## Joint UOttawa/Carleton

## Algebra Seminar

### Fall 2022

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

coffee 2:30-2:45; talk 2:45-3:45 (no Zoom, sorry)

**September 30**

Kirill Zaynullin (UOttawa)

Title: Coproduct on structure algebras and cohomology of a group

Abstract: The notion of a structure algebra of a moment graph is a key technical combinatorial tool to compute the equivariant cohomology of a T-variety (Braden-MacPherson, Fiebig, Lanini). We explain how to construct the coproduct map on such structure algebra and how to extend this construction to K-theory and any generalized cohomology. This is joint work with Martina Lanini and Rui Xiong.

## Algebra Day

#### October 6 (UOttawa)

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)

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.

**October 14**

Rui Xiong (UOttawa)

Reading course: Toric varieties and its applications. Introductory lecture.

Abstract: We intend to organize the reading course with talks given during available Algebra seminar time-slots. Graduate students and postdocs are welcome to participate.

The final purpose is to understand the proof of Read’s conjecture stating that the absolute value of coeﬀicients of the chromatic polynomial of a graph is unimodal (sinkless). Note that June Huh was awarded the Fields Medal in 2022 due to the mentioned work. More details are available HERE.

**October 21**

Rui Xiong (UOttawa)

Reading course: Toric varieties and its applications. Lecture 1.

Abstract: This is the reading course with talks given during available Algebra seminar time-slots. Graduate students and postdocs are welcome to participate.

The final purpose is to understand the proof of Read’s conjecture stating that the absolute value of coeﬀicients of the chromatic polynomial of a graph is unimodal (sinkless). More details are available HERE.

**October 28**

Jethro van Ekeren (Universidade Federal Fluminense)

Title: Unimodular lattices and vertex algebras of rank 24

Abstract: Classical results in lattice theory assert that unimodular even integral lattices must have rank divisible by 8, that there is a unique such lattice of rank 8 (namely the E8 lattice), that in rank 24 there are exactly 24 such lattices – the Niemeier lattices – and that these are indexed by their root systems. In this talk I will describe a similar story for a natural generalisation of unimodular lattices, namely the holomorphic vertex algebras. We find a complete classification of such algebras in rank 24 (modulo an outstanding conjecture about the monstrous moonshine module), in which the list of 24 indexing root systems is extended to a list of 71 reductive Lie algebras. Joint work with various subsets of: CH. Lam, S. Moeller, N. Scheithauer and H. Shimakura.

**November 4**

Rui Xiong (UOttawa)

Reading course: Toric varieties and its applications. Lecture 2.

Abstract: This is the reading course with talks given during available Algebra seminar time-slots. Graduate students and postdocs are welcome to participate.

The final purpose is to understand the proof of Read’s conjecture stating that the absolute value of coeﬀicients of the chromatic polynomial of a graph is unimodal (sinkless). More details are available HERE.

**November 11**

Rui Xiong (UOttawa)

Reading course: Toric varieties and its applications. Lecture 3.

Abstract: This is the reading course with talks given during available Algebra seminar time-slots. Graduate students and postdocs are welcome to participate.

The final purpose is to understand the proof of Read’s conjecture stating that the absolute value of coeﬀicients of the chromatic polynomial of a graph is unimodal (sinkless). More details are available HERE.

**November 25**

Rui Xiong (UOttawa)

Reading course: Toric varieties and its applications. Lecture 4.

**December 2 **

Allan Merino (UOttawa)

Title: Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup

Abstract: One of the main problems in representation theory is to determine the set of equivalence classes of irreducible unitary representations of a Lie group. Using the Weil representation, Roger Howe established a one-to-one correspondence (known as the local theta correspondence) between particular representations of two subgroups G and G’ forming a dual pair in Sp(W). This correspondence provides a nice way to construct unitary representations of small Gelfand-Kirillov dimension.

In his wonderful paper “Remarks on classical invariant theory”, Roger Howe suggested that his classical duality should be extendable to superalgebras/supergroups. In a recent work with Hadi Salmasian, we obtained a classification of irreducible reductive dual pairs in a real or complex orthosymplectic Lie supergroup SpO(V). Moreover, we proved a “double commutant theorem” for all dual pairs in a real or complex orthosymplectic Lie supergroup.

In my talk, I will spend quite some time explaining how the Howe duality works in the symplectic case and then talk about the results we obtained in our paper with H. Salmasian.

**December 9 **

Alistair Savage (UOttawa)

Title: Keeping it real

Abstract: Taking Schur’s lemma as our point of departure, we will begin with a stroll through the world of division superalgebras. This will take us down the threefold way and the tenfold way, before arriving at a classification of real forms of important Lie superalgebras. We will then describe how the representation theory of these real Lie superalgebras is controlled by certain natural string diagrams. This is joint work with Saima Samchuck-Schnarch.

**December 15 **(Thursday at 4:00pm !!!)

Philippe Gille (University of Lyon)

When is a reductive group scheme linear?

TBA