The structure of normal algebraic monoids

Local structure of algebraic monoids

We describe the local structure of an irreducible algebraic monoid $M$ at an idempotent element $e$. When $e$ is minimal, we show that $M$ is an induced variety over the kernel $MeM$ (a homogeneous space) with fibre the two-sided stabilizer $M_e$ (a connected affine monoid having a zero element and a dense unit group). This yields the irreducibility of stabilizers and centralizers of idempotents when $M$ is normal, and criteria for normality and smoothness of an arbitrary $M$. Also, we show that $M$ is an induced variety over an abelian variety, with fiber a connected affine monoid having a de…

Observable subgroups of algebraic monoids

A closed subgroup H of the affine, algebraic group G is called observable if G/H is a quasi-affine algebraic variety. In this paper we define the notion of an observable subgroup of the affine, algebraic monoid M. We prove that a subgroup H of G is observable in M if and only if H is closed in M and there are "enough" H-semiinvariant functions in K[M]. We show also that a closed, normal subgroup H of G (the unit group of M) is observable in M if and only if it is closed in M. In such a case there exists a determinant $χ: M \to K$ such that $H\subset ker(χ)$. As an application, we show that in …

Classification of noncommutative monoid structures on normal affine surfaces

In 2021, Dzhunusov and Zaitseva classified two-dimensional normal affine commutative algebraic monoids. In this work, we extend this classification to noncommutative monoid structures on normal affine surfaces. We prove that two-dimensional algebraic monoids are toric. We also show how to find all monoid structures on a normal toric surface. Every such structure is induced by a comultiplication formula involving Demazure roots. We also give descriptions of opposite monoids, quotient monoids, and boundary divisors.

The ring of regular functions of an algebraic monoid

Let M be an irreducible normal algebraic monoid with unit group G. It is known that G admits a Rosenlicht decomposition, G=G_antG_aff, where G_ant is the maximal anti-affine subgroup of G, and G_aff the maximal normal connected affine subgroup of G. In this paper we show that this decomposition extends to a decomposition M=G_antM_aff, where M_aff is the affine submonoid M_aff=\bar{G_aff}. We then use this decomposition to calculate $\mathcal{O}(M)$ in terms of $\mathcal{O}(M_aff)$ and G_aff, G_ant\subset G. In particular, we determine when M is an anti-affine monoid, that is when $\mathcal{O}(…

Algebraic monoids with affine unit group are affine

In this short note we prove that any irreducible algebraic monoid whose unit group is an affine algebraic group is affine.

On Algebraic Semigroups and Monoids

We present some fundamental results on (possibly nonlinear) algebraic semigroups and monoids. These include a version of Chevalley's structure theorem for irreducible algebraic monoids, and the description of all algebraic semigroup structures on curves and complete varieties.

仿射代数幺半群作为有限维代数的自同态幺半群

Affine algebraic monoids as endomorphisms' monoids of finite-dimensional algebras

本文证明了任意仿射代数幺半群均可实现为某个有限维非结合代数的自同态幺半群。

Algebraic K-theory and Grothendieck-Witt theory of monoid schemes

We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $K$-theory space of an integral monoid scheme $X$ in terms of its Picard group $\operatorname{Pic}(X)$ and pointed monoid of regular functions $Γ(X, \mathcal{O}_X)$ and a description of the Grothendieck-Witt space of $X$ in terms of an additional involution on $\operatorname{Pic}(X)$. We also prove space-level projective bundle formulae in both settings.

Manifold structures in regular irreducible algebraic monoids

In this paper we study regular irreducible algebraic monoids over $\fldc$ equipped with the euclidean topology. It is shown that, in such monoids, the Green classes and the spaces of idempotents in the Green classes all have natural manifold structures. The interactions of these manifold structures and the semigroup structures in these monoids have been investigated. Relations between these manifolds and Grassmann manifolds have been established. A generalisation of a result on the dimension of the manifold of rank $k$ idempotents in the semigroup of linear endomorphisms over $\fldc$ has been …

Picard Groups and Class Groups of Monoid Schemes

We define and study the Picard group of a monoid scheme and the class group of a normal monoid scheme. To do so, we develop some ideal theory for (pointed abelian) noetherian monoids, including primary decomposition and discrete valuations. The normalization of a monoid turns out to be a monoid scheme, but not always a monoid.

