Automorphisms of Chevalley groups of type $B_l$ over local rings with 1/2
Elena I. Bunina
本文证明了1/2交换局部环上$B_l$型Chevalley群$l\geqslant 2$的每个自同构是标准的，即.例如，它是环自同构、内自同构和中心自同构复合。
In the given paper we prove that every automorphism of a Chevalley group of type $B_l$, $l\geqslant 2$, over a commutative local ring with 1/2 is standard, i. e., it is a composition of ring, inner and central automorphisms.
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Automorphisms of Chevalley groups of types B_2 and G_2 over local rings
本文证明了B_2型或G_2型伴随Chevalley群的每个自同构都是标准的，即它是“内”自同构、环自同构和中心自同构的合成。
Automorphisms of Chevalley groups of type $F_4$ over local rings with 1/2
本文证明了交换局部环上$F_4$型Chevalley群的每个自同构都是标准的，即.例如，它是环与内自同构复合。
Automorphisms of Chevalley groups of types $A_l, D_l, E_l$ over local rings without 1/2
本文证明了不含1/2的交换局部环上的$A_l$，$D_l$或$E_l$，$l\geqslant 3$型Chevalley群的每个自同构都是标准的，即.例如，它是环自同构、内自同构、中心自同构和图自同构的组合。
Automorphisms of Chevalley groups of types $A_l, D_l, E_l$ over local rings with 1/2
本文证明了1/2交换局部环上的$A_l, D_l$型（或$E_l$，$l\geqslant 2$）Chevalley群的每个自同构是标准的，即.例如，是内自同构、环自同构、图自同构和中心自同构的合成。
Automorphisms and isomorphisms of Chevalley groups of type $G_2$ over local rings with 1/2 and 1/3
证明了1/2和1/3交换局部环上$G_2$型Chevalley群的每个同构是标准的，即.例如，它是一个环同构和一个内自同构复合。
Automorphisms of Chevalley groups of different types over commutative rings
本文证明了交换环上的（初等）伴随Chevalley群的每一个自同构都是标准的，即：交换环上的（初等）伴随Chevalley群的每一个自同构的秩为$>1$（对于系统$A_2$，$F_4$，$B_l$，$C_l$为1/2;对于系统$G_2$为1/2和1/3）。例如，它是环自同构、内自同构、中心自同构和图自同构的组合。
Automorphisms of a Chevalley group of type G_2 over a commutative ring R with 1/3 generated by the all invertible elements and 2R
该研究刻画了一类含 1/3 交换环上 G_2 型 Chevalley 群的自同构结构，证明其均可分解为环自同构与内自同构的复合。
Automorphisms of Chevalley groups over commutative rings
本文证明交换环上秩大于1的 Chevalley 群及其初等子群的所有自同构均为环、内、中心与图自同构的标准复合，并给出模型论应用。
Automorphisms and isomorphisms of Chevalley groups and algebras
有理代数（或类似环）上秩至少为2的伴随Chevalley群，其基本子群和相应的李环具有相同的自同构群。这些自同构被明确地描述。
Automorphisms of elementary adjoint Chevalley groups of types $A_l, D_l, E_l$ over local rings with 1/2
本文证明了1/2局部交换环上$A_l, D_l$型或$E_l$型的初等伴随Chevalley群的每个自同构都是环自同构与$\GL(V)$（$V$是伴随表示空间）中Chevalley群的正规化子的某个矩阵共轭的复合.
