$\mathbb{K}$-algebra. Let $X_{\sigma}$ be the spectrum of the algebra
$\mathbb{K}[S_{\sigma}]$.

Consider $\mathbb{K}$ as a semigroup with respect to multiplication. Any
semigroup homomorphism $S_{\sigma}\to\mathbb{K}$ defines an algebra
homomorphism $\mathbb{K}[X_{\sigma}]\to\mathbb{K}$ that corresponds to a point
in $X$. Conversely, any point in $X$ is defined by some semigroup homomorphism
$S_{\sigma}\to\mathbb{K}$. The torus $\mathbb{T}$ is identified with the
subset $X_{0}$ of $X$ as the restrictions of group homomorphisms
$M\to\mathbb{K}^{\times}$ to semigroup homomorphisms
$S_{\sigma}\to\mathbb{K}$.

In fact, $X_{0}$ is open in $X_{\sigma}$, so any character $\chi^{u}$ of the
torus $\mathbb{T}$ can be identified with a rational function on $X_{\sigma}$.
We have the decomposition

$\mathbb{K}[X_{\sigma}]=\\!\bigoplus\limits_{u\in
S_{\sigma}}\\!\mathbb{K}\chi^{u}.$ (3)

An element $t$ of the torus $\mathbb{T}$ acts on $\chi^{u}$ by multiplication
by $\chi^{u}(t)$. This defines the action of $\mathbb{T}$ on
$\mathbb{K}[X_{\sigma}]$, and whence the action of $\mathbb{T}$ on
$X_{\sigma}$.

###  3.2. Faces and toric open subsets

Recall that a face of a cone is the intersection of the cone with a supporting
hyperplane. There is a bijection between faces of the dual cones $\sigma$ and
$\sigma^{\vee}$: for any $k$-dimensional face $\tau$ of the cone $\sigma$ the
set

$\tau^{\perp}\cap\sigma^{\vee}=\\{u\in M_{\mathbb{Q}}\mid\langle
v,u\rangle=0\;\;\forall v\in\tau\\}\cap\sigma^{\vee}$

is an $(n-k)$-dimensional face of the cone $\sigma^{\vee}$, and, conversely,
for any face $\gamma$ of the cone $\sigma^{\vee}$ of dimension $k$ the set
$\gamma^{\perp}\cap\sigma$ is a face of the cone $\sigma$ of dimension $n-k$.
Faces of dimension one are called _rays_ , and faces of codimension one are
called _facets_.

Any homomorphism of semigroups $S\to S^{\prime}$ determines a homomorphism
$\mathbb{K}[S]\to\mathbb{K}[S^{\prime}]$ of algebras and a morphism
$\operatorname{Spec}\mathbb{K}[S^{\prime}]\to\operatorname{Spec}\mathbb{K}[S]$.
In particular, if $\tau\subseteq\sigma$, then $S_{\tau}\supseteq S_{\sigma}$,
which determines a morphism $X_{\tau}\to X_{\sigma}$. It can be proved that
the morphism $X_{\tau}\to X_{\sigma}$ is an open embedding if and only if
$\tau$ is a face of the cone $\sigma$. Moreover, in this case
$S_{\tau}=S_{\sigma}+\mathbb{Z}(-u^{\prime})$, where $u^{\prime}\in M$ and
$\tau=\sigma\cap u^{\prime\perp}$, or, equivalently, $u^{\prime}\in M$ is from
the relative interior of the dual face $\tau^{\perp}\cap\sigma^{\vee}$ of the
cone $\sigma^{\vee}$. Then $\mathbb{K}[S_{\tau}]$ is the localization of
$\mathbb{K}[S_{\sigma}]$ at $\chi^{u^{\prime}}$, which determines an embedding
$X_{\tau}\subseteq X_{\sigma}$ as a principal open subset. In terms of
semigroup homomorphisms, $X_{\tau}$ consists of points $S\to\mathbb{K}$ such
that the image of $u^{\prime}$ is nonzero, or, equivalently, the image of any
$u^{\prime}\in M$ from the relative interior of
$\tau^{\perp}\cap\sigma^{\vee}$ is nonzero. In particular, the open orbit of
$\mathbb{T}$ is $X_{0}$, and the actions of $\mathbb{T}$ on $X_{\tau}$ defined
above are compatible for all faces $\tau$ of $\sigma$.

###  3.3. Torus orbits

It is known that for toric varieties $\mathbb{T}$-orbits correspond to faces
of the cone $\sigma\subseteq N_{\mathbb{Q}}$, see [10, Section 3.1]. More
precisely, for a face $\tau\subseteq\sigma$ denote by $x_{\tau}$ the
distinguished point in $X_{\sigma}$ given by the following semigroup
homomorphism $S_{\sigma}\to\mathbb{K}$:

$u\mapsto\begin{cases}1\;\text{ if }u\in\tau^{\perp}\\\ 0\;\text{
otherwise}\end{cases}.$

By $O_{\tau}$ denote the $\mathbb{T}$-orbit of the point $x_{\tau}$. It has
dimension $n-\dim\tau$. In terms of semigroup homomorphisms it consists of
homomorphisms $S_{\sigma}\to\mathbb{K}$ such that

$u\mapsto\begin{cases}\mathbb{K}^{\times}\;\text{ if }u\in\tau^{\perp}\\\
0\;\text{ otherwise}\end{cases}.$

It is known that all the orbits of $\mathbb{T}$ on $X_{\sigma}$ are of the
form $O_{\tau}$ for a face $\tau$ of $\sigma$. If $\gamma$ is a face of
$\tau$, then $O_{\tau}\subseteq\overline{O_{\gamma}}$. For more details see
[5, Theorem 3.2.6]. It follows also that $O_{\gamma}$ is closed if and only if
$\gamma$ is a maximal face.

###  3.4. Locally nilpotent derivations, $\mathbb{G}_{a}$-actions and Demazure
roots

For an algebra $A$, a linear operator $\delta\colon A\to A$ is a _derivation_
if it satisfies the Leibniz rule:

$\delta(fg)=\delta(f)g+f\delta(g)$

for any $f,g\in A$. A derivation $\delta$ is called _locally nilpotent_ (LND)
if for any $f\in A$ there exists a number $n\in\mathbb{Z}_{>0}$ such that
$\delta^{n}(f)=0$. The exponential map defines a bijection between locally
nilpotent derivations on an algebra $A$ and rational $\mathbb{G}_{a}$-actions
on $A$, see [9, Section 1.5]. More precisely, an LND $\delta$ on $A$ defines
the $\mathbb{G}_{a}$-action $\alpha\cdot f=\exp(\alpha\delta)(f)$ for
$\alpha\in\mathbb{G}_{a}$, $f\in A$, and, conversely, given an action
$\xi\colon\mathbb{G}_{a}\times A\to A$, $(\alpha,f)\mapsto\alpha\cdot f$, we
can get the LND $\delta$ as $\left(\frac{d\xi}{d\alpha}\right)|_{\alpha=0}$.
For an affine algebraic variety $X$, $\mathbb{G}_{a}$-actions on $X$ are dual
to $\mathbb{G}_{a}$-actions on $\mathbb{K}[X]$, so LNDs on the algebra
$\mathbb{K}[X]$ are in bijection with $\mathbb{G}_{a}$-actions on $X$.

