Titles and Abstracts

Speaker ： Vladimir Bavula (University of Sheffield)
Title: Classical left regular left quotient ring of a ring and its semisimplicity criteria
Abstract: Goldie's Theorem (1958, 1960) is a semisimplicity criterion for the classical left quotient ring of a ring. Semisimplicity criteria are given for the classical left regular left quotient ring. As a corollary, two new semisimplicity criteria are obtained for the classical left quotient ring of a ring (in the spirit of Goldie). Applications are given for the algebra of polynomial integro-differential operators.

Speaker ： Frederik Caenepeel (University of Antwerp)
Title: Fragments and cofragments in group and Lie algebra theory
Abstract: My PhD topic deals with the development of fragment theory, which can be seen as a generalization of representation or module theory. In this talk I will present some results of fragment theory in the area of groups and Lie algebras. As I will point out, the starting point to consider fragments is a positive algebra filtration. Interesting situations are obtained by considering a chain of finite groups or Lie algebras with associated algebra filtration by looking at the group algebras or universal enveloping algebras. The step from groups and Lie algebras to Hopf algebras is a natural one, hence we propose the definition of a cofragment. If time permits I will also present some first results on cofragments. All this is joint work with Fred Van Oystaeyen.

Speaker ： Stefaan Caenepeel (Free University of Brussels )
Title: Descent and Galois theory for Hopf categories
Abstract: A semi-Hopf category is a category enriched over the category of coalgebras over a field $k$; otherwise stated, it is a $k$-linear category in which every vectorspace of homomorphisms between two objects is a coalgebra, with some extra compatibility conditions. An antipode on a semi-Hopf category is a collection of maps $S_{x,y}:\ {\rm Hom}(y,x)\to {\rm Hom}(x,y)$, for all object $x,y$, satisfying the appropriate convolution rules. Hopf categories are semi-Hopf categories with an antipode. Hopf categories with one object correspond bijectively to ordinary Hopf algebras. The notion is related to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. The fundamental theorem for Hopf modules and some of its applications to Hopf categories, opening the way to the development of Galois theory.\\ First we develop descent theory for linear category. Given a linear category as an extension of a diagonal category, we introduce descent data, and the category of descent data is isomorphic to the category of representations of the diagonal category, if some flatness assumptions are satisfied. Then Hopf-Galois descent theory for linear Hopf categories, the Hopf algebra version of a linear category, is developed. This leads to the notion of Hopf-Galois category extension. We have a dual theory, where actions by dual linear Hopf categories on linear categories are considered. Hopf-Galois category extensions over groupoid algebras correspond to strongly graded linear categories.\\ The first part of the talk is joint work with Eliezer Batista and Joost Vercruysse. The second part is joint work with Timmy Fieremans.

Speaker ： Kenny De Commer (Free University of Brussels )
Title: Quantum actions on discrete quantum spaces
Abstract: S.L. Woronowicz introduced the notion of a compact quantum group, which on the purely algebraic level consists of a co-semisimple Hopf algebra with *-structure for which the associated normalized invariant functional is positive. One can make sense of actions of such compact quantum groups on operator algebras, i.e. algebras of bounded operators on some Hilbert space, closed under a certain topology. We will be interested in actions on operator algebras which are a (possibly infinite) direct sum of finite-dimensional matrix algebras over some index set I, which we interpret as the quantum analogue of a discrete space. We show that to any such action on a discrete quantum space we can associate an equivalence relation on I, which classically corresponds to the partition of a space into orbits of the action, and we show that the orbits of this equivalence relation are finite. We then apply these results to generalize the classical theory of Clifford, concerning the restrictions of representations to normal subgroups, to the framework of quantum subgroups of discrete quantum groups. This is joint work with Pawel Kasprzak, Adam Skalski and Piotr Soltan.

Speaker ： Ji-Wei He (Hangzhou Normal University )
Title: Finite group actions on Koszul Artin-Schelter regular algebras and BGG correspondence.
Abstract: In this talk, I will report some recent progress on finite group actions on Koszul Artin-Schelter regular algebras. Some new invariants related to the group actions were introduced in the study of group actions, which were applied to study the singularities of the invariant subalgebras (both in the isolated and non-isolated cases).

Speaker ： Nai-Hong Hu (East China Normal University )
Title: Some families of finite-dimensional Hopf algebras without Chevalley property
Abstract: The classification of finite dimensional Hopf algebras of given dimensions is a hard open problem. In this talk we will introduce some related background and some existing theories, and some classes we obtained recently. This is a joint work with Rongchuan Xiong.

Speaker ： Hua-Lin Huang (Huaqiao University )
Title: The Hurwitz problem on compositions of quadratic forms
Abstract: We shall give an introduction to the Hurwitz problem on compositions of quadratic forms and report some recent progress via an approach of Hopf algebras and tensor categories.

Speaker ： Fang Li (Zhejiang University )
Title: The positivity and D-vector positivity of cluster algebras and generalized Laurent phenomenon algebras
Abstract: In this talk, we will introduce the notion of generalized Laurent phenomenon algebras and try to use it to oversee the essence of cluster theory. Following this method, the conjectures on d-vectors and g-vectors respectively of cluster algebras are solved. This is a joint work with Peigen Cao.

Speaker ： Gong-Xiang Liu (Nanjing University)
Title: Quasi-Frobenius-Lusztig kernels
Abstract: Quasi-Frobenius-Lusztig kernels can be regarded as quasi-Hopf analogues of Frobenius-Lusztig kernels. We will compare the developments of Hopf algebras and quasi-Hopf algebras, and from which we gave the construction of these quasi-Hopf analogues. Some parts of this talk are based on a joint work with professors Fred van Oystaeyen and Yinhuo Zhang.

