Speaker: Yuri Bahturin, Memorial University of Newfoundland
Time/Date: May 22, 2024, 1 p.m.
Room: HH-3017
Title: On generic properties of nilpotent algebras

Abstract:
We study general nilpotent algebras. The results obtained are new even for the classical algebras, such as associative or Lie algebras. We single out certain generic properties of finite-dimensional algebras, mostly over infinite fields. The notion of being generic in the class of n-generated algebras of an arbitrary primitive class of c-nilpotent algebras appears naturally in the following way. On the set of the isomorphism classes of such algebras one can introduce the structure of an algebraic variety. As a result, the subsets are endowed with the dimensions as algebraic varieties. A subset Y of a set X of lesser dimension can be viewed as negligible in X. For example, if n >> c, we determine that an automorphism group of a generic algebra P consists of the automorphisms, which are scalar modulo P2. Generic ideals are in I(P), the annihilator of P. In the case of classical nilpotent algebras as above, the generic algebras are always graded by degrees with respect to some generating sets.

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Speaker: Felipe Yukihide Yasumura, University of Sao Paulo, Brazil
Time/Date: Wednesday, March 20, 2024, 1 p.m.
Room: HH-3017
Title: On *-homogeneous-graded polynomial identities of upper triangular matrices

Abstract:
We will start this talk with an introduction to the basic theory of polynomial identities. Afterward, we will explain the concept of homogeneous involution and polynomial identities with homogeneous involution. Finally, we will describe the *-homogeneous-graded polynomial identities of the algebra of upper triangular matrices. This is joint work with Thiago Castilho (UNIFESP, Brazil).

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Speaker: Cameron Ruether, Memorial University of Newfoundland
Time/Date: Friday, February 16, 2024, 4 p.m.
Room: HH-3017
Title: Triality without octonions

Abstract:
Triality, the phenomena arising from the extra symmetries of the Dynkin diagram D4, is traditionally explained using octonion algebras. We will review this typical approach before discussing an alternative using Chevalley generators. We will discuss how the elementary subgroup generated by Chevalley generators may differ from its algebraic group and how these differences can be managed by working with sheaves. This allows us to use computations with the generators of the Spin group to identify trialitarian phenomena. Finally, we will discuss how, using either approach, many standard facts about triality over fields also hold more generally over schemes.

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Speaker: Phil Branagan, Memorial University of Newfoundland
Time/Date: Friday, January 26, 2024, 12 p.m.
Room: HH-3017
Title: Merge and why a Hopf algebra may matter for syntactic theory

Abstract:
In a series of recent papers, Marcolli, Chomsky and Berwick have proposed that the syntactic operation of Merge should be modelled with a Hopf algebra. This informal presentation will describe what Merge does in Linguistic theory and what types of problems these new proposals may help to solve.

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Speaker: Tom Baird, Memorial University of Newfoundland
Time/Date: 
Wednesday, January 17, 2024, 1 p.m. 
Wednesday, January 24, 2024, 1 p.m.
Room: HH-3017
Title: Cohomology of fixed point sets of anti-symplectic involutions I & II

Abstract:
Let S be a smooth quasi-projective complex surface. The Hilbert scheme of n points in S, S^[n], is a smooth variety of dimension 2n which contains the variety of n distinct unordered points as a dense open subvariety. This variety has many applications in algebraic geometry, combinatorics, and mathematical physics. If S is a symplectic variety, then S^[n] is naturally symplectic. Given an anti-symplectic involution of S, there is an induced involution on S^[n]. The fixed point set of this involution is a smooth Lagrangian subvariety of dimension n in S^[n]. In this talk I explain how to calculate the cohomology of this subvariety. For the special case S = C², this is done using a “Morse theory” argument borrowed from Ellingsrud-Stromme. For the general case, we adapt an argument due to Gottsche-Soergel involving the BBD Decomposition Theorem for semismall morphisms.

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Speaker: Felipe Yukihide Yasumura, University of Sao Paulo, Brazil
Time/Date: Wednesday, November 22, 2023, 1 p.m.
Room: HH-3017
Title: Group gradings on the algebra of infinite triangular matrices

Abstract:
In this talk, we present the classification of isomorphism classes of group gradings on an infinite-dimensional upper triangular matrix algebra. For this purpose, we discuss some connections between the grading and the natural topology of the algebra. This is joint work with Waldeck Schutzer (UFSCar).

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Speaker: Cameron Ruether, Memorial University of Newfoundland
Time/Date: Wednesday, November 15, 2023, 1 p.m.
Room: HH-3017
Title: Quadratic pairs on Clifford algebras - Part 2

Abstract:
We will begin by recalling the definition of quadratic pair and the construction of the Clifford algebra associated with a central simple algebra with quadratic pair from the seminar two weeks ago. The Clifford algebra has its own canonical involution, which in some cases is orthogonal, and when it is orthogonal the Clifford algebra has recently been given a canonical quadratic pair by Dolphin and Quéguiner-Mathieu. We will present their construction, which is formulated over fields of characteristic 2. Secondly, we will discuss the extension of these notions to Clifford algebras over a scheme where the naïve generalization of Dolphin and Quéguiner-Mathieu’s construction requires some modification.