Let $A=\bigoplus\limits_{u\in S}A_{u}$ be graded by a semigroup $S$. A
derivation $\partial\colon A\to A$ is called _homogeneous_ if it maps
homogeneous elements to homogeneous ones. From the Leibniz rule it follows
that for a domain $A$ a homogeneous derivation has the _degree_
$\deg\partial\in\mathbb{Z}S$ such that $\partial(A_{u})\subseteq
A_{u+\deg\partial}$ for any $u\in S$.

Let $X_{\sigma}$ be an affine toric variety and let $p_{i}\in N,\;1\leqslant
i\leqslant m$, be primitive vectors on the rays of a cone $\sigma\subseteq
N_{\mathbb{Q}}$. For any $1\leqslant i\leqslant m$ denote

$\mathfrak{R}_{i}=\\{e\in M\mid\langle p_{i},e\rangle=-1,\;\langle
p_{j},e\rangle\geqslant 0\;\text{ for all }j\neq i,\;1\leqslant j\leqslant
m\\}.$

The elements of the set $\mathfrak{R}=\\!\\!\\!\bigsqcup\limits_{1\leqslant
i\leqslant m}\\!\\!\\!\mathfrak{R}_{i}$ are called the _Demazure roots_ of the
toric variety $X_{\sigma}$. It is easy to see that if $e\in\mathfrak{R}_{i}$,
$\gamma$ is a face of $\sigma$ and $e\in\gamma^{\perp}$, then the cone
generated by $\gamma$ and $p_{i}$ is a face of $\sigma$ as well.

The Demazure roots of $X_{\sigma}$ are in one-to-one correspondence with
homogeneous LNDs on the algebra $\mathbb{K}[X_{\sigma}]$ with respect to
grading (3) and hence with $\mathbb{G}_{a}$-actions on $X_{\sigma}$ normalized
by the acting torus $\mathbb{T}$, see [11, Theorem 2.7]. The Demazure root $e$
is the degree of the corresponding LND, and the action of the torus
$\mathbb{T}$ on $\mathbb{G}_{a}$ by conjugation is the multiplication by
$\chi^{e}$.

##  4\. Classification results

Let $X$ be an affine monoid with a multiplication $X\times X\to X$. The dual
morphism is the _comultiplication_
$\mathbb{K}[X]\to\mathbb{K}[X]\otimes\mathbb{K}[X]$. The comultiplication
determines the multiplication, so affine monoids can be described in these
terms.

Let us formulate the main result.

######  Theorem 1.

Let $X$ be an affine monoid of corank one. Then $X=X_{\sigma}$ is toric, and
the comultiplication
$\mathbb{K}[X_{\sigma}]\to\mathbb{K}[X_{\sigma}]\otimes\mathbb{K}[X_{\sigma}]$
has the form

$\chi^{u}\mapsto\chi^{u}\otimes\chi^{u}\,(1\otimes\chi^{e_{1}}+\chi^{e_{2}}\otimes
1)^{\langle p,u\rangle},$ (4)

where $\mathbb{K}[X_{\sigma}]=\\!\bigoplus\limits_{u\in
S_{\sigma}}\\!\mathbb{K}\chi^{u}$ as in equation (3), $p$ is the primitive
vector on a ray of the cone $\sigma$, and $e_{1},e_{2}$ are Demazure roots
corresponding to $p$. Conversely, for any affine toric variety $X_{\sigma}$,
any primitive vector $p$ on a ray of the cone $\sigma$, and any Demazure roots
$e_{1},e_{2}$ corresponding to the same $p$, formula (4) defines a monoid
structure of corank one on $X_{\sigma}$.

More precisely, the group of invertible elements is the toric open subset
$X_{\rho}\subseteq X_{\sigma}$ isomorphic to the group $G_{\chi}$ for
$\chi=\chi^{e_{2}-e_{1}}$.

######  Example 3.

Let us find all monoid structures of rank 2 on the quadratic cone

$X=\\{\alpha\beta=\gamma\delta\\}\subseteq\mathbb{A}^{4};$

cf. [8, Example 5] for the commutative case. It is an affine toric variety
given by the cone $\sigma$ with $p_{1}=(1,0,0)$, $p_{2}=(0,1,0)$,
$p_{3}=(1,0,1)$ and $p_{4}=(0,1,1)$. Indeed, the semigroup
$S_{\sigma}=M\cap\,\sigma^{\vee}$ is generated by vectors $(1,0,0)$,
$(0,1,0)$, $(0,0,1)$ and $(1,1,-1)$, and for the functions
$\alpha=\chi^{(1,0,0)}$, $\beta=\chi^{(0,1,0)}$, $\gamma=\chi^{(0,0,1)}$ and
$\delta=\chi^{(1,1,-1)}$ we obtain
$\mathbb{K}[X]=\mathbb{K}[\alpha,\beta,\gamma,\delta]$ with relation
$\alpha\beta=\gamma\delta$.

$\displaystyle p_{1}=(1,0,0)$$\displaystyle
p_{2}=(0,1,0)$$N$$M$${\color[rgb]{0,0,0.78}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.78}\sigma}$${\color[rgb]{0,0,0.78}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.78}\sigma^{\vee}}$$0$$\displaystyle
p_{3}=(1,0,1)$$\displaystyle p_{4}=(0,1,1)$$0$

All vectors $p_{i}$ are equivalent up to an automorphism of the lattice $N$,
let $p=p_{1}$. The set of corresponding Demazure roots equals

$\mathfrak{R}_{1}=\\{(-1,k,l)\mid k\in\mathbb{Z}_{\geqslant
0},l\in\mathbb{Z}_{>0}\\},$

take $e_{1}=(-1,k_{1},l_{1})$ and $e_{2}=(-1,k_{2},l_{2})$. Then according to
Theorem 1 the multiplication of $x,y\in X$ is defined by formula

$\chi^{u}(x*y)=\chi^{u}(x)\chi^{u}(y)\,(\chi^{e_{1}}(y)+\chi^{e_{2}}(x))^{\langle
p,u\rangle},$

i.e.

