Related papers: Regular bi-interpretability of Chevalley groups ov…

Given a directed graph $E$ and an associative unital ring $R$ one may define the Leavitt path algebra with coefficients in $R$, denoted by $L_R(E)$. For an arbitrary group $G$, $L_R(E)$ can be viewed as a $G$-graded ring. In this article,…

Let $G$ be an elementary abelian $p$-group, $G\cong{\mathbb F}_p^r$ and let $s_1,\ldots,s_r$ be a basis of $G$ over ${\mathbb F}_p$. Let $V$ be the dual of $G$, $V={\rm Hom}(G,{\mathbb F}_p)=H^1(G,{\mathbb F}_p)$. Let $x_1,\ldots,x_r$ be…

Let $\Phi$ be a root system of type $A_l$ or $D_l$. Let $K$ be a field of characteristic not $2$. Let $\delta$ be the maximal root of $\Phi$ and set $\Phi_0 = \{\alpha\in\Phi; \delta\perp\alpha\}$. We describe orbits of the group…

Suppose $\mathbb{F}$ is a field of prime characteristic $p$ and $E$ is a finite subgroup of the additive group $(\mathbb{F},+)$. Then $E$ is an elementary abelian $p$-group. We consider two such subgroups, say $E$ and $E'$, to be equivalent…

Let $R$ be a commutative $F$-algebra, where $F$ is a field of characteristic 0, satisfying the following conditions: $R$ is equidimensional of dimension $n$, every residual field with respect to a maximal ideal is an algebraic extension of…

Let $A$ be a ring with $1\neq 0$, not necessarily finite, endowed with an involution~$*$, that is, an anti-automorphism of order $\leq 2$. Let $H_n(A)$ be the additive group of all $n\times n$ hermitian matrices over $A$ relative to $*$.…

Given a finitely-generated group $\pi$ and a linear algebraic group $G$, the representation variety Hom$(\pi,G)$ has a natural filtration by the characteristic varieties associated to a rational representation of $G$. Its algebraic…

Let $E/F$ be a unramified quadratic extension of non-archimedean local fields of odd characteristic $p$, and $G$ be the unramified unitary group $U(2, 1)(E/F)$. For an irreducible smooth representation $\pi$ of $G$ over…

Let $D$ be a division ring with center $F$, and $G$ a subnormal subgroup of $D^*$. We show that if $G$ is a locally solvable group such that $G^{(i)}$ is algebraic over $F$, then $G$ must be central. Also, if $M$ is non-abelian locally…

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…

(Translation from the original French:) We characterize the rings $A$ and groups $G$ for which the group rings $A [G]$ are local, semi-local, or left perfect. The recent work of M. P. Malliavin and J. L. Pascaud permits the completion of…

We completely calculate the $RO(G)$-graded coefficients of ordinary equivariant cohomology where $G$ is the dihedral group of order $2p$ for a prime $p>2$ both with constant and Burnside ring coefficients. The authors first proved it for…

Let $(R,M,k)$ be a complete local ring (not necessarily Noetherian). As the first main result of this article, we prove that in the unequal characteristic case $\Char(R)\neq\Char(k)$, the natural surjective map between the groups of units…

In the given paper we prove that every automorphism of a Chevalley group of type $A_l$, $D_l$, or $E_l$, $l\geqslant 3$, over a commutative local ring without 1/2 is standard, i. e., it is a composition of ring, inner, central and graph…

The classical Waring problem deals with expressing every natural number as a sum of g(k) k-th powers. Similar problems were recently studied in group theory, where we aim to present group elements as short products of values of a given…

In this paper we define and study quasipolar general rings (with or without identity) and extend many of the basic results to the wider class. We obtain some new characterizations of quasipolar and strongly $\pi$-regular elements by using…

The main aim of this paper is to prove $R$-triviality for simple, simply connected algebraic groups with Tits index $E_{8,2}^{78}$ or $E_{7,1}^{78}$, defined over a field $k$ of arbitrary characteristic. Let $G$ be such a group. We prove…

We compute the cobordism ring $MU^*(BG)\otimes \mathbb{F}_p$ when $G$ is a Chevalley group. In the particular case of the general linear group, we prove that it agrees with the Chow ring (as defined by Totaro).

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…

We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl's question whether or not $R$ being nil-clean implies that $\mathbb{M}_n(R)$ is…