The Ottawa-Carleton joint algebra seminar meets during the academic year approximately once per week, either at the University of Ottawa or at Carleton University.
| Date
| Speaker
| Title (click on titles to show/hide abstracts)
|
| Sept 13 |
Pedro Vaz (U C Louvain) |
An overview on categorification of Verma modules and the case of sl(2) via a symmetric group action
Abstract:
In this talk I will first give an overview of the program of categorification of Verma modules and its applications.
I will describe the construction for sl(2) via an action of the symmetric group on a supercommutative ring.
In the last part of the talk I will explain how the construction can be modified to the general case of parabolic Verma modules for symmetrizable Kac--Moody algebras and their tensor products with integrable representations and how to apply them to Khovanov-Rozansky link homology theories.
An emphasis will be given to open problems.
|
| Oct 4 |
Cameron Ruether (Memorial University) |
Quadratic pairs over schemes and the canonical quadratic pair of a Clifford algebra
Abstract:
Quadratic pairs on central simple algebras over a field were introduced in The Book of Involutions in order to manage the problems which arise with orthogonal involutions in characteristic 2. They are used throughout the book to give a characteristic agnostic treatment of semisimple groups of type D. However, a notable exception occurs in the final chapters on triality where the base field is assumed to be of characteristic different from 2. The authors make this assumption because, as they write, "we did not succeed in giving a rational definition of the quadratic pair on [the Clifford algebra]." This problem was recently solved by Dolphin and Quéguiner-Mathieu, who defined the canonical quadratic pair while working over fields of characteristic 2. The notion of quadratic pairs has been extended by Calmés and Fasel beyond working over fields to the setting over schemes. We show that there exists a canonical quadratic pair on Clifford algebras in this setting as well, which extends the definition of Dolphin and Quéguiner-Mathieu. However, due to some quirks of working over schemes, the approach of Dolphin and Quéguiner-Mathieu requires some modification. We will review the basic definitions and properties of quadratic pairs, over fields and over schemes, and then describe the modified construction of the canonical pair on Clifford algebras We will outline the key properties which earn it the name "canonical" and which allow for the theory of triality to proceed over schemes.
|
| Oct 26/27 |
Chelsea Walton (Rice University) |
Modernizing Modern Algebra, I,II: Category Theory is coming, whether we like it or not
Abstract:
Part I: Inspired by my past stays at the University of Hamburg, I will chat about the development of the field, Modern Algebra, starting with the foundations set in Germany in the 1920s. Through the work of E. Artin and E. Noether, and through the writings of B. L. van der Waerden, Modern Algebra launched onto the mathematical scene as a sort of Haute Couture fashion in the 1930s. These days, this field is certainly presented as fashion for the masses, as is included in any standard undergraduate curriculum in mathematics. In fact, the contents of van der Waerden’s landmark 1931 textbook “Moderne Algebra” is much in line with our syllabi for algebra courses today. Now at the 100 year mark since the emergence of Modern Algebra, one might wonder: What’s next? I believe it’s Category Theory, whether we like it or not. To support this belief, I’ll present a case study for “algebras” in various settings.
Part II: Continuing Part I of the talk, “Modernizing Modern Algebra”, I will discuss the history of “representations” in modern algebra, and will lead into the categorical versions of these structures. Recent joint work with Robert Laugwitz and Milen Yakimov (arXiv:2307.14764) on categorical representations will be presented.
|
| Nov 3 |
Erhard Neher (Ottawa) |
Knebusch's norm principles for quadratic forms
Abstract:
Let K/F be a field extension of finite degree d and let q : V -> F be a quadratic form. We thus have the set D(q) = q(V) \ {0} of non-zero values of q and the corresponding set D( qK) for the extended quadratic form qK : V⊗ F K -> K. Knebusch's norm principle says that N K/F(D( qK)) ⊂ F is a product of at most d elements of D( q).
I will discuss this principle, replacing the field F by a semilocal ring R and the field extension K/F by a finite étale extension of R of constant degree. Time permitting, I will describe consequences for spinor norms and norm groups of quadrics.
The talk is based on joint work with Philippe Gille (Lyon).
|
| Jan 25 |
Ben Webster (Waterloo/Perimeter) |
Gelfand-Tsetlin modules and other fun with Coulomb branches
Abstract:
As discussed in my colloquium, the theory of Coulomb branches sheds new light on familiar objects in representation theory, in particular, on the universal enveloping algebra of gl(n). I’ll discuss in a bit more detail how this gives us a new perspective on category O for this algebra and on the larger category of Gelfand-Tsetlin modules.
|
| Jan 26 |
Keivan M. Karai (CU Bremen) |
Optimal linear sofic approximations of countable groups
Abstract:
A finitely generated group is called linear sofic if it can be approximated (in a precise sense) by the family of metric groups GLd( C), where the distance between matrices A and B is defined by the normalised rank: d(A,B):=rank(A-B)/d. This family of groups which generalises the class of sofic groups, was introduced and studied by Arzhantseva and Paunescu.