An analogue of a reductive algebraic monoid, whose unit group is a Kac-Moody group

By a generalized Tannaka-Krein reconstruction we associate to the admissible representations of the category O of a Kac-Moody algebra, and its category of admissible duals a monoid with a coordinate ring. The Kac-Moody group is the Zariski open dense unit group of this monoid. The restriction of the coordinate ring to the Kac-Moody group is the algebra of strongly regular functions introduced by Kac and Peterson. This monoid has similar structural properties as a reductive algebraic monoid. In particular it is unit regular, its idempotents related to the faces of the Tits cone. It has Bruhat a…

$\mathbb A^1$-connectivity on Chow monoids v.s. rational equivalence of algebraic cycles

Let $k$ be a field of characteristic zero, and let $X$ be a projective variety embedded into a projective space over $k$. For two natural numbers $r$ and $d$ let $C_{r,d}(X)$ be the Chow scheme parametrizing effective cycles of dimension $r$ and degree $d$ on the variety $X$. An effective $r$-cycle of minimal degree on $X$ gives rise to a chain of embeddings of $C_{r,d}(X)$ into $C_{r,d+1}(X)$, whose colimit is the connective Chow monoid $C_r^{\infty }(X)$ of $r$-cycles on $X$. Let $BC_r^{\infty }(X)$ be the motivic classifying space of this monoid. In the paper we establish an isomorphism bet…

代数磁性

Algebraic Magnetism

本文针对可对角化幺半群概形作用于代数空间的情形，通过引入子幺半群对应的吸引子空间，系统研究了幺半群吸引子的性质。

The endomorphisms monoid of a homogeneous vector bundle

In this paper we give some properties of the algebraic and geometric structure of the endomorphisms monoid of a homogeneous vector bundle.

On Algebraic Semigroups and Monoids, II

Consider an algebraic semigroup $S$ and its closed subscheme of idempotents, $E(S)$. When $S$ is commutative, we show that $E(S)$ is finite and reduced; if in addition $S$ is irreducible, then $E(S)$ is contained in a smallest closed irreducible subsemigroup of $S$, and this subsemigroup is an affine toric variety. It follows that $E(S)$ (viewed as a partially ordered set) is the set of faces of a rational polyhedral convex cone. On the other hand, when $S$ is an irreducible algebraic monoid, we show that $E(S)$ is smooth, and its connected components are conjugacy classes of the unit group.

Twisted forms of commutative monoid structures on affine spaces

In this paper, we study affine commutative algebraic monoid structures on affine spaces over an arbitrary field of characteristic zero. We obtain full classification of such structures on $\mathbb{A}_K^2$ and $\mathbb{A}_K^3$ and describe some generalizations on $\mathbb{A}_K^n$ for any dimension $n$.

余秩一的仿射幺半群

Affine monoids of corank one

本文分类了正规仿射簇上可逆元群连通可解且具有一维幂幺根的非交换代数幺半群结构，描述了其幂等元与中心，并给出零元存在的判据。

Toric varieties, monoid schemes and $cdh$ descent

We give conditions for the Mayer-Vietoris property to hold for the algebraic K-theory of blow-up squares of toric varieties in any characteristic, using the theory of monoid schemes. These conditions are used to relate algebraic K-theory to topological cyclic homology in characteristic p. To achieve our goals, we develop for monoid schemes many notions from classical algebraic geometry, such as separated and proper maps.

A universality theorem for Voevodsky's algebraic cobordism spectrum

An algebraic version of a theorem due to Quillen is proved. More precisely, for a ground field k we consider the motivic stable homotopy category SH(k) of P^1-spectra equipped with the symmetric monoidal structure described in arXiv:0709.3905v1 [math.AG]. The algebraic cobordism P^1-spectrum MGL is considered as a commutative monoid equipped with a canonical orientation. For a commutative monoid E in the category SH(k) we identify the set of monoid homomorphisms from MGL to E in the motivic stable homotopy category with the set of all orientations of E. This result was stated originally in a s…