Automorphisms of Twisted Chevalley Groups of type ${}^2 D_\ell \ (\ell \geq 4)$ over Local Rings
本文作为交换环上扭Chevalley群同构与自同构分类系列研究的一部分，证明了含1/2的局部环上${}^2 D_\ell$（$\ell \geq 4$）型扭Chevalley群的所有自同构均为标准自同构。
Automorphisms of Twisted Chevalley Groups of type ${}^2 A_\ell \ (\ell \geq 5)$ over Local Rings
研究局部环上${}^2 A_\ell \ (\ell \geq 5)$型扭曲谢瓦莱群自同构，证明在2可逆局部环上其自同构是标准的，为后续相关群的自同构和同构分类研究奠定基础。
The irreducible Brauer characters of the automorphism group of the Chevalley group F_4(2)
本研究利用计算方法推导谢瓦莱群F_4(2)自同构群的不可约Brauer特征标，结果仍存在一个参数和一处一致性问题待完善。
Chevalley groups of type $G_2$ as automorphism groups of loops
设$M^*(q)$是在$GF(q)$上构造的唯一非结合有限单Moufang环.通过将$M^*(2)$的乘法自同构推广到GF（2）上唯一分裂八元数代数的线性自同构，证明了$Aut(M^*(2))$是Chevalley群$G_2(2)$.我们的许多辅助结果适用于一般情况。在证明过程中，我们证明了分裂八元数代数的每个元素都可以写成两个范数为1的元素之和。
Automorphisms of twisted Chevalley groups of type ${}^2E_6$ over local rings
本文分类了带对合且2可逆的交换局部环上，两种等义类型的${}^2E_6$型扭Chevalley群的抽象自同构，证明其均为标准型，明确了伴随与单连通情形下自同构的结构，通过基变换替换等步骤完成分类。
Automorphisms of Twisted Chevalley Groups of type ${}^2 A_3$ and ${}^2 A_4$ over Local Rings
本文对含1/2的局部环上${}^2A_3$、${}^2A_4$型扭Chevalley群的自同构完成完整分类，是交换环上扭Chevalley群自同构与同构分类系列研究的一部分，填补了低秩情形的空白。
We prove that Chevalley group over the field $F$ of zero characteristic possess $R_{\infty}$ property, if $F$ has torsion group of automorphisms or $F$ is an algebraically closed field which has finite transcendence degree over $\mathbb{Q}$. As a consequence we obtain that the twisted conjugacy class $[e]_φ$ of unit element is a subgroup of Chevalley group if and only if $φ$ is central automorphism.
Let $\hat G$ be the finite simply connected version of an exceptional Chevalley group, and let $V$ be a nontrivial irreducible module, of minimal dimension, for $\hat G$ over its field of definition. We explore the overgroup structure of $\hat G$ in $\mathrm{GL}(V)$, and the submodule structure of the exterior square (and sometimes the third Lie power) of $V$. When $\hat G$ is defined over a field of odd prime order $p$, this allows us to construct the smallest (with respect to certain properties) $p$-groups $P$ such that the group induced by $\mathrm{Aut}(P)$ on $P/Φ(P)$ is either $\hat G$ or…
An estimate on the commutator width is given for Chevalley groups over rings of stable rank 1, and the general method suitable for other rings of small dimension.
Centrality of the congruence kernel for elementary subgroups of Chevalley groups of rank >1 over noetherian rings
本文证明了任意诺特环R上秩>1的Chevalley群G的基本子群的同余核的中心性（如果G是C_n或G_2型，则对R作一些小的限制）.
We prove that Chevalley groups of the classical series $B_l, C_l, D_l$ over an integral domain of zero characteristic, which has torsion automorphism group, possess the $R_{\infty}$-property.
We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras.