$(\alpha_{x},\beta_{x},\gamma_{x},\delta_{x})*(\alpha_{y},\beta_{y},\gamma_{y},\delta_{y})=(\alpha_{x}\beta_{y}^{k_{1}}\gamma_{y}^{l_{1}}+\alpha_{y}\beta_{x}^{k_{2}}\gamma_{x}^{l_{2}},\,\beta_{x}\beta_{y},\,\gamma_{x}\gamma_{y},\,\delta_{x}\beta_{y}^{k_{1}+1}\gamma_{y}^{l_{1}-1}+\delta_{y}\beta_{x}^{k_{2}+1}\gamma_{x}^{l_{2}-1}).$

The proof of Theorem 1 is given in the next section. Let us specialize the
result to the case of affine space.

######  Theorem 2.

Any monoid structure of corank one on $\mathbb{A}^{n}$ up to a polynomial
change of variables is given by

$(x_{1},\ldots,x_{n})*(y_{1},\ldots,y_{n})=(x_{1}y_{1},\,\ldots,\,x_{n-1}y_{n-1},\,x_{1}^{a_{1}}\ldots
x_{n-1}^{a_{n-1}}y_{n}+y_{1}^{b_{1}}\ldots y_{n-1}^{b_{n-1}}x_{n}),$

where $a_{1},\ldots,a_{n-1},b_{1},\ldots,b_{n-1}\in\mathbb{Z}_{\geqslant 0}$.

###### Proof.

The cones $\sigma\subseteq N_{\mathbb{Q}}$ and $\sigma^{\vee}\subseteq
M_{\mathbb{Q}}$ of the affine space $\mathbb{A}^{n}$ are the positive octants
in the corresponding rational vector spaces. Since all rays of $\sigma$ are
equivalent up to automorphism, we can consider only one ray $\rho$ with the
primitive vector $p=(0,\ldots,0,1)\in N$. It remains to choose two Demazure
roots $e_{1}=(b_{1},\ldots,b_{n-1},-1)$, $e_{2}=(a_{1},\ldots,a_{n-1},-1)$,
$a_{1},\ldots,a_{n-1},b_{1},\ldots,b_{n-1}\in\mathbb{Z}_{\geqslant 0}$, and
use formula (4) for all basis vectors $u=u_{i}$, corresponding to coordinates
$x_{i}$ on $\mathbb{A}^{n}$, $1\leqslant i\leqslant n$. For $1\leqslant
i\leqslant n-1$ we have
$\chi^{u_{i}}(x*y)=\chi^{u_{i}}(x)\chi^{u_{i}}(y)=x_{i}y_{i}$, and for $i=n$
we obtain

$\chi^{u_{n}}(x*y)=\chi^{u_{n}}(x)\chi^{u_{n}}(y)(\chi^{e_{1}}(y)+\chi^{e_{2}}(x))=x_{n}y_{n}(y_{1}^{b_{1}}\ldots
y_{n-1}^{b_{n-1}}y_{n}^{-1}+x_{1}^{a_{1}}\ldots x_{n-1}^{a_{n-1}}x_{n}^{-1}).$

∎

##  5\. Proof of Theorem 1

The proof is divided into four subsections below. The first three subsections
contain the proof of the direct implication of the theorem, and the last
subsection provides to the converse implication.

###  5.1. Inclusion of cones

Let $X$ be an affine monoid of corank one. By Proposition 1, the variety $X$
is toric with respect to the torus $\mathbb{T}$, and the open subset
$G_{\chi}\subseteq X$ is invariant with respect to the torus $\mathbb{T}$,
i.e. is a toric variety as well. Let $X=X_{\sigma}$ for a cone
$\sigma\subseteq N_{\mathbb{Q}}$ and $G_{\chi}=X_{\rho}$ for a cone
$\rho\subseteq N_{\mathbb{Q}}$. As an algebraic variety, the group
$G_{\chi}=\mathbb{G}_{a}\leftthreetimes T$ is isomorphic to the direct product
of the line $\mathbb{G}_{a}$ and the torus $T$. Then the cone $\rho\subseteq
N_{\mathbb{Q}}$ is a ray and the dual cone $\rho^{\vee}\subseteq
M_{\mathbb{Q}}$ is a halfspace.

Notice that the inclusion $G_{\chi}=X_{\rho}\hookrightarrow X=X_{\sigma}$
restricts to the identity map on the open orbit of the torus action. So it
comes from the inclusion of semigroups $S_{\sigma}\subseteq S_{\rho}$ and of
cones $\rho\hookrightarrow\sigma$ as in subsection 3.2. Since
$X_{\rho}\subseteq X_{\sigma}$ is an open embedding, then $\rho$ is a face of
the cone $\sigma$, see subsection 3.2. So $\rho$ is a ray of the cone
$\sigma$.

The action of $\mathbb{T}=\mathop{\text{Im}}\theta$ on $G_{\chi}$ comes from
left and right multiplications of $T$ on $G_{\chi}$, see Proposition 1. If we
act on a point $(\alpha,t]\in G_{\chi}$ by an element
$\theta(t_{1},t_{2})\in\mathbb{T}$, $t_{1},t_{2}\in T$, we obtain a point

$\theta(t_{1},t_{2})\bigl{(}(\alpha,t]\bigr{)}=(0,t_{1}](\alpha,t](0,t_{2}^{-1}]=(\chi(t_{1})\alpha,\,t_{1}tt_{2}^{-1}].$
(5)

Notice that $\chi(t_{1})=\chi(t_{1}^{\prime})$ does not depend on a choice of
a representative
$\theta(t_{1},t_{2})=\theta(t_{1}^{\prime},t_{2}^{\prime})\in\mathbb{T}$ since
$\operatorname{Ker}\theta=\operatorname{diag}\operatorname{Ker}\chi$. So
$\theta(t_{1},t_{2})\mapsto\chi(t_{1})$ is a character of $\mathbb{T}$. It
follows that the coordinate function
$\boldsymbol{\alpha}\in\mathbb{K}[G_{\chi}]$ is homogeneous with respect to
grading (3) corresponding to the action of $\mathbb{T}$. The function
$\boldsymbol{\beta}=\chi^{-1}(t)\boldsymbol{\alpha}\in\mathbb{K}[G_{\chi}]$ is
homogeneous as well. Multiplying $\boldsymbol{\alpha}$ and
$\boldsymbol{\beta}$ by an appropriate scalar we may assume that
$\boldsymbol{\alpha}=\chi^{a}$ and $\boldsymbol{\beta}=\chi^{b}$ for some
$a,b\in S_{\rho}$. Notice that $\boldsymbol{\alpha}\in\mathbb{K}[G_{\chi}]$
and $1/\boldsymbol{\alpha}\notin\mathbb{K}[G_{\chi}]$, whence $a\in S_{\rho}$
does not belong to the face $\rho^{\perp}$, i.e. $\langle p,a\rangle>0$.

