# Seminars 11/12

**2011/2012 Seminars**

**August 13, 2012**

Ian Payne (University of Waterloo) "A Short Introduction to Universal Algebra"

Abstract: I will begin by defining universal algebras, and describing how they generalize familiar things like groups and rings. I will then explain how the notions of homomorphisms, subalgebras, and powers all can be defined in a natural way so that they agree with the corresponding notions for groups and rings. Finally, I will define and discuss varieties of algebras, which are a central object of study in universal algebra.

*HH-3017, 1:00-2:00p.m.*

**July 31, 2012**

Leandro Vendramin (University of Buenos Aires, Argentina) "Nichols algebras with many quadratic relations"

Abstract: We classify Nichols algebras of irreducible Yetter-Drinfeld

modules over groups such that the homogeneous component of

degree two of the Nichols algebra satisfies a given inequality. This

assumption turns out to be equivalent to a nice factorization

assumption on the Hilbert series. The talk is based on a joint work

with M. Graña and I. Heckenberger.

*HH-3017, 1:00-2:00p.m.*

**July 27, 2012**

Nicolas Andruskiewitsch (University of Cordoba, Argentina)

"From Hopf algebras to tensor categories"

Abstract: Tensor categories are the natural setting for the understanding of

many applications of Hopf algebras. In this talk I will introduce them and then I will discuss how to obtain tensor categories from spherical Hopf algebras and how we hope to discover new examples in this way.

*HH-3017, 1:00-2:00p.m.*

**July 25, 2012 Departmental Colloquium**

Nicolas Andruskiewitsch (University of Cordoba, Argentina) "The classification problem for finite-dimensional Hopf algebras"

Abstract: Hopf algebras were introduced by Cartier in the middle of the 1950's

as an axiomatization of the work of Dieudonne on algebras of distributions on algebraic groups in positive characteristic. Independently, an analogous notion appeared in the works of Hopf and A. Borel. A milestone in the development of the theory was the discovery by Drinfeld and Jimbo of quantum groups, which related

Hopf algebras to other areas of mathematics. In the first part of the talk, I will introduce Hopf algebras from scratch, give some basic examples and briefly sketch some aspects of their history. In the second part, I will discuss the classification program of finite-dimensional complex Hopf algebras. An overview of the current

state of the art will be given.

*HH-3017, 1:00-2:00p.m.*

**July 19, 2012**

Csaba Schneider (CAUL, Universidade de Lisboa, Portugal) "Constructive membership testing in classical groups"

Abstract: Computational matrix group theory has been the fastest growing area of

computational group theory during the last 20 years. Efficient

computations in finite matrix groups require that we are able to solve

some basic tasks in groups generated by a finite set of matrices over

a finite field. One of these tasks, known as the "Constructive

Membership Problem", requires that we write an arbitrary element of a

quasisimple matrix group as a product of its generators. I will

present several approaches to solve this problem in different contexts.

The implementations of the existing algorithms are written in Magma

and I will also show how these implementations perform in practice.

*HH-3017, 1:00-2:00p.m.*

**June 13, 2012**

Alexey Gordienko (Memorial University of Newfoundland) "Amitsur's conjecture for polynomial H-identities of H-module Lie algebras"

Abstract: In the 1980's (or even earlier), a conjecture about the

asymptotic behaviour of codimensions of ordinary polynomial identities

was made by S.A. Amitsur. Amitsur's conjecture was proved in 1999 by

A. Giambruno and M.V. Zaicev for associative algebras, in 2002 by M.V.

Zaicev for finite dimensional Lie algebras.

Alongside with ordinary polynomial identities of algebras, graded

polynomial identities, G- and H-identities are important too.

Usually, to find such identities is easier than to find the ordinary

ones. Furthermore, the graded polynomial identities, G- and

H-identities completely determine the ordinary polynomial identities.

Therefore the question arises whether the conjecture holds for graded

codimensions, G- and H-codimensions.

We consider a finite dimensional H-module Lie algebra L over a field

of characteristic 0 where H is a finite dimensional semisimple Hopf

algebra, and prove the analog of Amitsur's conjecture on asymptotic

behavior for codimensions of polynomial H-identities of L. As a

consequence, we obtain the analog of Amitsur's conjecture for graded

codimensions of any finite dimensional Lie algebra graded by a finite

group not necessarily Abelian.

This result is a generalization of the result of the author where he

proved the analog of Amitsur's conjecture for G-codimensions for a

finite group G and graded codimensions for a finite Abelian group.

*HH-3017, 1:00-2:00p.m.*

**May 2, 2012**

Alexey Gordienko (Memorial University of Newfoundland) "Structure of H-(co)module Lie algebras"

Abstract:

The applications of Lie and associative algebras with an additional

structure, e.g. graded, H-(co)module, or G-algebras, gave rise to the

studies of the objects and decompositions that have nice properties with

respect to these structures. I will discuss the H-(co)invariant analog of

the Levi theorem that is one of the main results of the structure Lie

theory. Also I am going to talk about the stability of the radicals and

about further decompositions of the solvable radical and a maximal

semisimple subalgebra. In the end of the talk I will discuss the

applications of the results obtained to graded Lie algebras and Lie

algebras with a rational action of a reductive affine algebraic group

by automorphisms.

*HH-3017, 1:00-2:00p.m.*

**March 6, 2012**

Alexey Gordienko (Memorial University of Newfoundland) "Graded polynomial identities, group actions, Hopf module algebras, and exponential growth"

Abstract: We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of different generalizations of polynomial identities, e.g. graded polynomial identities and G-identities for any finite not necessarily Abelian group G, and H-identities for finite dimensional associative algebras with an action of a finite dimensional semisimple Hopf algebra H.

*SN-1019, 1 – 2 p.m.*

**January 18, 2012**

Mikhail Kotchetov (Memorial University of Newfoundland) "Weyl groups of fine gradings"

*HH-3017, 1 – 2 p.m.*

**February 15, 2012**

Edgar Goodaire (Memorial University of Newfoundland) "Jordan loops and loop rings:

a mix of combinatorics and algebra"

Abstract:

Think of a loop as a group which is not necessarily associative. So the multiplication table of a loop is nothing but a standard Latin square. In this talk, we study loops that are Jordan in the sense that they are commutative and satisfy the Jordan identity:(x^2y)x=x^2(yx). We also discuss a subclass of Jordan loops we call RJ for ''ring Jordan,'' these being loops whose ''loop rings'' are also Jordan. We describe ways to construct finite Jordan loops and finite RJ loops. Our method involves finding Latin squares whose entries satisfy certain functional equations on an abelian group. We find some solutions, but wish we had more. There are a number of open combinatorial problems associated with this work.

*HH-3017, 1 – 2 p.m.*

**November 30 & December 7, 2011**

Alexey Gordienko (Memorial University of Newfoundland) "Graded polynomial identities, group actions and exponential growth of Lie algebras"

*HH-3017, from 1 – 2 p.m.*

**November 23, 2011**

Jonny Lomond (Memorial University of Newfoundland) "Growth functions of G-sets. II"

*HH-2010, 3-4 p.m.*

**September 28, 2011**

Toma Albu (Simion Stoilow Institute of Mathematics of Romanian Academy, Romania) "The Hopkins-Levitzki Theorem: old and new. II"

*HH-3017, 1 – 2 p.m.*

**September 21, 2011**

Alexey Gordienko (Memorial University of Newfoundland) "Codimensions of polynomial identities of representations of Lie algebras"

*HH-3017, 1 – 2 p.m.*

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