Let $p$ be an odd prime. We construct a $p$-group $P$ of nilpotency class two, rank seven and exponent $p$, such that $\mathrm{Aut}(P)$ induces $N_{\mathrm{GL}(7,p)}(G_2(p)) = Z(\mathrm{GL}(7,p)) G_2(p)$ on the Frattini quotient $P/Φ(P)$. The constructed group $P$ is the smallest $p$-group with these properties, having order $p^{14}$, and when $p = 3$, our construction gives two nonisomorphic $p$-groups. To show that $P$ satisfies the specified properties, we study the action of $G_2(q)$ on the octonion algebra over $\mathbb{F}_q$, for each power $q$ of $p$, and explore the reducibility of the…
Sha-rigidity of Chevalley groups over local rings
本文研究局部环上 Chevalley 群的刚性问题，证明在特定根系及可逆元条件下，其所有局部内自同态均为内自同态，从而确立这些群的 Sha 刚性。
We prove that Chevalley groups over polynomial rings $\mathbb F_q[t]$ and over Laurent polynomial $\mathbb F_q[t,t^{-1}]$ rings, where $\mathbb F_q$ is a finite field, are boundedly elementarily generated. Using this we produce explicit bounds of the commutator width of these groups. Under some additional assumptions, we prove similar results for other classes of Chevalley groups over Dedekind rings of arithmetic rings in positive characteristic. As a corollary, we produce explicit estimates for the commutator width of affine Kac--Moody groups defined over finite fields. The paper contains als…
On linear representations of Chevalley groups over commutative rings
设$G$是对应于秩为$\geq 2$的既约不可约根系的泛Chevalley-Demazure群概型，$R$是交换环.本文分析了初等子群$G(R)^+ \subset G(R).$的代数闭域$K$上的线性表示$ρ\colon G(R)^+ \to GL_n (K)$，主要结果是在一定条件下，任何这样的表示都有一个标准的描述，即存在一个交换的有限维$K$ -代数$B$，一个具有Zerkiki-稠密象的环同态$f \colon R \to B$，以及代数群$σ\colon G(B) \to GL_n (K)$的态射，使得$ρ$与$σ\circ F$在合适的有限指数子群$Γ\subset G(R)^+,$上重合，其中$F \colon G(R)^+ \to G(B)^+$是由$f.$诱导的群同态特别地，这证实了Borel和Tits关于特征为零的域上Chevalley群的猜想。
Let R be a Dedekind domain, and let G be a simply connected Chevalley-Demazure group scheme of rank =>2. We prove that G(R[x_1,...,x_n])=G(R)E(R[x_1,...,x_n]) for any n=>1. This extends the corresponding results of A. Suslin and F. Grunewald, J. Mennicke, and L. Vaserstein for G=SL_n, Sp_2n. We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.
In this paper we prove that if $G(R)=G_π(Φ,R)$ $(E(R)=E_π(Φ, R))$ is an (elementary) Chevalley group of rank $> 1$, $R$ is a local ring (with $\frac{1}{2}$ for the root systems ${\mathbf A}_2, {\mathbf B}_l, {\mathbf C}_l, {\mathbf F}_4, {\mathbf G}_2$ and with $\frac{1}{3}$ for ${\mathbf G}_{2})$, then the group $G(R)$ (or $(E(R)$) is regularly bi-interpretable with the ring~$R$. As a consequence of this theorem, we show that the class of all Chevalley groups over local rings (with the listed restrictions) is elementary definable, i.\,e., if for an arbitrary group~$H$ we have $H\equiv G_π(Φ, …
Finiteness Properties of Chevalley Groups over the Laurent Polynomial Ring over a Finite Field
证明了：若G是秩为n的Chevalley群，F_q[t，t^{-1}]是有限域上的Laurent多项式环，则G（F_q[t，t^{-1}]）是F_{2n-1}型的.这个界限是最优的，因为它是已知的-我们再次表明-该集团是不是类型F_{2n}。
Normalizer of Twisted Chevalley Groups over Commutative Rings