On the algebraic cobordism spectra MSL and MSp

We construct algebraic cobordism spectra MSL and MSp. They are commutative monoids in the category of symmetric T^{2}- spectra. The spectrum MSp comes with a natural symplectic orientation given either by a tautological Thom class th^{MSp} in MSp^{4,2}(MSp_{2}), a tautological Borel class b_{1}^{MSp} in MSp^{4,2}(HP^{\infty}) or any of six other equivalent structures. For a commutative monoid E in the category SH(S) we prove that assignment g -> g(th^{MSp}) identifies the set of homomorphisms of monoids g : MSp -> E in the motivic stable homotopy category SH(S) with the set of tautological Tho…

Non-normal affine monoids, modules and Poincaré series of plumbed 3-manifolds

We construct a non-normal affine monoid together with its modules associated with a negative definite plumbed $3$-manifold $M$. In terms of their structure, we describe the $H_1(M,\mathbb{Z})$-equivariant parts of the topological Poincaré series. In particular, we give combinatorial formulas for the Seiberg--Witten invariants of $M$ and for polynomial generalizations defined in a previous paper of the authors.

Monoid Embeddings of Symmetric Varieties

We determine when an antiinvolution on an adjoint semisimple linear algebraic group extends to an antiinvolution on a $J$-irreducible monoid. Using this information, we study a special class of compactifications of symmetric varieties. Extending the work of Springer on involutions, we describe the parametrizing sets of Borel orbits in these special embeddings.

代数 $n$ 值幺半群在 $\nathbb{C}P^1$ 上、判别式与射影对偶

Algebraic $n$-Valued Monoids on $\mathbb{C}P^1$, Discriminants and Projective Duality

本文建立代数 $n$ 值幺半群与射影对偶的联系，证明对偶与莫比乌斯变换复合给出移位运算，并将费马曲线映到由加法多项式定义的曲线，且三次曲线构造的陪集加法法则均为多项式。

关于矩阵合同相关的线性代数幺半群

On the Linear Algebraic Monoids Associated to Congruence of Matrices

本文研究任意域上广义合同方程，通过幺半群作用完全刻画了使解集成为李群的矩阵，并给出了幂零矩阵情形下李群结构、李代数及不变量的有限生成性。

From monoids to hyperstructures: in search of an absolute arithmetic

We show that the trace formula interpretation of the explicit formulas expresses the counting function N(q) of the hypothetical curve C associated to the Riemann zeta function, as an intersection number involving the scaling action on the adele class space. Then, we discuss the algebraic structure of the adele class space both as a monoid and as a hyperring. We construct an extension R^{convex} of the hyperfield S of signs, which is the hyperfield analogue of the semifield R_+^{max} of tropical geometry, admitting a one parameter group of automorphisms fixing S. Finally, we develop function th…

Algebraic homotopy classes of rational functions

We compute the set of naive pointed homotopy classes of endomorphisms of the projective line P^1 over the spectrum of a field. Our computation compares well with Fabien Morel's one of the motivic pointed homotopy classes of endomorphisms of P^1: there is an a priori monoid structure on the set of naive homotopy classes such that canonical map from this monoid to group of motivic homotopy classes is a group completion.

Algebraic cones

A characterization of algebraic cones in terms of actions of the one-dimensional multiplicative algebraic monoid ${\bf M}_{\rm m}$ and the algebraic group ${\bf G}_{\rm m}$ are given.

The monoid of families of quiver representations

A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, i.e. subvarieties of the varieties of representations. The study of this monoid leads to interesting interactions between representation theory, algebraic geometry and quantum group theory. For example, it produces a wealth of interesting examples of families of quiver representations, which can be analyzed by representation theoretic and geometric methods. Conversely, results from representation theory, in particular A. Schofield's work on general properties of quiv…

Incidence Monoids: Automorphisms and Complexity

The algebraic monoid structure of an incidence algebra is investigated. We show that the multiplicative structure alone determines the algebra automorphisms of the incidence algebra. We present a formula that expresses the complexity of the incidence monoid with respect to the two sided action of its maximal torus in terms of the zeta polynomial of the poset. In addition, we characterize the finite (connected) posets whose incidence monoids have complexity $\leq 1$. Finally, we determine the covering relations of the adherence order on the incidence monoid of a star poset.