Affine algebraic monoids as endomorphisms' monoids of finite-dimensional algebras
Alexander Perepechko
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。
在这篇短文中，我们证明任何仿射代数幺半群均可作为一个有限维（非结合）代数的自同态幺半群获得。
In this note we prove that any affine algebraic monoid can be obtained as the endomorphisms' monoid of a finite-dimensional (nonassociative) algebra.
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In this paper we give some properties of the algebraic and geometric structure of the endomorphisms monoid of a homogeneous vector bundle.
In this paper we prove the Dynamical Mordell-Lang Conjecture for polynomial endomorphisms of the affine plane.
Let G be a connected complex reductive group. A well known theorem of I. Losev's says that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper, we use the combinatorial theory of spherical varieties and a smoothness criterion of R. Camus to characterize the weight monoids of smooth affine spherical varieties.
Let A^2 denote the affine plane over an algebraically closed field of arbitrary characteristic. Besides contributing several new results in the general theory of birational endomorphisms of A^2, this article describes certain classes of birational endomorphisms f defined by requiring that the missing curves or contracting curves of f are lines. The last part of the article is concerned with the monoid structure of the set of birational endomorphisms of A^2.
In this short note we prove that any irreducible algebraic monoid whose unit group is an affine algebraic group is affine.
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.
We study Alexeev and Brion's moduli scheme $M_Γ$ of affine spherical varieties with weight monoid $Γ$ under the assumption that $Γ$ is free. We describe the tangent space to $M_Γ$ at its `most degenerate point' in terms of the combinatorial invariants of spherical varieties and deduce that the irreducible components of $M_Γ$, equipped with their reduced induced scheme structure, are affine spaces.
We classify commutative algebraic monoid structures on normal affine surfaces over an algebraically closed field of characteristic zero. The answer is given in two languages: comultiplications and Cox coordinates. The result follows from a more general classification of commutative monoid structures of rank 0, n-1 or n on a normal affine variety of dimension n.
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 $n$.
Algebraic monoid structures on the affine 3-space
本文研究了仿射3空间上代数独异结构的分类，并探讨了相关代数性质。
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。
We show that two varieties X and Y with isomorphic endomorphism semigroups are isomorphic up to field automorphism if one of them is affine and contains a copy of the affine line. A holomorphic version of this result is due to the first author.
In this paper we prove the following theorem. Let $f:\mathbb{A}^2\rightarrow \mathbb{A}^2$ be a dominate polynomial endomorphisms defined over an algebraically closed field $k$ of characteristic $0$. If there are no nonconstant rational function $g:\mathbb{A}^2-rightarrow \mathbb{P}^1$ satisfying $g\circ f=g$, then there exists a point $p\in \mathbb{A}^2(k)$ whose orbit under $f$ is Zariski dense in $\mathbb{A}^2$. This result gives us a positive answer to a conjecture of Amerik, Bogomolov and Rovinsky ( and Zhang) for polynomial endomorphisms on the affine plane.
On the dynamics of endomorphisms of affine surfaces
将Favre-Jonsson的赋值理论从复仿射平面推广到任意域上的仿射曲面，构造特征赋值并证明第一动力学次数是次数≤2的代数整数，进而得到loxodromic自同构第一动力学次数的刚性结果。
We study commutative associative polynomial operations $\mathbb{A}^n\times\mathbb{A}^n\to\mathbb{A}^n$ with unit on the affine space $\mathbb{A}^n$ over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.
The affine Hecke category is a monoidal colimit
该研究证明仿射Hecke范畴等价于标准抛物子群对应的有限型Hecke子范畴的幺半稳定∞范畴余极限，其方法可推广到环群D-模等类似问题。
Endomorphisms of Weyl algebras are studied using bimodules. Initially, for a Weyl algebra over a field of characteristic zero, Bernstein's inequality implies that holonomic bimodules finitely generated from the right or left form a monoidal category. The most important bimodule in this paper is the graph of an endomorphism. We prove that the graph of an endomorphism of a Weyl algebra over a field of characteristic zero is a simple bimodule. The simplicity of the tensor product of the dual graph and the graph is equivalent to the Dixmier conjecture. It is also shown how the graph construction l…
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.
Making the motivic group structure on the endomorphisms of the projective line explicit
本文在 Jouanolou 装置到射影直线的带点朴素同伦类上构造显式群结构，并证明 Cazanave 幺半群可经自然态射群完备化到由简单态射生成的子群。
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}(…
On endomorphisms of affine spaces and the Jacobian problem
本文研究仿射空间平展自同态的纤维基数、像补集及几何次数，构造反例否证若干猜想，并证明雅可比猜想在特定复合条件下成立。
On the DML(1) property for regular endomorphisms of affine spaces: multiplicities of points at infinity
本文研究仿射空间正则自同态的动力学Mordell-Lang猜想，在无穷远点重数几何平均小于代数度的条件下，证明该猜想对曲线成立。
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.