本文研究了交换环上扭曲Chevalley群的正规化子，证明在特定条件下，这些正规化子与群本身相等。
Twisted conjugacy classes in twisted Chevalley groups
该研究聚焦扭共轭类与R_{∞}-性质相关问题，证明了满足特定域条件的特征零域上的扭Chevalley群同时具备R_{∞}-性质和S_{∞}-性质。
Verbal width in arithmetic Chevalley groups
该研究证明了数域整数环局部化环上秩≥2的单连通Chevalley群中任意词语宽度有仅依赖根系和数域次数的常数界，推进了算术群词语宽度问题的研究。
Bounded generation by root elements is a property which has been widely studied for various types of linear algebraic groups defined over rings of integers in algebraic number fields. However, when considering global function fields, there are not many results beyond the treatment of special cases due to Nica and Queen. In this paper, we use model theoretic methods due to Carter, Keller and Paige written up by Morris to prove bounded generation by root elements for simply connected, split Chevalley groups defined over the ring of all integers in a global function field. We further apply this b…
Regular bi-interpretability and finite axiomatizability of Chevalley groups
本文证明在环含1/2（G_2型还含1/3）的条件下，秩大于1的Chevalley群的中心商与对应环正则双可解释，且该类群可初等定义并有限公理化，同时给出无限中心情形的反例。
Finiteness Properties of Chevalley Groups over a Polynomial Ring over a Finite Field
它是由H. Abels和P. Abramenko证明了对秩为n的经典Fq-群G，若q足够大，则Fq[t]-有理点的算术格G（Fq[t]）是Fn-1型的.我们证明了在不对q作任何假设的情况下，对于定义在Fq上的任何迷向的、绝对几乎单群G，上述陈述都是成立的。
Sha-rigidity of adjoint Chevalley groups of types $A_1$, $A_2$, $B_2$, $G_2$ over commutative rings
本文证明了在特定环上，A1、A2、B2和G2类型的伴随Chevalley群及其初等子群的局部内自同构是内自同构，从而这些群具有Sha-刚性。
Lattices of minimum covolume in Chevalley groups over local fields of positive characteristic
本文证明了：若G是秩大于1的经典型或E6型单连通Chevalley群，且q > 9是一个素数p > 5的幂，则G = G（F_q（（1/t）在自同构下有唯一的最小余体积格G（F_q[t]）.
Let $F$ be a non-Archimedean local field with the ring of integers $\mathcal{O}$ and the prime ideal $\mathfrak{p}$ and let $G={\bf G}\left(\mathcal{O}/\mathfrak{p}^n\right)$ be the adjoint Chevalley group. Let $m_f(G)$ denote the smallest possible dimension of a faithful representation of $G$. Using the Stone-von Neumann theorem, we determine a lower bound for $m_f(G)$ which is asymptotically the same as the results of Landazuri, Seitz and Zalesskii for split Chevalley groups over $\mathbb{F}_q$. Our result yields a conceptual explanation of the exponents that appear in the aforementioned res…
Reflection quotients in Riemannian Geometry. A Geometric Converse to Chevalley's Theorem
Chevalley定理和它的逆，谢泼德-托德定理，断言有限反射群的区别在于不变多项式的环是自由生成的。我们表明，在欧几里德的情况下，一个较弱的条件足以表征有限反射群，即一个自由生成的多项式子环是封闭的梯度产品。
The goal of this paper is to establish a general rigidity statement for abstract representations of elementary subgroups of Chevalley groups of rank at least 2 over a class of commutative rings that includes the localizations of 1-generated rings and the coordinate rings of affine curves. This is achieved by developing the approach introduced in our previous work, and in particular by verifying condition (Z) over the class of rings at hand. Our main result implies, for example, that any finite-dimensional representation of SL_n(Z[X]) (for n at least 3) over an algebraically closed field of cha…