###  5.2. Comultiplication on $\mathbb{K}[G_{\chi}]$

Let $G$ be the group of invertible elements in $X$. Then $G$ is open in $X$
and $\mathbb{K}[X]$ is a subalgebra of $\mathbb{K}[G]$, and the
comultiplication on $\mathbb{K}[X]$ is the restriction of the comultiplication
$\mathbb{K}[G]\to\mathbb{K}[G]\otimes\mathbb{K}[G]$ corresponding to the
multiplication in the group of $G$.

Thus, we are interested in the comultiplication on $\mathbb{K}[G_{\chi}]$.

######  Lemma 3.

The comultiplication
$\mathbb{K}[G_{\chi}]\to\mathbb{K}[G_{\chi}]\otimes\mathbb{K}[G_{\chi}]$ is
given by

$\chi^{u}\mapsto\chi^{u}\otimes\chi^{u}\,(1\otimes\boldsymbol{\alpha}^{-1}+\boldsymbol{\beta}^{-1}\otimes
1)^{\langle p,u\rangle},$ (6)

where $u\in S_{\rho}$, $\mathbb{K}[G_{\chi}]=\\!\bigoplus\limits_{u\in
S_{\rho}}\\!\mathbb{K}\chi^{u}$.

###### Proof.

Consider the orbits of the action of the torus $\mathbb{T}$ on
$X_{\rho}=G_{\chi}=\mathbb{G}_{a}\leftthreetimes T$. The cone $\rho$ has two
faces $0$ and $\rho$, so according to subsection 3.3 there are two orbits
$X_{\rho}=O_{0}\cup O_{\rho}$. By formula (5), we see that
$O_{0}=\\{\boldsymbol{\alpha}\neq 0\\}$ is the open orbit and
$O_{\rho}=\\{\boldsymbol{\alpha}=0\\}=T$ is its complement. The torus
$\operatorname{Hom}(M(\rho),\mathbb{K}^{\times})$ for the sublattice
$M(\rho)=\rho^{\perp}\cap M$ in $M$ is identified with $O_{\rho}$, so
$\mathbb{K}[T]=\mathbb{K}[M(\rho)]$.

Since $\mathbb{K}[G_{\chi}]=\mathbb{K}[T][\boldsymbol{\alpha}]$, the semigroup
$S_{\rho}$ is generated by the lattice $M(\rho)$ and $a\in S_{\rho}$, where
$\boldsymbol{\alpha}=\chi^{a}$. Notice that $\langle
p,u_{1}+u_{2}\rangle=\langle p,u_{1}\rangle+\langle p,u_{2}\rangle$ and
$\chi^{u_{1}+u_{2}}=\chi^{u_{1}}\chi^{u_{2}}$. Then it is sufficient to check
formula (6) for $u\in M(\rho)$ and for $u=a$.

For $u\in M(\rho)$ we obtain $\chi^{u}\mapsto\chi^{u}\otimes\chi^{u}$ since
$\chi^{u}\in\mathbb{K}[T]$ are multiplied as the characters in $T\subseteq
G_{\chi}$. Notice that for $u\in M(\rho)$ formula (6) also turns into
$\chi^{u}\mapsto\chi^{u}\otimes\chi^{u}$ since $\langle p,u\rangle=0$.

By equation (1) we also have
$\boldsymbol{\alpha}\mapsto\boldsymbol{\alpha}\otimes
1+\chi(t)\otimes\boldsymbol{\alpha}$. Let us check that it coincides with
formula (6) for $\chi^{u}=\boldsymbol{\alpha}$, i.e. for $u=a$. Notice that

$\boldsymbol{\alpha}\otimes
1+\chi(t)\otimes\boldsymbol{\alpha}=\boldsymbol{\alpha}\otimes\boldsymbol{\alpha}\,(1\otimes\boldsymbol{\alpha}^{-1}+\chi(t)\boldsymbol{\alpha}^{-1}\otimes
1),$

where $\chi(t)\boldsymbol{\alpha}^{-1}=\boldsymbol{\beta}^{-1}$. The exponent
$\langle p,a\rangle$ is equal to $1$ since the sublattice $M(\rho)=\\{u\in
M\mid\langle p,u\rangle=0\\}$ and the element $a\in\rho^{\vee}$ generate a
semigroup $S_{\rho}$ in $M$ such that $S_{\rho}-S_{\rho}=M$. ∎

###  5.3. Derivations and Demazure roots

By Lemma 3, the comultiplication on $\mathbb{K}[X_{\sigma}]$ is the
restriction of the comultiplication on $\mathbb{K}[G_{\chi}]$, see equation
(6):

$\chi^{u}\mapsto\chi^{u}\otimes\chi^{u}\,(1\otimes\boldsymbol{\alpha}^{-1}+\boldsymbol{\beta}^{-1}\otimes
1)^{\langle p,u\rangle}.$

Denote $e_{1}=-a$ and $e_{2}=-b$, where $\boldsymbol{\alpha}=\chi^{a}$ and
$\boldsymbol{\beta}=\chi^{b}$, $a,b\in S_{\rho}$. We have seen in the proof of
Lemma 3 that $\langle p,a\rangle=1$, so $\langle p,e_{1}\rangle=-1$. In the
same way $\langle p,e_{2}\rangle=-1$.

Consider two $\mathbb{G}_{a}$-actions on $G_{\chi}$ coming from left and right
multiplication by $\mathbb{G}_{a}\subseteq G_{\chi}$:

$\displaystyle\mathbb{G}_{a}\times G_{\chi}\to G_{\chi},$
$\displaystyle\quad\alpha^{\prime}\cdot(\alpha,\,t]=(\alpha^{\prime}+\alpha,\,t];$
$\displaystyle G_{\chi}\times\mathbb{G}_{a}\to G_{\chi},$
$\displaystyle\quad[t,\beta)\cdot\beta^{\prime}=[t,\beta+\beta^{\prime}).$

To obtain the corresponding LNDs on
$\mathbb{K}[G_{\chi}]=\mathbb{K}[T][\boldsymbol{\alpha}]=\mathbb{K}[T][\boldsymbol{\beta}]$,
we differentiate dual actions of $\mathbb{G}_{a}$ on $\mathbb{K}[G_{\chi}]$ at
$\alpha^{\prime}=0$ and $\beta^{\prime}=0$:

$\delta_{l}=\frac{d}{d\boldsymbol{\alpha}};\quad\delta_{r}=\frac{d}{d\boldsymbol{\beta}}.$

Left and right multiplications on $G_{\chi}$ can be extended to $X_{\sigma}$.
Since the algebra $\mathbb{K}[X_{\sigma}]$ is invariant with respect to left
and right multiplications, the algebra $\mathbb{K}[X_{\sigma}]$ is also
invariant with respect to derivations $\delta_{l}$ and $\delta_{r}$. Notice
that $\delta_{l}$ and $\delta_{r}$ are homogeneous LNDs on $\mathbb{K}[X]$
with respect to the grading $\mathbb{K}[X_{\sigma}]=\\!\bigoplus\limits_{u\in
S_{\sigma}}\\!\mathbb{K}\chi^{u}$ with degrees $e_{1}$ and $e_{2}$
respectively since $\boldsymbol{\alpha}^{-1}=\chi^{e_{1}}$ and
$\boldsymbol{\beta}^{-1}=\chi^{e_{2}}$. It follows that $e_{1}$ and $e_{2}$
are Demazure roots of $X_{\sigma}$. Since $\langle p,e_{1}\rangle=\langle
p,e_{2}\rangle=-1$, they correspond to the ray $\rho$. Thus we proved that the
comultiplication on $\mathbb{K}[X_{\sigma}]$ is given by formula (4).