In this talk, we will discuss a question of Arzhantseva and Paunescu regarding a quantitative aspect of linear sofic approximations and discuss its connection to random walks on groups, and then talk about stability questions with respect to this metric. This talk is based on a joint work with Maryam Mohammadi Yekta.
|
| Feb 28 |
Curtis Wendlandt (Saskatchewan) |
Affine Yangians and toroidal Lie bialgebras
Abstract:
The theory of Yangians arose in the 1980's as an algebraic framework for
systematically producing rational solutions of the celebrated quantum
Yang-Baxter equation (qYBE) from theoretical physics. Roughly speaking,
the mechanism by which these solutions arise is as follows: Starting
from any simple Lie algebra g, one can construct a Hopf algebra, called
the Yangian of g, which quantizes the Lie bialgebra g[t] of polynomials
in a single variable with coefficients in g. This Hopf algebra comes
equipped with a remarkable formal series R(z), called its universal
R-matrix, which provides a universal, formal solution to the qYBE. The
desired rational solutions are then obtained by evaluating R(z) on the
tensor product of any two finite-dimensional irreducible representations
of the Yangian. This construction, and the series R(z) itself, play an
integral role in the representation theory of affine quantum groups and
its many intersections with geometry and mathematical physics. In this
talk, I will present a non-trivial generalization of this story to the
setting where g is an affine Lie algebra and try to explain how to
interpret this from the point of view of quantization of toroidal Lie
bialgebras. This is based on joint work with Andrea Appel, Sachin
Gautam, Alex Weeks, and Dat Minh Ha.
|
| Mar 1 |
Khoa Nguyen (Queen's) |
A family of U(h)-free modules of rank 2 over sl(2)
Abstract:
The study of simple sl(2)-modules can be divided into two categories. The first category consists of weight modules, namely, those that decompose into direct sums of their weight space with respect to a fixed Cartan subalgebra $h$. Such modules have been classified explicitly. The other category consists of non-weight modules. In particular, simple non-weight sl(2)-modules are U(h)-free modules of finite rank when restricted to U(h). Block showed that there is a bijection between simple non-weight modules and irreducible elements in a non-commutative ring. However, an explicit realization of these modules has yet to be discovered.
In this talk, I will introduce a new family of U(h)-free modules of rank 2 over sl(2). Nonetheless, we will also discuss other families of U(h)-free modules in the literature. This is based on a joint work with D. Grantcharov and K. Zhao.
|
| Mar 18 |
Sara Dijols (Calgary) |
Generic representations and ABV packets for p-adic groups
Abstract:
I will first explain why tempered and generic Langlands parameters are open, and derive a number of consequences, among which stands the enhanced genericity conjecture of Shahidi for quasi-split classical groups and its analogue in terms of ABV packets. This is a joint work with Clifton Cunningham, Andrew Fiori, and Qing Zhang.
|
| Mar 22 |
Erhard Neher (Ottawa) |
Corestriction
Abstract:
Given a Galois extension K'/K of fields, corestriction associates a K-vector space with any K'-vector space. It preserves associative algebras, even central-simple algebras, and gives rise to a homomorphism of Brauer groups. In my talk, I will explain corestriction in the setting of Galois extensions of rings. At the end, I will give a very sketchy glimpse on my recent work with Philippe Gille and Cameron Ruether, in which we generalize the ring setting to schemes and stacks of quasi-coherent modules.
|
| April 5 |
Susanne Pomplün (Ottawa) |
Generalizing classical results from central simple algebras to the nonassociative setting, and introducing the semiassociative Brauer monoid.
Abstract:
Recently, the theory of semiassociative algebras and their Brauer monoid was introduced by Blachar, Haile, Matri, Rein, and Vishne as a canonical generalization of the theory of associative central simple algebras and their Brauer group: together with the tensor product semiassociative algebras over a field form a monoid that contains the classical Brauer group as its unique maximal subgroup.
I would like to introduce to you some classes of semiassociative algebras that are canonical generalizations of cyclic simple algebras and explore their behaviour in the Brauer monoid. Time permitting, we can also discuss some - hopefully interesting - particularities of this newly defined Brauer monoid.
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This seminar is sponsored by a grant from the Tutte Institute for Mathematics and Computing.
