Sushil Bhunia, Ayan Mahalanobis, Pralhad Shinde, Anupam Singh
This paper presents some algorithms in linear algebraic groups. These algorithms solve the word problem and compute the spinor norm for orthogonal groups. This gives us an algorithmic definition of the spinor norm. We compute the double coset decomposition with respect to a Siegel maximal parabolic subgroup, which is important in computing infinite-dimensional representations for some algebraic groups.
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Algebraic cohomotopy groups, algebraic fundamental groups and stably free modules
本文研究了代数上同调群与稳定自由模的关系，证明了拓扑结果的代数对应，并展示了非自由稳定自由模的例子。
We study topological group theoretic properties of algebraic groups over local fields. In particular, we find conditions under which such groups have closed images under arbitrary continuous homomorphisms into arbitrary topological groups.
Lie groups, algebraic groups and lattices
这是对李群、代数群及其离散子群的理论的简要介绍，这是基于2011年夏季在波兰巴拿赫中心举行的暑期学校期间给出的系列讲座。
On minimal epimorphic subgroups in simple algebraic groups of rank $2$
本文研究了秩 $2$ 单代数群中的极小满射子群问题，证明了在代数闭域上其闭满射子群的极小维数为 $3$。
The $R_{\infty}$ and $S_{\infty}$ properties for linear algebraic groups
本文研究了约化线性代数群的自同构的扭共轭类和扭等距类。证明了零特征域上的约化线性代数群具有$R_\infty$和$S_\infty$性质.
For simple algebraic groups defined over algebraically closed fields of good characteristic, we give upper bounds on the covering numbers of unipotent conjugacy classes in terms of their (co)ranks and in terms of their dimensions.
The main purpose of this paper is to give a new definition for the notion of group-groupoid. Also, several basic properties of group-groupoids are established.
Let G be an algebraic group over a complete separable valued field k. We discuss the dynamics of the G-action on spaces of probability measures on algebraic G-varieties. We show that the stabilizers of measures are almost algebraic and the orbits are separated by open invariant sets. We discuss various applications, including existence results for algebraic representations of amenable ergodic actions. The latter provides an essential technical step in the recent generalization of Margulis-Zimmer super-rigidity phenomenon due to Bader and Furman.
We show that any adjoint absolutely simple linear algebraic group over a field of characteristic zero is the automorphism group of some projector on a central simple algebra. Projective homogeneous varieties can be described in these terms; in particular, we reproduce quadratic equations by Lichtenstein defining them.
Computing in unipotent and reductive algebraic groups
幂幺群是一类重要的代数群。我们表明，用于计算的技术与生成的幂零群结转到幂幺群。我们特别集中在分裂还原群的最大幂幺子群上，并展示了这如何提高还原群本身的计算能力。
A note on real algebraic groups
证明了利用复化来理解实代数群的结构的有效性。特别地，证明了如下结果：a）若L是连通实代数群G的代数子群，使得L的复化包含G的复化的极大环面，则L包含G的Cartan子群b）设G是特征值全为实的可解实代数群.如果G的复化代数地作用在复簇V上，并且某个G轨道是紧的，则这个轨道是一个点。
Let $k$ be an algebraically closed field, $G$ a linear algebraic group over $k$ and $φ\in Aut(G)$, the group of all algebraic group automorphisms of $G$. Two elements $x, y$ of $G$ are said to be $φ$-twisted conjugate if $y=gxφ(g)^{-1}$ for some $g\in G$. In this paper we prove that for a connected non-solvable linear algebraic group $G$ over $k$, the number of its $φ$-twisted conjugacy classes is infinite for every $φ\in Aut(G)$.
Let $G$ be a linear algebraic group over an algebraically closed field $k$ and $\mathrm{Aut}_{\mathrm{alg}}(G)$ the group of all algebraic group automorphisms of $G$. For every $φ\in \mathrm{Aut}_{\mathrm{alg}}(G)$ let $\mathcal{R}(φ)$ denote the set of all orbits of the $φ$-twisted conjugacy action of $G$ on itself (given by $(g,x)\mapsto gxφ(g^{-1})$, for all $g,x\in G$). We say that $G$ has the algebraic $R_\infty$-property if $\mathcal{R}(φ)$ is infinite for every $φ\in \mathrm{Aut}_{\mathrm{alg}}(G)$. In \citep{bb} we have shown that this property is satisfied by every connected non-solva…
We determine all finite subgroups of simple algebraic groups that have irreducible centralizers - that is, centralizers whose connected component does not lie in a parabolic subgroup.