###  5.4. The inverse implication

First let us provide the proof of the following lemma.

######  Lemma 4.

Let $G$ be an affine algebraic group, and $G\hookrightarrow X$ be an open
embedding into an affine algebraic variety $X$. Let
$\mu\colon\mathbb{K}[G]\to\mathbb{K}[G]\otimes\mathbb{K}[G]$ be the
comultiplication on $\mathbb{K}[G]$ and
$\mu(\mathbb{K}[X])\subseteq\mathbb{K}[X]\otimes\mathbb{K}[X]$. Then the
restriction of $\mu$ to $\mathbb{K}[X]$ defines a comultiplication on
$\mathbb{K}[X]$ that corresponds to some monoid structure on $X$.

###### Proof.

If $\mu(\mathbb{K}[X])\subseteq\mathbb{K}[X]\otimes\mathbb{K}[X]$, then the
dual map $X\times X\to X$ is a morphism, which restricts to the group
multiplication $G\times G\to G$. Since the former is associative and admits a
unity, the same holds for $X\times X\to X$. ∎

Now let $X_{\sigma}$ be an affine toric variety, $p$ be the primitive vector
on a ray $\rho$ of the cone $\sigma\subseteq N_{\mathbb{Q}}$, and
$e_{1},e_{2}\in M$ be Demazure roots corresponding to $p$. First let us prove
that formula (4) defines a map to
$\mathbb{K}[X_{\sigma}]\otimes\mathbb{K}[X_{\sigma}]$. We have

$\chi^{u}\mapsto\chi^{u}\otimes\chi^{u}\,(1\otimes\chi^{e_{1}}+\chi^{e_{2}}\otimes
1)^{\langle p,u\rangle}=\sum\limits_{i+j=\langle
p,u\rangle}\begin{pmatrix}\langle p,u\rangle\\\
i\end{pmatrix}\chi^{u+ie_{2}}\otimes\chi^{u+je_{1}}.$ (7)

Notice that according to the definition of Demazure root $\langle
p,u+ie_{2}\rangle=\langle p,u\rangle-i\geqslant 0$, and for the primitive
vector $p^{\prime}$ on any other ray of the cone $\sigma$ we have $\langle
p^{\prime},u+ie_{2}\rangle=\langle p^{\prime},u\rangle+i\langle
p^{\prime},e_{2}\rangle\geqslant 0+0=0$ as well. So
$\chi^{u+ie_{2}}\in\mathbb{K}[X_{\sigma}]$. In the same way
$\chi^{u+je_{1}}\in\mathbb{K}[X_{\sigma}]$.

Consider the natural open embedding of the toric variety $X_{\rho}$ with the
cone $\rho$ into $X_{\sigma}$, see subsection 3.2. Let us show that the unity
of the given multiplication is the point $x_{\rho}$, see subsection 3.3.
Recall that by the definition of this point for $u\in S_{\rho}$ we have
$\chi^{u}(x_{\rho})=1$ if $u\in\rho^{\perp}$ and $\chi^{u}(x_{\rho})=0$
otherwise. Suppose $x\in X_{\sigma}$, $u\in S_{\sigma}$. Then

$\chi^{u}(x*x_{\rho})=\sum\limits_{i+j=\langle
p,u\rangle}\begin{pmatrix}\langle p,u\rangle\\\
i\end{pmatrix}\chi^{u+ie_{2}}(x)\chi^{u+je_{1}}(x_{\rho})=\chi^{u}(x).$

Indeed, the last equality holds since $\langle p,u+je_{1}\rangle=0$ if and
only if $j=\langle p,u\rangle$.

Let us prove that the set of invertible elements in the monoid $X_{\sigma}$ is
a subset of $X_{\rho}$. Consider a point $x\in X_{\sigma}$, and assume that
there exists the inverse $y=x^{-1}\in X_{\sigma}$. Then for any $u\in
S_{\sigma}$ in the relative interior of the facet
$\rho^{\perp}\cap\sigma^{\vee}$ we have $\langle p,u\rangle=0$, so formula (7)
gives

$\chi^{u}(x*y)=\chi^{u}(x)\chi^{u}(y),$

that has to equal $1$ since $\chi^{u}(x_{\rho})=\begin{cases}1\;\text{ if
}u\in\rho^{\perp}\\\ 0\;\text{ otherwise}\end{cases}$. So
$\chi^{u}(x)\in\mathbb{K}^{\times}$ if $u$ is in the relative interior of the
facet $\rho^{\perp}\cap\sigma^{\vee}$. According to the description of the
open subset $X_{\rho}$ in subsection 3.2, we obtain $x\in X_{\rho}$.

Finally, since $e_{1},e_{2}$ are Demazure roots with $\langle
p,e_{1}\rangle=\langle p,e_{2}\rangle=-1$, it follows that
$-e_{1},-e_{2}\in\rho^{\vee}$, so the characters
$\boldsymbol{\alpha}:=\chi^{-e_{1}}$ and $\boldsymbol{\beta}:=\chi^{-e_{2}}$
are regular functions on $X_{\rho}$. The restriction of comultiplication (4)
to the open subset $X_{\rho}$ coincides with formula (6), so all points in
$X_{\rho}$ are invertible and $X_{\rho}\cong G_{\chi}$, where
$\chi=\boldsymbol{\alpha}/\boldsymbol{\beta}$. Thus, $X_{\sigma}$ is a monoid
with the group of invertible elements $X_{\rho}$ of corank one by Lemma 4.
This completes the proof of Theorem 1.

##  6\. Algebraic properties

The classification of affine monoids of corank one from Theorem 1 allows us to
study algebraic properties of such monoids.