Speaker ： Shahn Majid (Queen Mary University of London)
Title: Double-Bosonization and Dual Bases of C_q[SL_2] and C_q[SL_3]
Abstract: The talk is based on my recent work with Ryan Aziz. We find a dual version of a previous double-bosonisation theorem whereby each finite-dimensional braided-Hopf algebra B in the category of comodules of a coquasitriangular Hopf algebra A has an associated coquasitriangular Hopf algebra coD_A(B). As an application we find new generators for c_q[SL2] reduced at q a primitive odd root of unity with the remarkable property that their monomials are essentially a dual basis to the standard PBW basis of the reduced Drinfeld-Jimbo quantum enveloping algebra u_q(sl2). Our methods apply in principle for general c_q[G] as we demonstrate for c_q[SL3] at certain odd roots of unity.

Speaker ： Ehud Meir (University of Hamburg )
Title: Hopf algebras, monoidal categories and geometric invariant theory
Abstract: In this talk I will describe a study of Hopf algebras by tools from symmetric monoidal categories and geometric invariant theory. I will explain how the study of finite dimensional semisimple Hopf algebras can be reduced into studying scalar invariants, and explain the connections of these scalars to some well known invariants (such as the Reshetikhin Turaev invariants of 3-manifolds and Frobenius-Schur indicators). I will also explain how one can receive some new finiteness results for Hopf algebras by using symmetric monoidal categories. More precisely, I will explain why every finite dimensional semisimple Hopf algebra admits at most finitely many Hopf orders over a given number ring.

Speaker ： Richard Ng (Louisiana State University )
Title: On spherical fusion categories of odd dimension
Abstract: A classical theorem of Burnside asserts that a finite group G has no nontrivial self-dual irreducible complex representation if and only if G has odd order. This result has been recently generalized to integral fusion categories. However, there exists nontrivial self-dual simple object in a non-integral fusion category of odd dimension. In this talk, we will discuss a relation satisfied by the self-dual simple objects of a modular tensor category of odd dimension in terms of their Frobenius-Schur indicators.

Speaker ： Lian-Gang Peng (Sichuan University)
Title: Modified Ringel-Hall algebras and derived Hall algebras
Abstract:In this talk, I will introduce modified Ringel-Hall algebras of complexes over hereditary abelian categories. It is shown that there is an embedding from the derived Hall algebra to the modified Ringel-Hall algebra. This is a joint work with Ji Lin.

Speaker ： Blas Torrecillas (University of Almeria )
Title: Cleft wreath algebras
Abstract: We will study this class of wreath algebras and its connections with cleft cowreath. We will present several situations where this algebras appears in a natural way: crossed product by a coalgebra, generalized crossed products and quasi-Hopf bimodules. This is a joint work with D. Bulacu.

Speaker ： Shuan-Hong Wang (Southeast University )
Title: Pontryagin Duality for Weak Multiplier Hopf Algebras with Integrals
Abstract: We generalize the main result of the first author Van Daele (1998) Adv. Math. 140 (1998), 323--366.） on the Pontryagin duality of multiplier Hopf algebras with integrals to weak multiplier Hopf algebras with integrals; we illustrate this dualityby considering the two natural weak multiplier Hopf algebras associated with a groupoid in detail and show that they are dual to each other in the sense of the above duality.

Speaker ：Quan-Shui Wu (Fudan University)
Title: Poisson Hopf algebras and co-Poisson Hopf algebras
Abstract: Co-Poisson structure (or coalgebra) is a dual concept of Poisson structure in categorial point of view. It arises also in mathematics and mathematical physics naturally.In the talk I will start from the defintions and basic properties of co-Poisson structures.The Hopf dual $H^\circ$ of any Poisson Hopf algebra $H$ is proved to be a co-Poisson Hopf algebra provided $H$ is noetherian. It is proved that there is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra.So the polynomial Hopf algebra, viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, is considered.The Poisson Hopf structures on polynomial Hopf algebras are exactly linear Poisson structures. The co-Poisson structures on polynomial Hopf algebras are characterized. Some correspondences between co-Poisson and Poisson structures are also established. This is a joint work with Lou Qi.

Speaker ： Jie Xiao (Tsinghua University )
Title: Ringel-Hall algebras beyond their quantum groups
Abstract: This talk is my recent joint work with Fan Xu and Minghui Zhao. We generalize the categorical constructions of a quantum group and its canonical basis introduced by Lusztig to the generic form of the whole Ringel-Hall algebra. We clarify the explicit relation between the Green formula and the restriction functor. By a geometric way to prove the Green formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztig's framework.

Speaker ：Pu Zhang (Shanghai Jiaotong University )
Title: Types of Serre subcategories of Grothendieck categories
Abstract: Every Serre subcategory $\mathcal S$ of an abelian category $\mathcal A$ is assigned a unique type. Roughly speaking, it is given by a pair $(m, -n)$ of numbers, where $m$ (resp. $n$) counts how many times one can form left (resp. right) adjoints starting from $i$ and $Q$, where $i: \mathcal S\rightarrow \mathcal A$ is the inclusion functor and $Q: \mathcal A\rightarrow \mathcal A/\mathcal S$ is the quotient functor. The main result gives a complete list of all the types of Serre subcategories of Grothendieck categories: $$(0, 0), (0, -1), (1, -1), (0, -2), (1, -2), (2, -1), (+\infty, -\infty).$$ The main tool used is (right, left) recollements of abelian categories. Two observations are technically crucial in proving the main result: the exactness of all the functors in a recollement of abelian categories forces the recollement to split; and any left recollement of a Grothendieck category can be extended to a recollement. This is a joint work with J. Feng.