
3.Place of Birth: Dingyuan, Anhui of China
1978.91982.7 Department of Mathematics, Anhui Normal University, undergraduate student.
1982.91985.7 Department of Mathematics, Beijing Normal University, graduate student for master degree.
1985.91988.7 Department of Mathematics, Beijing Normal University, graduate student for doctor degree.
1991.71993.1 University of Antwerp, Belgium, postdoctor.
5. Academic Degree
Ph.D, issued at Beijing Normal University, date: Sept. of 1988. superviser: Liu Shaoxue
1988.91990.6 Department of Mathematics, Beijing Normal University, Lecturer.
1990.61993.6 Department of Mathematics, Beijing Normal University, Associate Professor.
1993.61999.8 Department of Mathematics, Beijing Normal University, Professor.
1999.9the present, Department of Mathematical Sciences, Tsinghua University, Professor.
1993.121995.7 Universitat Bielefeld, Germany, Research member for the program SFB343.
1996.81996.10 Universitat Bielefeld, Germany, Visiting member for the program SFB343.
1998.61998.8 Universitat Bielefeld, Germany, Visiting researcher by Volkswagen Foundation.
1999.71999.9 Universitat Bielefeld, Germany, Visiting member for the program SFB343.
2000.12000.2 Universitat Bielefeld, Germany, Visiting member for the program SFB343.
2001.12001.5 Kansaz State University, USA, Visiting Professor.
2002.12002.5 North Carolina State University, USA, Visiting Professor.
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 wellknown 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 RingelHall algebra is canonically isomorphic to the quantized enveloping algebra of a generalized KacMoody algebra. A further investigation shows that not only RingelHall algebra is a very efficient approach to study the quantum group and its representation theory, but also, the knowledge of Lie theory for this RingelHall algebra gives us more information on representations of the quiver. Among many questions, the Kac conjecture is still open.
7. Preprints
1.B.Deng and J.Xiao, On RingelHall algebras, 01Apr.2003.
1. B.Deng and J.Xiao, A new approach to Kac’s theorem on representations of valued quivers, Math.Z
5.B. Deng and J. Xiao, On double RingelHall algebras, Journal of Algebra 251,110149 (2002).
11.J.Zhang, J.Wang, J.Xiao and N.Ding, The representation Theory of groups and algebras,Chn.Adv.Math.30(6),2001,481488.
13. L.Peng and J.Xiao, Root categories and simple Lie algebras, J. Algebra 198,1956 (1997).
14. J.Xiao, Drinfeld double and RingelGreen theory of Hall algebras, J. Algebra 190, 100144(1997).
21. J.Xiao, Restricted representations of U(sl(2))quantizations, Algebra Colloq. 1:1 (1994) 5566.
32.J.Xiao, Indecomposable modules over selfinjective algebras of finite representation Dntype (II):threecornered algebras,Acta Math.Sinica (in Chinese)1990(2),214232.
33.J.Xiao, Indecomposable modules over selfinjective algebras of finite representation Dntype (I):twocornered algebras,Acta Math.Sinica (in Chinese)1989(5),659677.
34.J.Xiao,J.Guo and Y.Zhang,Loewy factors of indecomposable modules over selfinjective algebras of class An, Science in China(A) 1990(8), 897908.
35.J.Xiao, A characterization of Artin algebras of finite representation type,Advance in Math.(China)1988(2), 169172.
36.J.Xiao, Two kind of Artin rings related to Schur Lemma.Chn.Sci.Bull.(in Chinese)1988(3), 165167.
37.S.Liu,Y.Luo and J.Xiao, Isomorphisms of path algebras,J.of Beijing Normal University 1986(3), 1319.