First recall some results of [3]. Let $X$ be an affine toric variety with an
action torus $\mathbb{T}$. For a Demazure root $e$ of $X$, denote by
$H_{e}\subseteq\operatorname{Aut}(X)$ the one-parameter subgroup normalized by
the torus $\mathbb{T}$ and corresponding $e$, i.e. to a homogeneous locally
nilpotent derivation with degree $e$. A pair of $\mathbb{T}$-orbits
$(O_{1},O_{2})$ is called $H_{e}$-connected if $H_{e}x\subseteq O_{1}\cup
O_{2}$ for some $x\in X\setminus X^{H_{e}}$.

######  Lemma 5 ([3, Lemma 2.2]).

A pair of $\mathbb{T}$-orbits $(O_{\tau_{1}},O_{\tau_{2}})$ is
$H_{e}$-connected if and only if $\langle\tau_{2},e\rangle\leqslant 0$,
$\tau_{1}$ is a facet of $\tau_{2}$ and $\tau_{1}=\tau_{2}\cap e^{\perp}$.

It follows that if a pair $(O_{1},O_{2})$ is $H_{e}$-connected, then up to
change of notation $\dim O_{1}=\dim O_{2}+1$,
$O_{2}\subseteq\overline{O_{1}}$, and one can take any $x\in O_{1}\cup O_{2}$.

Let $X$ be an affine monoid of corank one as in Theorem 1. The group of
invertible elements equals $G_{\chi}=\mathbb{G}_{a}\leftthreetimes\mathbb{T}$,
where left and right multiplications by $\mathbb{G}_{a}$ on $X$ correspond to
$e_{1}$ and $e_{2}$. So the orbits of the natural action of the group
$G_{\chi}\times G_{\chi}$ on $X$ are the unions of $H_{e_{1}}$\- and
$H_{e_{2}}$-connected $\mathbb{T}$-orbits.

Now, let us give the classification of idempotents in $X$, i.e. elements $x\in
X$ such that $x*x=x$.

######  Theorem 3.

Let $X=X_{\sigma}$ be an affine monoid of corank one as in Theorem 1. Then for
any face $\gamma$ of the cone $\sigma$ the set of idempotents in the orbit
$O_{\gamma}$ is as follows:

  1. (a)

$\\{x_{\gamma}\\}$ if $\rho$ is a ray of $\gamma$;

  2. (b)

$\varnothing$ if $\rho$ is not a ray of $\gamma$ and
$e_{1},e_{2}\notin\gamma^{\perp}$;

  3. (c)

$\varnothing$ if $\rho$ is not a ray of $\gamma$ and
$e_{1},e_{2}\in\gamma^{\perp}$;

  4. (d)

$O_{\gamma}\cap\,\\{\chi^{u}=1\;\;\forall
u\in\operatorname{cone}(\gamma,\rho)^{\perp}\cap S_{\sigma}\\}$ otherwise.

######  Remark 2.

Let $e$ be a Demazure root corresponding to the ray $\rho$ of the cone
$\sigma$, which is not a ray of the cone $\gamma$, and $e\in\gamma^{\perp}$.
Then $\operatorname{cone}(\gamma,\rho)$ is a face of $\sigma$, see subsection
3.4, and the $\mathbb{T}$-orbits corresponding to faces
$\operatorname{cone}(\gamma,\rho)$ and its facet $\gamma$ are
$H_{e}$-connected.

######  Example 4.

Let us consider an example, where all the cases of Theorem 3 are realized. Let
$X=\mathbb{A}^{4}$ and the multiplication rule is as follows:

$(x_{1},x_{2},x_{3},x_{4})*(y_{1},y_{2},y_{3},y_{4})=(x_{1}y_{1},x_{2}y_{2},x_{3}y_{3},x_{3}^{a}y_{4}+y_{2}^{b}y_{3}^{c}x_{4}),\quad
a,b,c>0.$ (8)

Denote a basis of $N$ by $s_{i}$, $1\leqslant i\leqslant 4$. According to the
proof of Theorem 2 the monoid corresponds to the cone
$\sigma=\operatorname{cone}(s_{1},s_{2},s_{3},s_{4})$, the ray $\rho$ with the
primitive vector ${p=s_{4}\in N}$ and Demazure roots $e_{1}=(0,b,c,-1)\in M$,
$e_{2}=(0,0,a,-1)\in M$.

For any subset $I\subseteq\\{1,2,3,4\\}$ we have the face
$\gamma=\gamma(I)=\operatorname{cone}\limits_{i\in I}s_{i}$ of the cone
$\sigma$, corresponding to the torus orbit

$O_{\gamma}=\\{(x_{1},x_{2},x_{3},x_{4})\mid x_{i}=0\text{ if }i\in I\text{\;
and \;}x_{i}\neq 0\text{ if }i\notin I\\}.$

The point $x_{\gamma}\in O_{\gamma}$ is the point with $i$-th coordinate $0$
if $i\in I$ and $1$ if $i\notin I$. Notice that $e_{1}\in\gamma^{\perp}$ if
and only if $I\subseteq\\{1\\}$, and $e_{2}\in\gamma^{\perp}$ if and only if
$I\subseteq\\{1,2\\}$.

Let us find the set of idempotents in an orbit $O_{\gamma}$,
$\gamma=\gamma(I)$, comparing results of Theorem 3 and direct computations. By
equation (8) a point $(x_{1},x_{2},x_{3},x_{4})\in\mathbb{A}^{4}$ is an
idempotent if

$(x_{1}^{2},x_{2}^{2},x_{3}^{2},x_{3}^{a}x_{4}+x_{2}^{b}x_{3}^{c}x_{4})=(x_{1},x_{2},x_{3},x_{4}).$
(9)

(a) Let $\rho$ be a ray of $\gamma$, i.e. $4\in I$ (there are eight such cones
$\gamma$). Then by Theorem 3 there exists exactly one idempotent
$x_{\gamma}=(x_{1},x_{2},x_{3},0)$ in $O_{\gamma}$, where $x_{i}=0$ if $i\in
I$ and $x_{i}=1$ otherwise. The same follows from equation (9) under the
assumption $x_{4}=0$.

(b) Let $\rho$ be not a ray of $\gamma$ and $e_{1},e_{2}\notin\gamma^{\perp}$,
i.e. $4\notin I$, $3\in I$. Then there are no idempotents in $O_{\gamma}$ by
Theorem 3. This agrees with formula (9): from $x_{4}\neq 0,x_{3}=0$ it follows
that $x_{3}^{a}x_{4}+x_{2}^{b}x_{3}^{c}x_{4}=0\neq x_{4}$.

(c) Let $\rho$ be not a ray of $\gamma$ and $e_{1},e_{2}\in\gamma^{\perp}$,
i.e. $I\subseteq\\{1\\}$. There are no idempotents in such $O_{\gamma}$ by
Theorem 3 as well. Also, from equation (9) and $x_{2},x_{3},x_{4}\neq 0$ it
follows that $x_{2}=x_{3}=1,\,x_{4}+x_{4}=x_{4},\,x_{4}\neq 0$, a
contradiction.