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Speaker: Erhard Neher, University of Ottawa
Time/Date: Tuesday, November 7, 2023, 1 p.m.
Room: HH-3017
Title: Knebusch’s norm principles for quadratic forms

Abstract:
I will explain what is Knebusch’s norm principle for quadratic forms, and how to prove it over semilocal rings, in particular fields. I will also discuss an application of the principle to the cohomology of spin groups. The talk is based on joint work with Philippe Gille (Lyon).

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Speaker: Cameron Ruether, Memorial University of Newfoundland
Time/Date: Wednesday, November 1, 2023, 1 p.m.
Room: HH-3017
Title: The canonical quadratic pair on Clifford algebras

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-free 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. We will review the basic definitions and properties of quadratic pairs and Clifford algebras and then describe the construction of the canonical pair on Clifford algebras. We will outline the key properties which earn it the name "canonical". Time permitting, we will discuss the extension of these notions to Clifford algebras over a scheme where the na ̈ıve generalization of Dolphin and Quéguiner-Mathieu’s construction requires some modification.

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Speaker: Bradley Howell (MSc thesis), Memorial University of Newfoundland
Time/Date: Wednesday, October 25, 2023, 2 p.m.
Room: HH-3017
Title: Coends and categorical Hopf algebras

Abstract:
We first review some basics of category theory, the construction of a coend as a categorical Hopf algebra, and duals and homomorphic images of categorical Hopf algebras. We then prove that, in the case of the category of finite-dimensional modules over a finite-dimensional quasitriangular ribbon Hopf algebra, the hypothesis that the field is algebraically closed can be dropped in the result in Theorem 1.1 of the paper "Non-degeneracy conditions for braided finite tensor categories" by Kenichi Shimizu, which states that in a braided finite tensor category over an algebraically closed field, the triviality of the Müger centre implies that the Hopf pairing is non-degenerate. The coend in this category can be constructed as the dual space of the quasitriangular Hopf algebra.

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Speaker: Yorck Sommerhäuser, Memorial University of Newfoundland
Time/Date: 
Wednesday, October 4, 2023, 1 p.m.  
Wednesday, October 18, 2023, 2 p.m.
Room: HH-3017
Title: Hopf algebras in categories and their extensions - Parts 1 & 2

Abstract:
Beginning with Schreier’s extension theory for groups, the theory of extensions has been developed for many algebraic system, in particular for Hopf algebras by the work of N. Andruskiewitsch, J. Devoto, M. Feth, I. Hofstetter, S. Majid, W. M. Singer, and others. Even more recently, this theory has been extended to Hopf algebras in quasisymmetric monoidal categories by Y. Bespalov and B. Drabant.

However, when applied to the quasisymmetric category of Yetter-Drinfel’d modules over a given Hopf algebra, it turns out that this latter extension theory does not cover important examples, which is due to the fact that in these examples the cleaving and the cocleaving maps are not morphisms in the Yetter-Drinfel’d category. As explained by the speaker in earlier talks, this problem can be addressed by introducing the so-called deviation and codeviation maps. In this talk, we give a categorical adaptation of these maps, making the theory applicable to categories that are more general than the Yetter-Drinfel’d category. The talk is based on joint work with Y. Kashina.

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Speaker: Sebastiano Argenti, Università degli Studi della Basilicata, Italy
Time/Date: Wednesday, September 20, 2023, 1 p.m.
Room: HH-3017
Title: Lie semisimple algebras of derivations and varieties of almost polynomial growth

Abstract:
In this talk we are interested in the growth of the differential identities of finite dimensional associative algebras, i.e., polynomial identities of algebras with an action of a Lie algebra by derivations.

In this context Gordienko and Kochetov proved that, over fields of characteristic zero, the exponent always exists and is a positive integer. As a consequence of their proof, the codimension sequence associated to the differential identities either grows polynomially or exponentially [1].

We are interested in characterizing associative algebras A of almost poly- nomial growth, that is the codimension sequence of A grows exponentially but for any finite dimensional algebra in the variety generated by A, generating a proper subvariety, the corresponding growth is only polynomial.

This problem appears naturally in the study of varieties of PI-algebras with (or without) some additional structure and was solved for associative algebras [2], graded algebras [3], algebras with involution [4], trace algebras [5], etc... A common flavor of these results is the existence of a finite number of such varieties and the indication of a concrete list of generating algebras. Our main
result concerns the characterization of varieties of almost polynomial growth in case the Lie algebra acting by derivations is sl2. In this case we obtain two infinite classes of algebras generating distinct differential varieties of almost polynomial growth.

We also discuss the more general case in which L is any finite dimensional semisimple Lie algebra. We exhibit a list of algebras such that the codimension growth of a variety is exponential if and only if it contains one of the algebras in the list.