Related papers: On the essential dimension of an algebraic group w…

This is a companion paper to our previous work, where we proved the finiteness of the Tate-Shafarevich group for an arbitrary torus $T$ over a finitely generated field $K$ with respect to any divisorial set $V$ of places of $K$. Here, we…

The faithful dimension of a finite group $\mathrm G$ over $\mathbb C$, denoted by $m_\mathrm{faithful}(\mathrm G)$, is the smallest integer $n$ such that $\mathrm G$ can be embedded in $\mathrm{GL}_n(\mathbb C)$. Continuing our previous…

Let $G$ be a connected real algebraic group. An unrefinable chain of $G$ is a chain of subgroups $G=G_0>G_1>...>G_t=1$ where each $G_i$ is a maximal connected real subgroup of $G_{i-1}$. The maximal (respectively, minimal) length of such an…

The Pr\"ufer rank $\mathrm{rk}(G)$ of a profinite group $G$ is the supremum, across all open subgroups $H$ of $G$, of the minimal number of generators $\mathrm{d}(H)$. It is known that, for any given prime $p$, a profinite group $G$ admits…

We show that a connected split reductive group G over a field of characteristic 0 is uniquely determined up to isomorphism by specifying a maximal torus T of G, the set of isomorphism classes of irreducible representations of G, and the…

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $D$ be a $p$-divisible group over $k$. Let $n_D$ be the smallest non-negative integer for which the following statement holds: if $C$ is a $p$-divisible group over $k$ of…

Recently there has been considerable interest in studying the length and the depth of finite groups, algebraic groups and Lie groups. In this paper we introduce and study similar notions for algebras. Let $k$ be a field and let $A$ be an…

Let $k$ be an algebraically closed field of characteristic $p \geq 0$, let $G$ be a simple simply connected classical linear algebraic group of rank $\ell$ and let $T$ be a maximal torus in $G$ with rational character group $X(T)$. For a…

Let G be a connected algebraic group over an algebraically closed field of characteristic p (possibly 0), and X a variety on which G acts transitively with connected stabilizers. We show that any \'etale Galois cover of X of degree prime to…

Algorithms working with linear algebraic groups often represent them via defining polynomial equations. One can always choose defining equations for an algebraic group to be of the degree at most the degree of the group as an algebraic…

Let $G$ be a simple and simply connected algebraic group over a field of characteristic $p>0$, and $G_r$ its $r$-th Frobenius kernel. In this paper, we initiate a general study of $\text{Dist}(G_r)^T$, the subalgebra of $\text{Dist}(G_r)$…

Let $E$ be a field, $p$ a prime number and $F/E$ a finitely-generated extension of transcendency degree $t$. This paper shows that if the absolute Galois group $\mathcal{G}_{E}$ is of nonzero cohomological $p$-dimension cd$_{p}(E)$, then…

We prove the Andruskiewitsch-Dumas conjecture that the automorphism group of the positive part of the quantized universal enveloping algebra $U_q({\mathfrak{g}})$ of an arbitrary finite dimensional simple Lie algebra g is isomorphic to the…

We endow a topological group $(G, \tau)$ with a coarse structure defined by the smallest group ideal $S_{\tau} $ on $G$ containing all converging sequences with their limits and denote the obtained coarse group by $(G, S_{\tau})$. If $G$ is…

Belk and Connelly introduced the realizable dimension $\textrm{rd}(G)$ of a finite graph $G$, which is the minimum nonnegative integer $d$ such that every framework $(G,p)$ in any dimension admits a framework in $\mathbb{R}^d$ with the same…

Let $G$ be a countable group that splits as a free product of groups of the form $G=G_1\ast\dots\ast G_k\ast F_N$, where $F_N$ is a finitely generated free group. We identify the closure of the outer space…

The representation dimension of a finite group $G$ is the minimal dimension of a faithful complex linear representation of $G$. We prove that the representation dimension of any finite group $G$ is at most $\sqrt{|G|}$ except if $G$ is a…

Let G be a compact p-adic analytic group. We study K-theoretic questions related to the representation theory of the completed group algebra kG of G with coefficients in a finite field k of characteristic p. We show that if M is a finitely…

Let \(G\) be a finite group, and let \(\Delta(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). It is well known that, whenever \(\Delta(G)\) is connected, the diameter of…

Let $\overline G$ be the wonderful compactification of a simple affine algebraic group $G$ defined over $\mathbb C$ such that its center is trivial and $G\not= {\rm PSL}(2,\mathbb{C})$. Take a maximal torus $T \subset G$, and denote by…