## Joint Ottawa/Carleton

## Algebra Seminar

### Winter 2022

#### Fridays, 2:30-4:00pm (UOttawa STEM201):

coffee 2:30-2:45; the talk 2:45-3:45

**March 4, 2:30-3:45pm (UOttawa, STEM201)**

Allan Francis Merino (UOttawa)

*Character Varieties of classical groups*

In my talk, I will define the concept of character varieties and explain how to compute it for classical complex groups (GL(n, C), Sp(2n, C), O(n, C)) by using results of Weyl and Procesi. Time permitting, I will say few words concerning the character varieties of G2 and Spin(n, C) (joint work with Simon Roby).

**March 11, 2:30-3:45pm (UOttawa, STEM201)**

Alistair Savage (UOttawa)

*Categories on cylinders*

We define the affinization of an arbitrary monoidal category, corresponding to the category of string diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to the category. The affinization formalizes and unifies many constructions appearing in the literature. We describe a large number of examples coming from Hecke-type algebras, braids, tangles, and knot invariants. This is joint work with Youssef Mousaaid.

**March 18, 2:30-3:45pm (UOttawa, STEM201)**

Kirill Zanullin (UOttawa)

*Unimodular degree of a root system*

We introduce a combinatorial invariant of a root system which we call a unimodular degree. In the geometric case (by the works of Karpenko) it gives an upper bound for the canonical dimension of a linear algebraic group. We produce a new algorithm to compute this invariant for some semisimple linear algebraic groups.

**March 25, 2:30-3:45pm (UOttawa, STEM201)**

Stefan Gille (UAlberta)

*Brauer invariants of finite groups*

Brauer invariants of finite groups and also more general

of algebraic groups are closely related to projective representations

of these groups. In my talk I present an elementary proof that

every Brauer invariant, i.e. a cohomological invariant with values in

the Brauer group, of a finite group G comes from a projective

representation of G.

**April 8, 3:00-4:00pm (UOttawa, STEM201)**

Cameron Ruether (UOttawa)

*Twisting Linear Algebraic Groups and Hopf Algebras*

We begin by reviewing the well known story that central simple algebras over a non-algebraically closed field F are twisted forms of matrix algebras. This procedure readily applies to linear algebraic groups which come from subgroups of M_n(F), such as SL_n or SO_n, to produce their twisted counterparts, SL(A) or SO(A,t), where t is an orthogonal involution on A, and the procedure also extends to all classical linear algebraic groups. In particular we show that we can twist Spin_n and HSpin_n to produce Spin(A,t) and HSpin(A,t). In addition, as it is needed for the result about HSpin, we discuss how the twisting story for linear algebraic groups is mirrored in their representing Hopf algebras.

**May 5, 3:15-4:30pm (UOttawa, STEM664)**

Philippe Gille (University of Lyon)

*Residue on affine Grasmannians*

This is a report on joint work with Mathieu Florence. For an affine algebraic group G/k, we will explain how a non integral point of G( k((t)) ) gives rise to a homomorphism H –> G where H is either the additive group or the multiplicative group.