(d) Let $\rho$ be not a ray of $\gamma$ and
$e_{1}\notin\gamma^{\perp},e_{2}\in\gamma^{\perp}$, i.e. $I\subseteq\\{1,2\\}$
and $2\in I$. Consider two possible cases.

(d1) Let $I=\\{1,2\\}$, i.e. $O_{\gamma}=\\{(0,0,x_{3},x_{4})\mid
x_{3},x_{4}\in\mathbb{K}^{\times}\\}$. Denote by $u_{i},\,1\leqslant
i\leqslant 4$, the basis of $M$ dual to $s_{i},\,1\leqslant i\leqslant 4$. The
semigroup of integer points in the face

$\gamma^{\perp}\cap\rho^{\perp}\cap\sigma^{\vee}=\operatorname{cone}(u_{3},u_{4})\cap\operatorname{cone}(u_{1},u_{2},u_{3})=\operatorname{cone}(u_{3})$

of the dual cone $\sigma^{\vee}=\operatorname{cone}(u_{1},u_{2},u_{3},u_{4})$
is generated by $u_{3}$, so by Theorem 3 the set of idempotents in
$O_{\gamma}$ equals

$O_{\gamma}\cap\,\\{\chi^{u}=1\;\;\forall
u\in\gamma^{\perp}\cap\rho^{\perp}\cap S_{\sigma}\\}=\\{(0,0,1,x_{4})\mid
x_{4}\in\mathbb{K}^{\times}\\}.$

This agrees with equation (9), which is equivalent to $x_{3}=1$ under the
assumptions $x_{1}=x_{2}=0$, $x_{3},x_{4}\neq 0$.

(d2) Let $I=\\{2\\}$. In the same way we obtain

$\gamma^{\perp}\cap\rho^{\perp}\cap\sigma^{\vee}=\operatorname{cone}(u_{1},u_{3},u_{4})\cap\operatorname{cone}(u_{1},u_{2},u_{3})=\operatorname{cone}(u_{1},u_{3})$

and

$O_{\gamma}\cap\,\\{\chi^{u}=1\;\;\forall
u\in\gamma^{\perp}\cap\rho^{\perp}\cap S_{\sigma}\\}=\\{(1,0,1,x_{4})\mid
x_{4}\in\mathbb{K}^{\times}\\}.$

Gathering all cases together, we obtain that the set of idempotents consists
of two lines $(0,0,1,x_{4})$ and $(1,0,1,x_{4})$, $x_{4}\in\mathbb{K}$, and
six points

$(0,0,0,0),(0,1,0,0),(0,1,1,0),(1,0,0,0),(1,1,0,0),(1,1,1,0).$

It is known that irreducible commutative algebraic monoid in characteristic
zero has finite number of idempotents [14, Section 3.5.3]. This agrees with
our classification.

######  Corollary 1.

If $X=X_{\sigma}$ is a commutative monoid of corank one corresponding to a ray
$\rho$ of the cone $\sigma$, then the number of idempotents in $X$ is finite
and equals the number of faces of the cone $\sigma$ that contain the ray
$\rho$.

###### Proof.

If $X$ is commutative, then $e_{1}=e_{2}$ and the case $(d)$ is impossible. ∎

###### Proof of Theorem 3.

Let $x\in O_{\gamma}$ be an idempotent of the monoid. From now on $u\in
S_{\sigma}$. First recall that according to (7) for any $x_{1},x_{2}\in
X_{\sigma}$ the multiplication is given by

$\chi^{u}(x_{1}*x_{2})=\sum\limits_{i+j=\langle
p,u\rangle}\begin{pmatrix}\langle p,u\rangle\\\
i\end{pmatrix}\chi^{u+ie_{2}}(x_{1})\chi^{u+je_{1}}(x_{2}).$ (10)

Informally, $u+ie_{1}$ and $u+ie_{2}$, $0\leqslant i\leqslant\langle
p,u\rangle$, form two segments in $S_{\sigma}$ between $u$ and the facet
$\rho^{\perp}\cap\sigma^{\vee}$, see the pictures below. As $i$ increases, an
element $u+ie_{k}$, $k=1,2$, becomes closer to the facet
$\rho^{\perp}\cap\sigma^{\vee}$ and not closer to all other facets by
definition of Demazure root, see the proof of the converse part of Theorem 1.
In particular, $u+ie_{k}\in\rho^{\perp}$ if and only if $i=\langle
p,u\rangle$, where $k=1,2$.

First consider $u\in\rho^{\perp}$. Then the sum in (10) has only one summand,
so for the idempotent $x$ we have
$\chi^{u}(x)=\chi^{u}(x*x)=\chi^{u}(x)\chi^{u}(x)$. Thus

$\chi^{u}(x)=0,1\quad\text{ for any }u\in\rho^{\perp}.$ (11)

Recall that from $x\in O_{\gamma}$ it follows that $\chi^{u}(x)\neq 0$ if and
only if $u\in\gamma^{\perp}$.

(a) Let $\rho$ be a ray of the face $\gamma$. Then
$\gamma^{\perp}\subseteq\rho^{\perp}$, so according to the above we obtain
that $\chi^{u}(x)=\begin{cases}1\text{ if }u\in\gamma^{\perp}\\\ 0\text{
otherwise}\end{cases}$, i.e. $x=x_{\gamma}$.

Conversely, let us check that $x=x_{\gamma}$ is an idempotent, i.e.
$\chi^{u}(x_{\gamma}*x_{\gamma})=\chi^{u}(x_{\gamma})$ for any $u\in
S_{\sigma}$. If $u\in\rho^{\perp}$, then the sum in equation (10) consists of
one summand and
$\chi^{u}(x_{\gamma}*x_{\gamma})=\chi^{u}(x_{\gamma})\chi^{u}(x_{\gamma})=\chi^{u}(x_{\gamma})$
since $\chi^{u}(x_{\gamma})=0,1$. If $u\notin\rho^{\perp}$, then the sum in
equation (10) consists of more than one summand, but in every summand
$u+ie_{2}$ or $u+ie_{1}$ do not belong to $\rho^{\perp}$, i.e. every summand
equals zero. So if $u\notin\rho^{\perp}$, then
$\chi^{u}(x_{\gamma}*x_{\gamma})=0=\chi^{u}(x_{\gamma})$.

