Short talks
Yasmeen Baki
Title: Obstructions towards defining nonabelian group-graded twisted Calabi–Yau algebras
Abstract: In this talk, we motivate the study of nonabelian group-graded twisted Calabi–Yau (tCY) algebras by examining a result of Crawford which gives an example of a 2-dimensional AS regular algebra graded by a finite nonabelian group. While historically the study of graded tCY algebras has meant the study of such algebras under N-gradings, recent results have given a suitable definition of G-graded tCY algebras for G an arbitrary abelian group. A natural next question is if such a suitable definition can be given if we are to drop the abelian assumption on G. We discuss the major obstructions and possible remedies towards providing a definition of graded tCY algebras in the nonabelian case.
Lucas Buzaglo
Title: Hopf coactions on quantum polynomial rings
Abstract: Hopf coactions on two-dimensional AS-regular algebras have been widely studied. For example, Crawford proved that if H = kG is a noncommutative group algebra which coacts inner faithfully on a two-dimensional AS-regular algebra A, then A = k<x,y>/(xy + yx) and G is a quotient of the group <a,b | a^2 = b^2>. On the other hand, little is known about the situation in three or more dimensions. I will talk about some of my recent work with Daniel Rogalski about Hopf coactions on three-variable quantum polynomial rings.
Sarafina Ford
Title: Stanley’s theorem for gentle algebras
Abstract: Stanley’s theorem is a result from commutative algebra which allows us to determine if an algebra is Gorenstein from its Hilbert series. For gentle algebras, a well-studied class of bound quiver algebra, there is an analogue of Stanley’s theorem that allows us to determine whether a gentle algebra is AS Gorenstein from symmetries of its matrix Hilbert series.
Peter Goetz
Title: Noncommutative Matrix Factorizations from Quadratic Normal Elements
Abstract: Motivated by recent work with Kirkman-Moore-Vashaw, I will explain how to construct noncommutative matrix factorizations from quadratic normal elements in a Koszul algebra. This work, joint with W. Frank Moore, builds upon work of Smith-van den Bergh, Mori-Ueyama and He-Ye.
Mengwei Hu
Title: On certain Lagrangian subvarieties in minimal resolutions of Kleinian singularities
Abstract: Kleinian singularities are quotients of C^2 by finite subgroups of SL_2(C). They are in bijection with the ADE Dynkin diagrams via the McKay correspondence. In this talk, I will introduce certain singular Lagrangian subvarieties in the minimal resolutions of Kleinian singularities that are related to the geometric classification of certain unipotent Harish-Chandra (g,K)-modules. The irreducible components of these singular Lagrangian subvarieties are P^1's and A^1's. I will describe how they intersect with each other through the realization of Kleinian singularities as Nakajima quiver varieties. Time permitting, we will also discuss the deformations of these singular Lagrangian subvarieties.
Thomas Lamkin
Title: Point Modules of Non-Connected Graded Algebras
Abstract: One of the earliest successes of noncommutative projective geometry was the use of (truncated) point modules---or more precisely, their (sequence of) fine moduli spaces---by Artin-Tate-Van den Bergh to study 3-dimensional Artin-Schelter regular algebras. Since then, there has been much interest in studying point modules of connected graded algebras. In this short talk, we discuss what we believe is the right notion of a (truncated) point module for a path algebra with relations, show they are parameterized by a quotient stack, and show the resulting sequence of stacks stabilizes for a family of 3-dimensional twisted graded Calabi-Yau algebras.
Ian Martin
Title: Bicharacter twists of quantum groups
Abstract: It has long been known that multiparameter quantum groups can be obtained from their one-parameter counterparts by twisting the algebra structure by a Hopf 2-cocycle. However, such a description is often not suitable for explicit computations, and it has further limitations for affine quantum groups, since there is no explicit description of the coproduct in the new Drinfeld realization. We shall indicate how a different method -- twisting a bigraded Hopf algebra by a skew-bicharacter -- applies to quantum groups, and show that this procedure allows one to prove fundamental results on multiparameter quantum groups by reducing to the corresponding one-parameter quantum groups. In particular, we will see how to establish isomorphisms between the Drinfeld-Jimbo, new Drinfeld, and R-matrix realizations of two-parameter quantum affine algebras in all classical types. This is based on joint work with Alexander Tsymbaliuk.
Shashank Singh
Title: A Quiver-Theoretic Description of Module Categories over Taft Algebras
Abstract: The Taft algebras $T_\ell(q)$ are a fundamental class of non-commutative, non-cocommutative finite-dimensional Hopf algebras. While the classification of exact module categories over $\text{Rep}(T_\ell(q))$ is established, explicit combinatorial realizations of these categories provide deeper structural insight. In this talk, I present a quiver-theoretic description of these module categories. We demonstrate how the action of the tensor category can be encoded into the path algebra of a directed quiver. This approach translates the abstract categorical data into concrete graph-theoretic properties, offering a clear visualization of the indecomposable module categories associated with Taft algebras.
Padmini Veerapen
Title: A potpourri of twists
Abstract: In this talk we explore twists of algebras and bialgebras/Hopf algebras and discuss how they can be explicitly computed.
Trung Vu
Title: On De Concini-Kac forms of quantum groups
Abstract: A quantum group over complex numbers is a Hopf algebra depending on a semisimple Lie algebra and a complex parameter q. When q is a root of unity, there are several forms that people consider. The De Concini-Kac form is the easiest to define, and it behaves like the enveloping algebra in positive characteristics; in particular, it is finitely generated over the center. When the order of q is odd, there is a nice description of the center, and its representation theory has been extensively studied in the works of De Concini-Kac-Procesi and others. In contrast, when the order of q is even, the center is more complicated, and its representation theory is less understood. I will discuss a modification to the De Concini-Kac form that places the cases of odd and even order roots of unity on equal footing.
Yunmeng Wu
Title: Quantum Laurent Phenomenon Algebras
Abstract: Laurent Phenomenon Algebras (LPAs), introduced by Lam and Pylyavskyy, generalize cluster algebras by allowing arbitrary exchange polynomials while preserving the Laurent Phenomenon property under mutation. Although LPAs provide a flexible framework for commutative mutation systems, yet a corresponding quantum theory has not been developed. In this talk, I outline ongoing work that defined Quantum Laurent Phenomenon Algebras (qLPAs)—a new class of noncommutative algebras that extend the structural ideas of LPAs to the quantum setting. I will describe how quasi-commutation relations and a generalized compatibility condition can be used to formulate mutation rules and establish a quantum Laurent phenomenon. This short talk aims to illustrate how the qLPA provides a unifying framework for certain import quantum algebras that are not quantum cluster algebras, including quantized Weyl algebra at roots of unity, allowing their representation theory to be treated analogously to quantum cluster algebras and LPAs. This is based on joint work with Charles Barth and Milen Yakimov.