Past Algebra Seminar

12 November 2013
17:00
Ben Martin
Abstract
Let $\Gamma$ be a group and let $r_n(\Gamma)$ denote the number of isomorphism classes of irreducible $n$-dimensional complex characters of $\Gamma$. Representation growth is the study of the behaviour of the numbers $r_n(\Gamma)$. I will give a brief overview of representation growth. We say $\Gamma$ has polynomial representation growth if $r_n(\Gamma)$ is bounded by a polynomial in $n$. I will discuss a question posed by Brent Everitt: can a group with polynomial representation growth have the alternating group $A_n$ as a quotient for infinitely many $n$?
5 November 2013
17:00
Andrei Jaikin-Zapirain
Abstract
I will review several known problems on the automorphism group of finite $p$-groups and present a sketch of the proof of the the following result obtained jointly with Jon Gonz\'alez-S\'anchez: For each prime $p$ we construct a family $\{G_i\}$ of finite $p$-groups such that $|Aut (G_i)|/|G_i|$ goes to $0$, as $i$ goes to infinity. This disproves a well-known conjecture that $|G|$ divides $|Aut(G)|$ for every non-abelian finite $p$-group $G$.
22 October 2013
17:00
Gunnar Traustason
Abstract
Let F be a field. A symplectic alternating algebra over F consists of a symplectic vector space V over F with a non-degenerate alternating form that is also equipped with a binary alternating product · such that the law (u·v, w)=(v·w, u) holds. These algebraic structures have arisen from the study of 2-Engel groups but seem also to be of interest in their own right with many beautiful properties. We will give an overview with a focus on some recent work on the structure of nilpotent symplectic alternating algebras.
15 October 2013
(All day)
Konstantin Ardakov
Abstract

Affinoid enveloping algebras arise as certain p-adic completions of ordinary enveloping algebras, and are closely related to Iwasawa algebras. I will explain how to use Beilinson-Bernstein localisation to compute their (non-commutative) Krull dimension. This is recent joint work with Ian Grojnowski.

4 June 2013
17:00
Alexey Sevastyanov
Abstract
<p>Let g be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, b a Borel subalgebra of g, h the Cartan sublagebra contained in b and N the unipotent subgroup corresponding to the nilradical n of b. Extremal projection operators are projection operators onto the subspaces of n-invariants in certain g-modules the action of n on which is locally nilpotent. Zhelobenko also introduced a family of operators which are analogues to extremal projection operators. These operators are called now Zhelobenko operators.<br />I shall show that the explicit formula for the extremal projection operator for g obtained by Asherova, Smirnov and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence (N, h) -&gt; b given by the restriction of the adjoint action. Simple geometric proofs of &nbsp;formulas for the ``classical'' counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.</p>
28 May 2013
17:00
Yves Cornulier
Abstract
We will first recall the known notion of commensurating actions and its link to actions on CAT(0) cube complexes. We define a group to have Property FW if every isometric action on a CAT(0) cube complex has a fixed point. We conjecture that every irreducible lattice in a semisimple Lie group of higher rank has Property FW, and will give some instances beyond the trivial case of Kazhdan groups.

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