$0$$\rho^{\perp}$$\gamma^{\perp}$$u$0,1$0$$\rho^{\perp}$$\gamma^{\perp}$$u\notin\rho^{\perp}$0$e_{1}$00$e_{2}$00(a)
$\gamma^{\perp}\subseteq\rho^{\perp}$$0$$\rho^{\perp}$$\gamma^{\perp}$$u\in\gamma^{\perp}\setminus\rho^{\perp}$$\neq
0$$e_{1}$000$e_{2}$000(b)
$e_{1},e_{2}\notin\gamma^{\perp}$$0$$\rho^{\perp}$$\gamma^{\perp}$$\rho^{\perp}\cap\gamma^{\perp}$$\langle
p,u\rangle=1$$u\in\gamma^{\perp}$$\neq 0$$e_{1}$1$e_{2}$1(c)
$e_{1},e_{2}\in\gamma^{\perp}$$0$$\rho^{\perp}$$\gamma^{\perp}$$u\notin\gamma^{\perp}$$0$$e_{2}$000$e_{1}$000(d)
$e_{1}\notin\gamma^{\perp},e_{2}\in\gamma^{\perp}$$0$$\rho^{\perp}$$\gamma^{\perp}$$\rho^{\perp}\cap\gamma^{\perp}$$u\in\gamma^{\perp}$$\neq
0$$e_{2}$1$e_{1}$000(d) $e_{1}\notin\gamma^{\perp},e_{2}\in\gamma^{\perp}$

(b) Let $\rho$ be not a ray of the face $\gamma$ and
$e_{1},e_{2}\notin\gamma^{\perp}$. Since $e_{1}\notin\gamma$, there exists a
ray $\rho^{\prime}$ of $\gamma$ with the primitive vector $p^{\prime}$ such
that $\langle p^{\prime},e_{1}\rangle\neq 0$. Moreover,
$\rho^{\prime}\neq\rho$ since $\rho$ is not a ray of $\gamma$, so $\langle
p^{\prime},e_{1}\rangle>0$.

Take any $u\in\gamma^{\perp}\setminus\rho^{\perp}$. Then for any $1\leqslant
j\leqslant\langle p,u\rangle$ we have $\langle
p^{\prime},u+je_{1}\rangle=j\langle p^{\prime},u\rangle>0$, so
$u+je_{1}\notin\gamma^{\perp}$ and $\chi^{u+je_{1}}(x)=0$. In the same way
$\chi^{u+ie_{2}}(x)=0$ for $1\leqslant i\leqslant\langle p,u\rangle$. Since
$u\notin\rho^{\perp}$, it follows that the sum in $\chi^{u}(x*x)$ has at least
two summands and equals zero. However, $\chi^{u}(x)\neq 0$ since
$u\in\gamma^{\perp}$. Thus, there is no idempotents in case (b).

(c) Let $\rho$ be not a ray of the face $\gamma$ and
$e_{1},e_{2}\in\gamma^{\perp}$. Take $u\in\gamma^{\perp}$ such that $\langle
p,u\rangle=1$. Such $u\in S_{\sigma}$ exists since the rational plane
$\gamma^{\perp}\cap\\{\langle p,\,\cdot\,\rangle=1\\}$ contains a lattice
element $-e_{1}$, so one can find a lattice element in the intersection of
this plane and the cone $\sigma^{\vee}$ as well.

Then $u+e_{1},u+e_{2}\in\gamma^{\perp}\cap\rho^{\perp}$, whence
$\chi^{u+e_{1}}(x)=\chi^{u+ie_{2}}(x)=1$ and formula (10) gives for an
idempotent $x$ the equation
$\chi^{u}(x)=\chi^{u}(x*x)=\chi^{u}(x)+\chi^{u}(x)$. It follows
$\chi^{u}(x)=0$, which contradicts $u\in\gamma^{\perp}$.

(d) Let $\rho$ be not a ray of the face $\gamma$ and
$e_{1}\notin\gamma^{\perp}$, $e_{2}\in\gamma^{\perp}$. From
$e_{1}\notin\gamma^{\perp}$ it follows that there exists a ray of $\gamma$
with the primitive vector $p^{\prime}$ such that $\langle
p^{\prime},e_{1}\rangle>0$, see case (b). Consider two cases.

Let $u\notin\gamma^{\perp}$. Since $p^{\prime}\in\gamma$ and
$u\in\sigma^{\vee}\setminus\gamma^{\perp}$, we have

$\displaystyle\langle p^{\prime},u+je_{1}\rangle$ $\displaystyle>0+j\langle
p^{\prime},e_{1}\rangle\geqslant 0\quad\text{ for any }0\leqslant
j\leqslant\langle p,u\rangle,$ $\displaystyle\langle
p^{\prime},u+ie_{2}\rangle$ $\displaystyle>0+i\langle
p^{\prime},e_{2}\rangle\geqslant 0\quad\text{ for any }0\leqslant
i\leqslant\langle p,u\rangle,$

so

$\displaystyle u+je_{1}$ $\displaystyle\notin\gamma^{\perp}\quad\text{ for any
}0\leqslant j\leqslant\langle p,u\rangle,$ $\displaystyle u+ie_{2}$
$\displaystyle\notin\gamma^{\perp}\quad\text{ for any }0\leqslant
i\leqslant\langle p,u\rangle.$

Then $\chi^{u}(x)=0=\chi^{u}(x*x)$ for any $u\notin\gamma^{\perp}$.

Let $u\in\gamma^{\perp}$. In this case we have

$\displaystyle\langle p^{\prime},u+je_{1}\rangle$ $\displaystyle=0+j\langle
p^{\prime},e_{1}\rangle>0\quad\text{ for any }1\leqslant j\leqslant\langle
p,u\rangle$ $\displaystyle\langle p^{\prime},u+ie_{2}\rangle$
$\displaystyle=0+i\langle p^{\prime},e_{2}\rangle=0\quad\text{ for any
}0\leqslant i\leqslant\langle p,u\rangle,$

i.e.

$\displaystyle u+je_{1}$ $\displaystyle\notin\gamma^{\perp}\quad\text{ for any
}1\leqslant j\leqslant\langle p,u\rangle,$ $\displaystyle u+ie_{2}$
$\displaystyle\in\gamma^{\perp}\quad\text{ for any }0\leqslant
i\leqslant\langle p,u\rangle,$

Then for $x\in O_{\gamma}$ the condition $\chi^{u}(x)=\chi^{u}(x*x)$ is
equivalent to $\chi^{u}(x)=\chi^{u}(x)\chi^{u+\langle p,u\rangle e_{2}}(x)$,
i.e. $\chi^{u+\langle p,u\rangle e_{2}}(x)=1$. Notice that $u+\langle
p,u\rangle
e_{2}\in\rho^{\perp}\cap\gamma^{\perp}=\operatorname{cone}(\gamma,\rho)^{\perp}$,
so the condition $\chi^{u+\langle p,u\rangle e_{2}}(x)=1$ is equivalent (11).
This completes the proof. ∎

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