## 75th Algebra Days

## uOttawa/Carleton

#### October 6 (uOttawa, STM 664)

3:30 Opening coffe

3:45-4:30 Yuri Bahturin (Memorial University of New Foundland)

Title: Nilpotent algebras, implicit function theorem, and polynomial

quasigroups (joint work with Alexander Olshanskii)

Abstract: We study finite-dimensional nonassociative algebras. We prove

the implicit function theorem for such algebras. This allows us to

establish a correspondence between such algebras and quasigroups in the

spirit of classical correspondence between divisible torsion-free

nilpotent groups and rational nilpotent Lie algebras. We study the

related questions of the commensurators of nilpotent groups, filiform

Lie algebras of maximal solvability length and partially ordered

algebras.

4:45-5:30 Charles Paquette (Royal Military College of Canada, Kingston)

Title: Bricks in representation theory of algebras

Abstract: Over a given algebra, a brick (or Schur module) is a module having its endomorphism algebra a division algebra. For a finite-dimensional algebra A, bricks form an interesting (and often proper) subset of the set of indecomposable modules. In this talk, we will see why bricks are important objects among all indecomposable modules, from geometric, combinatorial and representation theoretic point of views. We will see how Fomin and Zelevinsky cluster algebras brought the representation theorists to study bricks. I will also mention some results and open problems on the distribution of bricks.

#### October 7 (Carleton University, HP 4331)

2:30 Opening coffee

2:45-3:30 Cameron Ruether (UOttawa)

Title: Obstructions to Quadratic Pairs over a Scheme

Abstract: Quadratic pairs on a central simple algebra over a field were introduced by Knus, Merkurjev, Rost, and Tignol in “The Book of Involutions” in order to work with semisimple linear algebraic groups of type D in characteristic 2. The concept was generalized by Calmés and Fasel, who defined quadratic pairs on Azumaya algebras over an arbitrary base scheme, also with groups of type D in mind. We will review these definitions and some of the motivations for using quadratic pairs before discussing recent work with Philippe Gille and Erhard Neher. We define two cohomological obstructions attached to an Azumaya algebra with orthogonal involution. The weak obstruction prevents the existence of a quadratic pair, and the strong obstruction prevents any potential quadratic pairs from having a certain convenient presentation. Interestingly, both these obstructions are trivial over affine schemes, and so quadratic pairs have noticeably different behaviour when working over arbitrary schemes. To demonstrate that this behaviour is possible, we will also present examples where one or both obstructions are non-trivial.

3:45-4:30 Emily Cliff (University of Sherbrook)

Title: Higher symmetries: smooth 2-groups and their principal bundles

Abstract: A 2-group is a categorical generalization of a group: it’s a category with a multiplication operation which satisfies the usual group axioms only up to coherent isomorphisms. In this talk I will introduce the category of Lie groupoids and bibundles between them, in order to provide the definition of a smooth 2-group. I will define principal bundles for such a smooth 2-group, and provide classification results that allow us to compare them to principal bundles for ordinary groups. This talk is based on joint work with Dan Berwick-Evans, Laura Murray, Apurva Nakade, and Emma Phillips. I will not assume any previous background on 2-groups or Lie groupoids.

4:45-5:30 Rui Xiong (UOttawa)

Title: Pieri Rules for CSM Classes

Abstract: In this talk, we will focus on Chern-Schwartz-MacPherson (CSM) classes over classic flag varieties. We will begin with a brief introduction to CSM classes. Then we will explain our main theorem – an equivariant CSM Pieri rule which includes as special cases many previously known formulas for CSM classes or Schubert classes. Lastly, we will discuss the application of the Murnaghan-Nakayama rule to the enumeration of domino tableaux. This talk is based on the joint work with Neil J. Fan and Peter L. Guo.