JIE XIAO

Professor

Department of mathematical sciences,

Tsinghua University,

Beijing100084, P.R.China

Tel:

Fax: +86(0)1062781785

Email: jxiao@math.tsinghua.edu.cn

My CV My research interests Papers and preprints

My CV

Date of Birth: 02-May-1962.
Place of Birth:Dingyuan, Anhui of China.
Educational Background :
- 1978.9-1982.7 Department of Mathematics, Anhui Normal University, undergraduate student.
- 1982.9-1985.7 Department of Mathematics, Beijing Normal University, graduate student for master degree.
- 1985.9-1988.7 Department of Mathematics, Beijing Normal University, graduate student for doctor degree.
- 1991.7-1993.1 University of Antwerp, Belgium, post-doctor.
Academic Degree:
- Ph.D, issued at Beijing Normal University, date: Sept. of 1988. superviser: Liu Shaoxue
Professional Background:
- 1988.9-1990.6 Department of Mathematics, Beijing Normal University, Lecturer.
- 1990.6-1993.6 Department of Mathematics, Beijing Normal University, Associate Professor.
- 1993.6-1999.8 Department of Mathematics, Beijing Normal University, Professor.
- 1999.9-the present, Department of Mathematical Sciences, Tsinghua University, Professor.
- 1993.12-1995.7 Universitat Bielefeld, Germany, Research member for the program SFB343.
- 1996.8-1996.10 Universitat Bielefeld, Germany, Visiting member for the program SFB343.
- 1998.6-1998.8 Universitat Bielefeld, Germany, Visiting researcher by Volkswagen Foundation.
- 1999.7-1999.9 Universitat Bielefeld, Germany, Visiting member for the program SFB343.
- 2000.1-2000.2 Universitat Bielefeld, Germany, Visiting member for the program SFB343.\
- 2001.1-2001.5 Kansaz State University, USA, Visiting Professor.
- 2002.1-2002.5 North Carolina State University, USA, Visiting Professor.

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

B.Deng and J.Xiao, A new approach to Kac’s theorem on representations of valued quivers, Math.Z
J.Xiao and B.Zhu, Relations for the Grothendieck groups of triangulated categories, Journal of Algebra 257,37-50 (2002).
J.Hua and J.Xiao, On Ringel-Hall algebras of tame hereditary algebras, Algebras and Representation Theory 5, 527-550 (2002).
B.Deng and J.Xiao, Ringel-Hall algebras and Lusztig’s symmetries, Journal of Algebra 255,357-372 (2002).
B. Deng and J. Xiao, On double Ringel-Hall algebras, Journal of Algebra 251,110-149 (2002).
J. Xiao and S. Yang, Coxeter transformations in quantum groups, Algebras and Representation Theory 4, 491-501 (2001).
J. Xiao and S. Yang, BGP-reflection functors and Lusztig’s symmetries: a Ringel-Hall algebra approach to quantum groups, Journal of Algebra 241, 204-246 (2001).
L. Peng and J. Xiao, Triangulated categories and Kac-Moody algebras, Invent. Math. 140, 563-603 (2000).
X.Chen and J.Xiao, Root vectors arising from Auslander-Reiten quivers, Journal of Algebra 222,328-356 (1999).
X. Chen and J. Xiao, Exceptional sequences in Hall algebras and quantum groups,Compositio Math. 117 (1999), 161-187.
J.Zhang, J.Wang, J.Xiao and N.Ding, The representation Theory of groups and algebras,Chn.Adv.Math,30(6),2001,481-488.
B.Deng and J.Xiao A quiver description of hereditary categories and its application to the first Weyl algebra, Can. Math. Soc. Conference Proceedings Vol 24,1998,125-137.
L.Peng and J.Xiao, Root categories and simple Lie algebras, J. Algebra 198,19-56 (1997).
J.Xiao, Drinfeld double and Ringel-Green theory of Hall algebras, J. Algebra 190, 100-144(1997).
J.Xiao, Finite dimensional representations of Ut(sl(2)) at root of unity, Can. J. Math. vol.49(4) (1997), 772-787.
L.Peng and J.Xiao, A realization of affine Lie algebras of type An-1 via the derived categories of cyclic quivers, Can. Math. Soc. Conference Proceedings vol 18, 1996, 539-554.
L.Peng and J.Xiao, Invariability of trivial extensions of tilted algebras under stable equivalence, J. London Math. Soc.(2)52 (1995), 61-72.
J.Xiao and Van Oystaeyen, Weight modules and their extensions over a class of algebras similar to the enveloping algebra of sl(2),J. Algebra 175, 844-864(1995).
L.Peng and J.Xiao, Invariablity of repetitive algebras of tilted algebras under stable equivalence, J. Algebra 170, 54-68 (1994).
J.Xiao, Generic modules over the quantum group Ut(sl(2)) at t a root of unit, Manuscripta Math. 83,1994, 75-98.
J.Xiao, Restricted representations of U(sl(2))-quantizations, Algebra Colloq. 1:1 (1994) 55-66.
J.Xiao and Van Oystaeyen, Extension of certain modules over the first Weyl algebras, Science in China(A) 37(5) 1994, 535-546.
J.Xiao and Van Oystaeyen, Projective modules and coverings of finitely generated algebras, Carleton- Ottawa Mathematical Lecture Note Series 14,1992.
Y.Zhang and J.Xiao, An introduction to representations of algebras,Chn.Adv. Math.(in Chinese)22(6), 481-501,1993.
J.Xiao and P.Zhang, One class of representations over trivial extensions of iterated tilted algebras, Tsukuba J.Math.17(1),1993,131-141.
L.Peng and J.Xiao, On the number of DTr-orbits containing directing modules, Proc.Amer.Math.Soc.118(3),1993,753-756.
B.Deng and J.Xiao, Dimension vector,Loewy factors,socle factors are the invariants of Auslander-Reiten quivers, Chinese Science Bulletin 37(10),1992,793-797.
J.Xiao and R.Yang, Construction of representation-finite selfinjective Artin algebra of class Bn and Cn, Acta.Math.Sinica,new series, 9(3),1993,290-306.
J.Xiao and R.Yang,The indecomposable representations over representation-finite selfinjective algebra of class Bn and Cn, Chinese edition in Chinese Ann.Math.13A:1(1992),76-90. English edition in Chinese J. of Contemporary Math. 13(1),1992, 29-46.
J.Xiao, On indecomposable modules over a representation-finite trivial extension algebra,Science in China(A) 1991(2), 129-137.
J.Xiao, Projective modules over a path algebra and its localization. Chinese Ann.Math.(in Chinese)1991 (supplementary issue), 144-148.
J.Xiao, Indecomposable modules over selfinjective algebras of finite representation Dn-type (II):three-cornered algebras,Acta Math.Sinica (in Chinese)1990(2),214-232.
J.Xiao, Indecomposable modules over selfinjective algebras of finite representation Dn-type (I):two-cornered algebras,Acta Math.Sinica (in Chinese).
J.Xiao,J.Guo and Y.Zhang, Loewy factors of indecomposable modules over self-injective algebras of class An, Science in China(A) 1990(8), 897-908.
J.Xiao, A characterization of Artin algebras of finite representation type,Advance in Math.(China)1988(2), 169-172.
J.Xiao, Two kind of Artin rings related to Schur Lemma.Chn.Sci.Bull.(in Chinese)1988(3), 165-167.
S.Liu,Y.Luo and J.Xiao, Isomorphisms of path algebras,J.of Beijing Normal University 1986(3), 13-19.

Links

Algebras and Representations at Tsinghua