A subgroup of a group $G$ is called algebraic if it can be expressed as a finite union of solution sets to systems of equations. We prove that a non-elementary subgroup $H$ of an acylindrically hyperbolic group $G$ is algebraic if and only if there exists a finite subgroup $K$ of $G$ such that $C_G(K) \leq H \leq N_G(K)$. We provide some applications of this result to free products, torsion-free relatively hyperbolic groups, and ascending chains of algebraic subgroups in acylindrically hyperbolic groups.
Algebraic monoids with affine unit group are affine
本文证明了单位群为仿射代数群的不可约代数幺半群是仿射的。
Linear Bounds for Cohomology of Algebraic Groups
本文在素特征代数闭域上，对单代数群的有理上同调维数给出了显式的线性上界。
The focus of this article is to define the descriptively approximations in proximal relator spaces. Afterwards, descriptive approximately algebraic structures such as groupoids, semigroups and groups in digital images endowed with descriptive Efremovic proximity relations were introduced.
Algebraic properties of profinite groups
近年来，关于投射有限群的研究取得了许多进展。我们调查一些新的结果，并讨论开放的问题。一个中心主题是有限群分解成各种子集的有界积，从而产生拓扑群的代数性质。
Extensions of Algebraic Groups
设$G$是一个连通复代数群，$A$是一个连通交换代数群，通过群自同构赋予$G$一个代数作用.本文用一个短的正合序列描述了$A$对$G$的代数群扩张的交换群$\Ext_{alg}(G,A)$，该正合序列将交换群的下群与相对的第二李代数上同调空间和交换群的基本群联系起来.我们的第二个主要结果是代数群上同调的Van-Est定理的一个类似物，即对于一个代数$G$模$\a$和$p \geq 0$，代数群上同调$H^p_{alg}(G,\a)$是由李代数$\g$相对于极大约化子群的李代数的相对上同调给出的。
Arithmetic of algebraic groups
本文研究线性代数群的算术。它涉及研究性质的线性代数群定义的全球领域，局部域和有限域，或更普遍的研究线性代数群定义的领域承认任意循环扩张。
Examples of algebraic groups of type G2 having the same maximal tori
提出了一个关于A. Rapinchuk，我们构造了G2型非同构半单代数群H1和H2的例子，它们具有极大k-环面的相干等价系统。
The additivity with respect to exact sequences is notoriously a fundamental property of the algebraic entropy of group endomorphisms. It was proved for abelian groups by deeply exploiting their structure. On the other hand, a solvable counterexample was recently found, showing that it does not hold in general. Nevertheless, we give a rather short proof of the additivity of the algebraic entropy for locally finite groups that are either quasihamiltonian or FC-groups.
Epimorphic subgroups of simple algebraic groups
本文研究了线性代数群的满射子群，为任意特征代数闭域上的单代数群构造了维数至多为五的有界维数满射子群。
We show that, in a weakly regular $p$-adic Lie group $G$, the subgroup $G_u$ spanned by the one-parameter subgroups of $G$ admits a Levi decomposition. As a consequence, there exists a regular open subgroup of $G$ which contains $G_u$.
Representatives of elliptic Weyl group elements in algebraic groups
外尔群W的元素w称为椭圆的，如果它在标准反射表示中没有特征值1.我们确定了椭圆元w的半单代数群G中的任意代表g在相应的外尔群W中的阶。特别地，若w的阶为d，且G是不同于Cn或F4的单形，则g在G中的阶为d.