Triangular and Unitriangular Factorization of Twisted Chevalley Groups
本文针对扭曲Chevalley群中未解决的${}^2A_{2n}$型情形，引入特殊稳定秩一条件和$\theta$-完备环两类新环类，证明了在该类环上三角与单位三角分解的存在性，涵盖所有域、局部环等。
Chevalley Warning type results on abelian groups
本文通过定义阿贝尔群间函数的次数概念，将Chevalley-Warning定理从域推广到非交换环及素数幂阶阿贝尔群。
On Normal Subgroups of Twisted Chevalley Groups over Commutative Rings
本文研究了扭曲Chevalley群在可换环上的结构，证明了初等同余子群的正规性及子群分类，同时探讨了半局部环下的群分解性质。
Uniform bounded elementary generation of Chevalley groups
本文证明秩至少为2的Chevalley群在任意算术型Dedekind环上具有均匀有界初等生成，给出了普适的宽度上界。
Strong boundedness of simply connected split Chevalley groups defined over rings
本文研究 S-算术分裂 Chevalley 群的强有界性，证明由共轭类定义的字度量直径仅依赖共轭类数量，并通过正规生成集示例说明界的尖锐性及小型生成集障碍。
To each automorphism of a spherical building there is naturally associated an "opposition diagram", which encodes the types of the simplices of the building that are mapped onto opposite simplices. If no chamber (that is, no maximal simplex) of the building is mapped onto an opposite chamber then the automorphism is called "domestic". In this paper we give the complete classification of domestic automorphisms of split spherical buildings of types $\mathsf{E}_6$, $\mathsf{F}_4$, and $\mathsf{G}_2$. Moreover, for all split spherical buildings of exceptional type we classify (i) the domestic homo…
A Survey on Automorphism Groups of Finite $p$-Groups
有限p-群的自同构群的研究主要集中在三个方面：对常见的有限p-群，如Chevalley群的特殊p-群和Sylow p-子群的显式计算;构造具有特定自同构群的p-群;发现其自同构群本身是或不是p-群的有限p-群。该材料以不同的详细程度呈现，其中一些示例给出了完整的细节。
We formulate and prove relative versions of several classical decompositions known in the theory of Chevalley groups over commutative rings. As an application we obtain upper estimates for the width of principal congruence subgroups in terms of several families of generators. Some of our results are new even in the absolute case and were previously studied only for groups over finite fields.
Bounded reduction for Chevalley groups of types $E_6$ and $E_7$
本文证明了小维数环上多项式环的$E_6$和$E_7$型Chevalley群元素可用有界数量的初等根元素约化到真子系统子群，并明确给出界，作为$K_1$-函子早期稳定化的有效版本，同时部分证明了$E_8$的类似假设。
Diameters of Chevalley groups over local rings
设G是秩为1的Chevalley群概型.我们证明了对于某个绝对常数d> 0和两个函数p_0 = p_0（l）和C = C（l，p），下列定理成立。设p> p_0是素数，G_n：= G（\Z/p ^n \Z）是n> 0的有限群族.则对于任何n> 0和生成G_n的任何子集S，我们有diam（G_n，S）<Cn ^d，即，G_n的任一元素都是来自S \cup S ^{-1}的Cn ^d个元素的乘积。特别地，对于某些C '= C'（1，p）和对于任何n> 0，我们有diam（G_n，S）<C 'log ^d（|G_n|）。我们的证明是初等的和有效的，因为常数d和函数p_0（l）和C（l，p）是显式计算的。此外，还存在一个计算群G_n中任意Cayley图中任意两点间最短路的有效算法。
The Table of the Structure Constants for the Complex Simple Lie Algebra of Type F_4 and its Application to the Calculation of Commutators in the Chevalley Group of Type F_4 over Fields and Rings
本文构造 F_4 型复单李代数结构常数表，推导 Chevalley 换位子公式，并利用有向加权图刻画相关计数与复换位子和。
