Susanne Pumpluen: A way to generalize classical results from central simple algebras to the nonassociative setting

11th December, 2023

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. We present classes of semiassociative algebras that are canonical generalizations of classes of certain central simple algebras and explore their behavior in the Brauer monoid. Time permitting, we also discuss some - hopefully interesting - particularities of this newly defined Brauer monoid.

Eric Marberg: From Klyachko models to perfect models

5th December, 2023

In this talk a "model" of a finite group or semisimple algebra means a representation containing a unique irreducible subrepresentation from each isomorphism class. In the 1980s Klyachko identified an elegant model for the general linear group over a finite field with q elements. There is an informal sense in which taking the q→1 limit of Klyachko's construction gives a model for the symmetric group, which can be extended to its Iwahori-Hecke algebra. The resulting Hecke algebra representation is a special case of a "perfect model", which is a more flexible construction that can be considered for any finite Coxeter group. In this talk, I will classify exactly which Coxeter groups have perfect models, and discuss some notable features of this classification. For example, each perfect model gives rise to a pair of related W-graphs, which are dual in types B and D but not in type A. Various interesting questions about these W-graphs remain open.

Senne Trappeniers: The interplay between skew braces, the Yang-Baxter equation and Hopf-Galois structures

27th November, 2023

In 2007, Wolfgang Rump introduced algebraic objects called braces, these generalize Jacobson radical rings and are related to involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation (YBE). These objects were subsequently generalized to skew braces by Leandro Guarnieri and Leandro Vendramin in 2017, and a similar relation was shown to hold for non-degenerate set-theoretic solutions of the YBE which are not necessarily involutive. In this talk, we will describe this interplay between skew braces and the YBE. We will also discuss their relation to Hopf-Galois structures and see how this extends the classical Galois theory in an elegant way.

Guodong Zhou: The homotopy theory of operated algebras

16th October, 2023

The talk is a survey of our recent results on the homotopy theory of operated algebras such as Rota-Baxter associative (or Lie) algebras and differential associative (or Lie) algebras etc. We make explicit the Kozul dual homotopy cooperads and the minimal models of the operads governing these operated algebras. As a consequence the L∞ structures on the deformation complexes are described as well.

Yann Palu: 0-Auslander extriangulated categories

13th October, 2023

Categorification of cluster algebras has instilled the idea of mutation in representation theory. Nice theories of mutation, for some forms of rigid objects, have thus been developed in various settings. In a collaboration with Mikhail Gorsky and Hiroyuki Nakaoka, we axiomatized the similarities between most of those settings under the name of 0-Auslander extriangulated categories. The prototypical example of a 0-Auslander extriangulated category is the category of two-term complexes of projectives over a finite-dimensional algebra. In this talk, we will give several examples of 0-Auslander categories, and explain how they relate to two-term complexes.

Alfilgen Sebandal: Finite graded classification conjecture for Leavitt path algebras

2nd October, 2023

Given a directed graph, one can associate two algebraic entities: the Leavitt path algebra and the talented monoid. The Graded Classification conjecture states that the talented monoid could be a graded invariant for the Leavitt path algebra, i.e., isomorphism in the talented monoids reflects as graded equivalence in the category of graded modules over the Leavitt path algebra of the corresponding directed graphs. In this talk, we shall see confirmations of this invariance in the ideal structure of the talented monoid with the so-called Gelfand-Kirillov Dimension of the Leavitt path algebra. The last part of the talk is an affirmation of the Graded classification conjecture in the finite-dimensional case.

Mykola Khrypchenko: Transposed Poisson structures

4th September, 2023

A transposed Poisson algebra is a triple (L,⋅,[⋅,⋅]) consisting of a vector space L with two bilinear operations ⋅ and [⋅,⋅], such that (L,⋅) is a commutative associative algebra; (L,[⋅,⋅]) is a Lie algebra; and the 'transposed' Leibniz law holds: 2z⋅[x,y]=[z⋅x,y]+[x,z⋅y] for all x,y,z∈L. A transposed Poisson algebra structure on a Lie algebra (L,[⋅,⋅]) is a (commutative associative) multiplication ⋅ on L such that (L,⋅,[⋅,⋅]) is a transposed Poisson algebra. I will give an overview of my recent results in collaboration with Ivan Kaygorodov (Universidade da Beira Interior) on the classification of transposed Poisson structures on several classes of Lie algebras.

Peng Shan: Modularity for W-algebras and affine Springer fibres

31st August, 2023

We will explain a bijection between admissible representations of affine Kac-Moody algebras and fixed points in affine Springer fibres. We will also explain how to match the modular group action on the characters with the one defined by Cherednik in terms of double affine Hecke algebras, and extensions of these relations to representations of W-algebras. This is based on joint work with Dan Xie and Wenbin Yan.

Lleonard Rubio y Degrassi: Hochschild cohomology groups under gluing idempotents

7th August, 2023

Stable equivalences occur frequently in the representation theory of finite-dimensional algebras; however, these equivalences are poorly understood. An interesting class of stable equivalences is obtained by ‘gluing’ two idempotents. More precisely, let A be a finite-dimensional algebra with a simple projective module and a simple injective module. Assume that B is a subalgebra of A having the same Jacobson radical. Then B is constructed by identifying the two idempotents belonging to the simple projective module and to the simple injective module, respectively. In this talk we will compare the first Hochschild cohomology groups of finite-dimensional monomial algebras under gluing two arbitrary idempotents (hence not necessarily inducing a stable equivalence). As a corollary, we will show that stable equivalences obtained by gluing two idempotents provide 'some functoriality' to the first Hochschild cohomology, that is, HH1(A) is isomorphic to a quotient of HH1(B).

Ziqing Xiang: Quantum wreath product

28th July, 2023

The classical wreath product G ≀ Σd is a semidirect product Gd ⋊ Σd with Σd acting on Gd by permutations. We deform this classical wreath product by deforming G into an associative algebra B, deforming Σd into a Hecke algebra, and deforming the action. The result is called a quantum wreath product B ≀ H(d). Many variants of Hecke algebras can be viewed as quantum wreath products, hence could be treated in a unified manner.

In this talk, we will discuss necessary and sufficient conditions for quantum wreath products to have a basis of suitable size. We will also discuss some other structural results, the Schur algebras of these quantum wreath products, and their representations.