Lucas Buzaglo
Title: Dual reflection groups for three-dimensional Artin–Schelter regular algebras.
Abstract: Recent efforts in noncommutative invariant theory have sought to generalize the Chevalley–Shephard–Todd theorem by replacing commutative polynomial rings with Artin–Schelter (AS) regular algebras and group actions with Hopf actions. One possibility of Hopf algebra is the dual of a group algebra, whose action on an algebra A is the same as a grading of A. In this framework, a finite group G is called a dual reflection group if there exists a G-graded AS regular algebra such that the identity component under the grading is also AS regular. While gradings on two-dimensional AS regular algebras are completely understood, higher-dimensional cases have previously yielded only scattered examples.
In this talk, I will provide a classification of group gradings on three-dimensional AS regular algebras, focusing specifically on nonabelian grading groups. This is because an abelian group grading is the same as an action by an abelian group, which is better understood in the literature. Our main result proves that there are 6 infinite families and 7 exceptional 3-dimensional quadratic AS regular algebras graded by nonabelian groups, alongside 4 infinite families and 2 exceptional 3-dimensional cubic algebras. The classification is achieved by analyzing the superpotential and the Nakayama automorphism associated with the algebras. Thanks to our classification, we provide a complete list of nonabelian dual reflection groups for 3-dimensional AS regular algebras.
Ben Elias
Title: Fusion quivers
Abstract: A classical result of Gabriel states that the ADE quivers are precisely those quivers with finitely many indecomposable representations, and that indecomposable representations are parametrized by positive roots. Edmund Heng found a generalization of the concept of quiver representations which broadens Gabriel's theorem to include all finite Coxeter types ABCDEFGH. This talk will give an overview of recent work of Elias-Heng, which provides the proper framework for Heng's generalization and others: the concept of a fusion quiver. Our main result is a classification of fusion quivers with finitely many indecomposable representations.
Daniel K. Nakano
Title: Category O for Lie superalgebras
Abstract: In this talk, the speaker will define a general Category $\mathcal{O}$ for any quasi-simple Lie superalgebra. Our construction encompasses (i) the parabolic Category ${\mathcal O}$ for complex semisimple Lie algebras, and (ii) the known constructions of Category ${\mathcal O}$ for specific examples of classical Lie superalgebras. In particular, the speaker will develop a parabolic Category ${\mathcal O}$ for classical Lie superalgebras.
Connections between the categorical cohomology and the relative Lie superalgebra cohomology will be firmly established. These results will be used to show that the Category ${\mathcal O}$ is costandardly stratified. The definition of costandardly stratified used in this context is generalized from original definition of Cline, Parshall, and Scott. An description of the categorical cohomology ring will be given, and finite generation results will be presented.
Furthermore, it will be shown that the complexity of modules in Category ${\mathcal O}$ is finite with an explicit upper bound given by the dimension of the subspace of the odd degree elements in the given Lie superalgebra. This result provides a generalization of prior work for $\mathfrak{gl}(m|n)$ due to Coulembier and Serganova.
This talk represents joint work with Chun-Ju Lai and Arik Wilbert.