We prove that every mapping torus of any free group endomorphism is residually finite. We show how to use a not yet published result of E. Hrushovski to extend our result to arbitrary linear groups. The proof uses algebraic self-maps of affine spaces over finite fields. In particular, we prove that when such a map is dominant, the set of its fixed closed scheme points is Zariski dense in the affine space.
Images of dominant endomorphisms of affine space
本文研究仿射空间自同态的像，证明对任意维数至多 n-2 的代数簇 Z，存在双有理簇 W 及满射态射 A^n 到 A^n\W，并给出猜想性方法。
Root monoids and active algebraic groups
本文通过正则函数代数上的余乘法刻画可逆元群为幂幺群与环面活跃半直积的仿射幺半群，并提出由 Demazure 根对构造的根幺半群概念。
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 …
Let $G$ be a connected reductive group over a perfect field $k$. We study a certain normal reductive monoid $\overline M$ associated to a parabolic $k$-subgroup $P$ of $G$. The group of units of $\overline M$ is the Levi factor $M$ of $P$. We show that $\overline M$ is a retract of the affine closure of the quasi-affine variety $G/U(P)$. Fixing a parabolic $P^-$ opposite to $P$, we prove that the affine closure of $G/U(P)$ is a retract of the affine closure of the boundary degeneration $(G \times G)/(P \times_M P^-)$. Using idempotents, we relate $\overline M$ to the Vinberg semigroup of $G$. …
On the DML(1) property for regular endomorphisms of affine spaces: the $\mathbb{G}_m$-case
本文证明在仿射空间正则自同态下，若不可约曲线与某点的轨道有无限交集且其正规化为$\mathbb{G}_m$，则该曲线是周期的，验证了动力Mordell-Lang猜想在该情形下的预期结论。
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 …
Given a connected reductive algebraic group $G$ and a finitely generated monoid $Γ$ of dominant weights of $G$, in 2005 Alexeev and Brion constructed a moduli scheme $\mathrm M_Γ$ for multiplicity-free affine $G$-varieties with weight monoid $Γ$. This scheme is equipped with an action of an `adjoint torus' $T_{\mathrm{ad}}$ and has a distinguished $T_{\mathrm{ad}}$-fixed point $X_0$. In this paper, we obtain a complete description of the $T_{\mathrm{ad}}$-module structure in the tangent space of $\mathrm M_Γ$ at $X_0$ for the case where $Γ$ is saturated. Using this description, we prove that t…
Let $G$ be a finite group acting effectively on the complex affine plane. If the $G$-action commutes with an étale endomorphism $f$ of the affine plane and the order of $G$ is even then the endomorphism $f$ is an automorphism.
Let $A$ be an abelian variety defined over $\bar{\mathbb{Q}}$, and let $φ$ be a dominant endomorphism of $A$ as an algebraic variety. We prove that either there exists a non-constant rational fibration preserved by $φ$, or there exists a point $x\in A(\bar{\mathbb{Q}})$ whose $φ$-orbit is Zariski dense in $A$. This provides a positive answer for abelian varieties of a question raised by Medvedev and the second author ("nvariant varieties for polynomial dynamical systems", Ann. of Math. (2) 179 (2014), no. 1, 81-177). We prove also a stronger statement of this result in which $φ$ is replaced by…
Let G be a complex connected reductive group. I. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical varieties to classify: (a) all such varieties when G is $\mathrm{SL}(2) \times \mathbb{C}^{\times}$ and (b) all such varieties for G simple which have a G-saturated weight monoid of full rank. We also use the characterization and F. Knop's classificat…
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…
We show that any normal algebraic monoid is an extension of an abelian variety by a normal affine algebraic monoid. This extends (and builds on) Chevalley's structure theorem for algebraic groups.
There is no Cazanave's Theorem for punctured affine space
本文证明卡扎纳夫关于射影直线自同态的朴素A¹-同伦类幺半群完备化定理不能推广到n≥2的穿孔仿射空间。
Cyclotomic integral points for affine dynamics
本文研究了高维仿射动力学中循环积分点的性质，证明了在特定条件下正则内射的迭代可表示为代数群内射的商，并应用结果分析了后向轨道和Hénon型自同构的周期点。
We study the geometry of algebraic monoids. We prove that the group of invertible elements of an irreducible algebraic monoid is an algebraic group, open in the monoid. Moreover, if this group is reductive, then the monoid is affine. We then give a combinatorial classification of reductive monoids by means of the theory of spherical varieties.
On injective endomorphisms of symbolic schemes
本文将 Gromov 的符号代数簇自同态理论推广至概形上的元胞自动机，研究闭像、满射、可逆等性质，并否定了逆非代数的双射复仿射代数元胞自动机的存在性。
Dynamical Mordell-Lang conjecture for split self-maps of affine curve times projective curve
本文证明了在仿射曲线与投影曲线乘积上，关于有理数闭包上内射映射的动力学Mordell-Lang猜想。
We compute the Andre-Quillen cohomology of an affine toric variety. The best results are obtained either in the general case for the first three cohomology groups, or in the case of isolated singularities for all cohomology groups, respectively.
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