Algebraic entropy of elementary amenable groups
证明了任意代数熵为零的初等顺从群都含有有限指数的幂零子群，等价地，任意指数增长性的初等顺从群都是一致指数增长性的.我们还证明了0是初等顺从群的熵集的一个聚点。
Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems
基于相对等周不等式，提出了一种研究相对双曲群的新方法。各种几何，代数和算法的属性进行了讨论。
Subgroups of simple algebraic groups containing regular tori, and irreducible representations with multiplicity 1 non-zero weights
我们的主要目标是在一定的限制下，确定包含正则环面的单代数群的极大闭连通子群。如果环面的中心化子是交换的，我们称之为正则环面。我们也得到了一些独立的利益的结果。特别是，我们确定的不可约表示的简单代数群的非零重量发生的重数为1。
Embeddings of algebraic groups in Kac-Moody groups
设$k_1,k_2$是特征为0的两个字段。设$G_1$是$k_1$上的分裂半单代数群，$G_2$是$k_2$上的分裂Kac-Moody群，$ϕ\colon G_1(k_1)\to G_2(k_2)$是抽象嵌入.证明了当$k_1$是有理数的代数扩张时，$\im ϕ$是有界子群，而当$k_1$在有理数上具有无穷超越度时，存在无界象嵌入.
Nonvanishing of algebraic entropy for geometrically finite groups of isometries of Hadamard manifolds
我们证明了捏挤Hadamard流形的任何几何有限（非初等）等距群都具有均匀的指数增长性。
Constructing algebraic groups from their Lie algebras
特征为0的连通代数群由它的李代数唯一确定。本文给出了一个构造特征为0的代数群的算法，给出了它的李代数。利用这一方法给出了求代数群的极大约化子群和幂幺根的算法。
Higher dimensional algebraic fiberings of group extensions
本文给出群扩张高维代数纤维化的存在条件，并由此推出群不连贯性推论及若干代数纤维的几何实例。
Let $G$ be a permutation group on a set $Ω$. A subset of $Ω$ is a base for $G$ if its pointwise stabilizer is trivial; the base size of $G$ is the minimal cardinality of a base. In this paper we initiate the study of bases for algebraic groups defined over an algebraically closed field. In particular, we calculate the base size for all primitive actions of simple algebraic groups, obtaining the precise value in almost all cases. We also introduce and study two new base measures, which arise naturally in this setting. We give an application concerning the essential dimension of simple algebraic…
Products and commutators of classes in algebraic groups
本文对几乎单代数群中的共轭类对进行分类，其乘积由多个类组成。这导致了几个有趣的家庭的例子，这是有关的推广贝尔-铃木定理的有限群。我们还回答了一个问题的帕维尔舒米亚茨基对单代数群的共轭类的分解。事实证明，所得到的例子正是那些产品也只包含1000个类的例子。
We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to the normalizer property or to uniqueness of Sylow subgroups. As a consequence, we show algebraic decompositions of o-minimal nilpotent groups, and we prove that a nilpotent Lie group is definable in an o-minimal expansion of the reals if and only if it is a linear algebraic group.
In this paper, we attempt to develop the Quillen Suslin theory for the algebraic fundamental group of a ring. We give a surjective group homomorphism from the algebraic fundamental group of the field of the real numbers to the group of integers. At the end of the paper, we also propose some problems related to the algebraic fundamental group of some particular type of rings.
Let $G$ be a group and let $V$ be an algebraic group over an algebraically closed field. We introduce algebraic group subshifts $Σ\subset V^G$ which generalize both the class of algebraic sofic subshifts of $V^G$ and the class of closed group subshifts over finite group alphabets. When $G$ is a polycyclic-by-finite group, we show that $V^G$ satisfies the descending chain condition and that the notion of algebraic group subshifts, the notion of algebraic group sofic subshifts, and that of algebraic group subshifts of finite type are all equivalent. Thus, we obtain extensions of well-known resul…
On the Genus number of Algebraic groups
我们计算了代数闭域上单连通单代数群以及紧单连通单李群的轨道类型数。我们还计算了这些群在它们的李代数上的伴随作用的轨道类型的数目。我们还证明了连通约化代数群的亏格数与其半单部分的亏格数一致。
Linear Algebraic Groups as Parameterized Picard-Vessiot Galois Groups
本文证明了对于一定的微分域k，线性代数群是k（x）的参数化Picard-Vessiot扩张的伽罗瓦群，x' = 1，当且仅当它的单位连通区没有一维商作为线性代数群.
