We generalize S. Gelfand's classical construction of a Novikov algebra from a commutative differential algebra to get a deformation family $(A,\circ_q)$ of Novikov algebras by an admissible commutative differential algebra, which ensures a bialgebra theory of commutative differential algebras, enriching the antisymmetric infinitesimal bialgebra. Moreover, a deformation family of Novikov bialgebras is obtained, under certain further condition. In particular, we obtain a bialgebra variation of $S$. Gelfand's construction with an interesting twist: every commutative and cocommutative differential antisymmetric infinitesimal bialgebra gives rise to a Novikov bialgebra whose underlying Novikov algebra is $(A,\circ_{-1/2})$ instead of $(A,\circ_0)$ which recovers the construction of S. Gelfand. This is the joint work with Yanyong Hong and Li Guo.

The cyclotomic Hecke algebra $H_{r,p,n}$ of type $G(r,p,n)$ (where $r=pd$) can be realized as the $\sigma$-fixed point subalgebra of certain cyclotomic Hecke algebra $H_{r,n}$ of type $G(r,1,n)$ with some special cyclotomic parameters, where $\sigma$ is an automorphism of $H_{r,n}$ of order $p$. In this talk I shall present a number of remarkable rational and symmetric properties on the $\gamma$-coefficients arising in the construction of the seminormal basis for the semisimple Hecke algebra $H_{r,n}$. As an application, a seminormal basis for the Hecke algebra $H_{r,p,n}$ is explicitly constructed. This talk is based on a joint work with Wang Shixuan.

In order to study the “modular” representation theory of quantum gl(m | n) at root of unity, we introduce the quantum Manin supersapce and quantum (dual) Grassmann superalgebra with quantum divided power structure, and develop a kind of quantum differential calculus over them, and construct two kinds of quantum de Rham super complexes: one is of infinite length which is the quantized version of the classical analogue due to Manin-Deligne-Morgan in their early study of supermanifolds from gauge field theory, another is of finite length which has no classical analogue to our knowledge. For the latter, we prove the Poincare lemma for nontruncated complex, while for the truncated case, in order to calculate all the quantum de Rham cohomologies we need to develop a specific technique to overcome the complicated difficulties encountered in the quantum supercase. If time permits, I'll also talk about the “l-adic phenomenon” occurred in a kind of indecomposable modules in the root of unity case which originally were irreducible modules in the generic case. This talk is based on a series of our joint work with Dr. Ge Feng, and Prof. Marc Rosso.

In this talk, we try to talk about Hopf algebras with Chevalley property of finite, tame and discrete corepresentation typies. This is a joint work with Yu Jing.

We introduce a new family of quivers with potential for triangulated marked surfaces with punctures. We show that the deformation of the associated 3-Calabi-Yau categories corresponds to the partial compactification (with orbifolding) of the associated moduli spaces. As an application, we calculate the fundamental groups of these moduli spaces (of framed quadratic differentials), which in particular produces Euclidean Artin braid groups of type A, B, C and D.

Guided by Koszul duality theory, we consider the graded Lie algebra of coderivations of the cofree conilpotent graded cocommutative cotrialgebra generated by a graded vector space $V$. We show that in the case of $V$ being a shift of an ungraded vector space $W$, Maurer-Cartan elements of this graded Lie algebra are exactly post-Lie algebra structures on $W$. The cohomology of a post-Lie algebra is then defined using Maurer-Cartan twisting. The second cohomology group of a post-Lie algebra has a familiar interpretation as equivalence classes of infinitesimal deformations. Next we define a post-Lie-infty algebra structure on a graded vector space to be a Maurer-Cartan element of the aforementioned graded Lie algebra. Post-Lie-infty algebras admit a useful characterization in terms of $L$-infty-actions (or open-closed homotopy Lie algebras). Finally, we introduce the notion of homotopy Rota-Baxter operators on open-closed homotopy Lie algebras and show that certain homotopy Rota-Baxter operators induce post-Lie-infty algebras. This is a joint work with Andrey Lazarev and Rong Tang.

Elliptic Lie algebras of maximal type are nullity two extended affine Lie algebras, which are generalization of the affine Kac-Moody Lie algebras. It is well known that the restricted modules for any untwisted affine Kac-Moody Lie algebra are isomorphic to the modules for the associated affine vertex algebra, while the restricted modules for the twisted affine Kac-Moody Lie algebra are isomorphic to the twisted modules for the affine vertex algebra. In this talk we will recall the classification of elliptic Lie algebras of maximal type, and the notion of $\Gamma$-vertex algebra and equivariant $\phi$-coordinated quasi-modules for vertex algebras. I will then claim that there exists a vertex algebra $V$ associated with any elliptic Lie algebra of maximal type and an automorphism group $G$ of $V$ equipped with a linear character $\chi$, such that the category of restricted modules for the elliptic Lie algebra is isomorphic to the category of $(G,\chi)$-equivariant $\phi$-coordinated quasi-modules for the vertex algebra $V$. The arguments will be divided into the case for fgc type and the case for non-fgc type.

We firstly investigate the ghost ideal and the phantom ideal, respectively, in the derived category of a DG-ring. This allows us to introduce and investigate the notions of Cartan-Eilenberg projective modules, Cartan-Eilenberg injective modules, and Cartan-Eilenberg flat modules. By this way, we develop a theory of Cartan-Eilenberg homological algebra for a DG-ring. An immediate application of the above theory is that we give an affirmative answer to a conjecture of Minamoto. As another direction of the application of this theory, we may investigate Noetherian DG-ring from this approach, and parallel to classical ring theory, introduce and investigate the notions of a hereditary DG-ring, a coherent DG-ring, and a perfect DG-ring. This work is joint with Xianhui Fu and Wei Hu

D.Quillen给出的与表示论相关的模型结构都是abelian. M.Hovey总结出 abelian范 畴上的abelian模型结构的定义,并通过两个完备的余挠对作成的 Hovey三元组得到abelian 模型结构的一般构造，即著名的Hovey对应. 这项工作被J.Gillespie和J.Stovicek发展为正 合范畴上的正合模型结构的理论. 另一方面, A.Beligiannis和I.Reiten通过一个核为反变有 限的遗传的完备的余挠对得到 abelian 范畴上的弱投射模型结构. 本报告将A.Beligiannis和I.Reiten的理论发展到正合范畴上;说明正合模型结构与弱投 射模型结构之间的关系. 更主要地，我们提出模型结构的第3种构造，它不依赖于完备余挠 对. 这3类模型结构一般都是不同的. 我们将讨论它们之间的关系. 本报告基于与陆雪松的合作工作.

E. C. Dade firstly initiated the study of block extensions. Block extensions are useful in reducing problems and conjectures in representation theory of finite groups. Extensions of nilpotent blocks due to B. Kuelshammer and L. Puig are typical. In this talk, I focus on various generalizations of extensions of nilpotent blocks and their application to some conjectures in representation theory of finite groups.