We consider the simply connected Chevalley group $G(E_7,R)$ of type $E_7$ in the 56-dimensional representation. The main objective of the paper is to prove that the following four groups coincide: the normalizer of the elementary Chevalley group $E(E_7,R)$, the normalizer of the Chevalley group $G(E_7,R)$ itself, the transporter of $E(E_7,R)$ into $G(\mathrm E_7,R)$, and the extended Chevalley group $\overline G(E_7,R)$. This holds over an arbitrary commutative ring $R$, with all normalizers and transporters being calculated in $GL(56,R)$. Moreover, we characterize $\overline G(E_7,R)$ as the …
Finiteness Properties of Chevalley Groups over the Ring of (Laurent) Polynomials over a Finite Field
本文给出了群G（O_S）的有限长，其中G是F_q-迷向的、连通的、非交换的、几乎单的F_q-群，O_S是F_q[t]、F_q[t^{-1}]和F_q[t，t^{-1}]之一。也就是说，k = F_q（t）和S包含分别对应于多项式p（t）= t到无穷远无穷远点的位置s_0和s_ \infty之一或两者。设G（O_S）的有限长为2n-1，其中n是G的F_q秩，若S包含其中一个位置，则G（O_S）的有限长为n-1。例如，群SL_3（F_q[t，t^{-1}]）是F_3类型的，但不是F_4类型的，这是以前未知的事实。
Unitriangular factorisations of Chevalley groups
最近，以下问题在各种情况下引起了很多关注：根据标准Borel子群$B=B(Φ,R)$的幂单根$U=U(Φ,R)$和相反Borel子群$B^-=B^-(Φ,R)$的幂单根$U^-=U^-(Φ,R)$，找到Chevalley群$G=G(Φ,R)$的最短分解$G=UU^-UU^-...U^{\pm}$。到目前为止，有限域上的记录是在2010年由Babai，Nikolov和Pyber的论文中建立的，他们证明了一个Lie型群允许长度为5的单三角分解$G=UU^-UU^-U$。他们的证明需要深入的分析和组合工具。在本文件中，我们注意到，从工作的低音和塔夫根之一，立即得到一个更普遍的结果，声称在任何环的稳定秩1有untriangular分解$G=UU^-UU^-$的长度为4。此外，我们给出了一个详细的调查三角因子分解，证明了一些相关的结果，讨论的前景推广到其他类别的环，并说明几个未解决的问题。本文的另一个主要结果是，在广义黎曼猜想的假设下，环$\Int\Big[\displaystyle{1\over p}\Big]$上的Chevalley群允许长度为6的单三角分解$G=UU^-UU^-UU^-$。否则，根据Cooke和Weinberger的工作，具有无限乘法群的Hasse域的最佳长度估计给出了9个因子。
The Diophantine problem in Chevalley groups
本文证明任意含1交换环上Chevalley群中的丢番图问题与环本身的丢番图问题Karp等价，并由此得到若干模型论推论。
Automorphisms of triality groups over local rings
本文对D₄型伴随与单连通形式下的初等及全三元组群的抽象自同构进行分类，证明其由伴随群元素与保持特定迹的环自同构唯一确定，并建立了相关同态的相对刚性定理。
Automorphism Groups of Schur Rings
1993年Muzychuk \cite{muzychuk}证明了循环群$Z_n$上的有理Schur环与$n$的因子格的子格一一对应，或者等价地，与$Z_n$的子群格的子格一一对应。这可以很容易地推广到表明，对于任何有限群$G$，$G$的特征子群格的子格以自然的方式产生$G$上的有理Schur环。我们的主要结果是：对于任意奇素数$p$，任意有限群都可以表示为交换$p$ -群上的有理Schur环的（代数）自同构群.相反，我们证明了在循环群上任何Schur环的自同构群都是交换的。我们还证明了一个逆著名的结果Muzychuk \cite{muzychuk2}，两个舒尔环在一个循环群是同构的当且仅当它们重合，即，我们表明，在一个群体是不是循环的，总是存在不同的同构舒尔环。
For a root system $Φ$ of type $E_l$ and arbitrary commutative ring $R$ we show that the group $K_2(Φ, R)$ is contained in the centre of the Steinberg group $St(Φ, R)$. In course of the proof we also demonstrate an analogue of Quillen---Suslin local-global principle for $K_2(Φ, R)$.
Given a universal elementary Chevalley-Demazure group $E_Φ^{sc}(R)$ for which its (standard) parabolic subgroups are finitely generated, we consider the problem of classifying which parabolics $P(R) \subset E_Φ^{sc}(R)$ are finitely presented. We show that, under mild assumptions, this is equivalent to the finite presentability of a suitable retract of $P$ which contains the Levi factor. If the base ring $R$ is a Dedekind domain of arithmetic type, we combine our results with well-known theorems due to Borel-Serre, Abels, Behr and Bux to give a partial classification of finitely presentable $S…
Some Properties of Twisted Chevalley Groups
本论文研究交换环上扭Chevalley群的结构性质，探讨了相对初等子群的正规性、被初等子群正规化的子群分类，以及在环扩张中正规化子的重合性，证明了伴随型群的正规化子恰好等于原群。
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