Algebraic groups generated by semisimple elements
本文刻画了特征零代数闭域上连通线性代数群中由所有半单元素生成的子群。
On extensions of algebraic groups with finite quotient
得到了代数群的有限代数元的一个提升性质，并应用于这类群的结构。
Algebraic groups over free and hyperbolic groups
研究自由群和双曲群上一致群的定义与性质，通过词映射描述群律，对不可约变种给出完整描述。
Alperin's Conjecture for Algebraic Groups
我们证明了类似物还原代数群的一些结果有限群由于诺尔和罗宾逊发挥了核心作用，他们的重新制定Alperin的猜想有限群。
Anti-affine algebraic groups
我们说域上的代数群$G$是反仿射的，如果$G$上的每个正则函数都是常数。我们得到这些群体的分类，与应用程序的结构代数群的积极特征，并建设许多反例希尔伯特的第十四个问题。
A Proof of the Barsotti-Chevalley Theorem on Algebraic Groups
Barsotti和Chevalley的一个基本定理指出，完满域上的每一个光滑代数群都是阿贝尔簇被光滑仿射代数群的扩展。在1956年Rosenlicht了一个简短的证明定理。在这篇简要的文章中，我们用现代代数几何的语言解释他的证明。
Reflection groups in algebraic geometry
这是一个关于代数几何中实反射群和复反射群出现的综述。我们还包括反射群理论的简要介绍。
We say that a smooth algebraic group $G$ over a field $k$ is very special if for any field extension $K/k$, every $G_K$-homogeneous $K$-variety has a $K$-rational point. It is known that every split solvable linear algebraic group is very special. In this note, we show that the converse holds, and discuss its relationship with the birational classification of algebraic group actions.
In this paper, we study group equations with occurrences of automorphisms. We describe equational domains in this class of equations. Moreover, we solve a number of open problem posed in universal algebraic geometry.
Jordan groups and algebraic surfaces
证明了关于一般线性群的有限子群的Jordan定理的一个类似定理对代数曲面的双正则自同构群也成立。这就肯定地回答了弗拉基米尔·L.波波夫
Regular Algebraic $K$-theory for groups -- Part I
本文提出群的正则代数K理论，作为离散群的一种新同调理论，与群同调相关但不同，并通过函子映射为环给出代数K理论版本。
On maximal tori of algebraic groups of type G_2
给定域k上的一个八元数代数，它的自同构群G是一个G_2型代数半单k-群。本文利用代数C研究了G的极大环面。
We show an analogue of Jordan's theorem for algebraic groups defined over a field $\mathbb k$ of arbitrary characteristic. As a consequence, a Jordan-type property holds for the automorphism group of any projective variety over $\mathbb k$.
Hyperstructures of Affine Algebraic Group Schemes
我们施加一个相当未知的代数结构称为“超结构”的仿射代数群计划的基本空间。这种代数结构推广了典型群结构，并由整体瓣的霍普夫代数的结构规范地定义。本文部分推广了A.Connes和C.Consani的结果
We describe an algorithm, which - given the characters of tilting modules and assuming that Donkin's tilting conjecture is true - computes the characters of simple modules for an algebraic group in any characteristic.
Infra-Solvmanifolds and Rigidity of Subgroups in Solvable Linear Algebraic Groups
给出了具有同构基本群的紧亚解流形是光滑亚纯的一个新的证明。更一般地说，我们证明了刚性的流形上使用仿射作用的虚拟多环群可解李群。我们的结果是由可解线性代数群中子群的刚性性质得到的。
Cohomology of algebraic groups, finite groups, and Lie algebras: Interactions and Connections
本文综述了2009年7月在华东师范大学举办的“李与表示论研讨会暨暑期班”上关于代数群、有限群的上同调与Frobenius核之间关系的研究成果.
On strong approximation for algebraic groups
这是我在“热门话题：薄群和超强近似”研讨会（MSRI，伯克利，2012年2月6日至10日）上的演讲的扩展版本。
Regular algebraic surfaces isogenous to a higher product constructed from group representations using projective planes
从具有分歧结构的有限群可以得到与曲线的高阶积同构的正则代数曲面。我们发现，非混合分歧结构的有限群构造为p-子的特定的无限群的特殊表示有关的有限射影平面。
Orbits of Real Algebraic Groups
建立了某些实代数空间上实代数群闭轨道的存在性。作为应用，证明了若$G$是具有Iwasawa分解$G=KAN$的实约化群，则$G$的所有幂幺子群与$N$的一个子群共轭.
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