Department of mathematical sciences,
My Research Interests
The research of Xiao Jie has mainly been on representation theory of algebras, and related topics in quantum groups, infinite dimensional Lie algebras and triangulated categories. In recent years he has concentrated on the interaction between Hall algebras and quantum groups. A quiver is just a directed graph, and a representation associates a vector space to each vertex and a linear map to each arrow. It was observed by P. Gabriel that the quivers with only finitely many indecomposable representations are exactly the ADE Dynkin diagrams occuring in Lie theory. Representations of quivers arise naturally in algebraic geometry and quantum groups and many elsewhere. The Hall algebra, which was introduced by C.M.Ringel from the representation category of a quiver, or more generally, from the module category of a hereditary algebra, is well-known now as a successful model for the realization of quantum groups. A new result of Deng and Xiao has provided an explicit formulation and its extension of the observation by Sevenhant and Van den Bergh , which claims that the Drinfeld double of a Ringel-Hall algebra is canonically isomorphic to the quantized enveloping algebra of a generalized Kac-Moody algebra. A further investigation shows that not only Ringel-Hall algebra is a very efficient approach to study the quantum group and its representation theory, but also, the knowledge of Lie theory for this Ringel-Hall algebra gives us more information on representations of the quiver. Among many questions, the Kac conjecture is still open.
My Papers and Preprints
LinksAlgebras and Representations at Tsinghua