Advertisement

Science in China Series A: Mathematics

, Volume 48, Issue 8, pp 1033–1045 | Cite as

BGP reflection functors in root categories

Article
  • 42 Downloads

Abstract

We define the BGP-reflection functors in the derived categories and the root categories. By Ringel’s Hall algebra approach, the BGP-reflection functor is applicable to obtain the classical Weyl group action on the Lie algebra.

Keywords

m-cycle complexes derived categories BGP-reflections Kac-Moody algebras 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bernstein, J., Gelfand, I. M., Ponomarev, V., Coxeter functors, and Gabriel theorem, Uspehi Mat. Nauk, 1973, 28: 19–33; Russian Math. Surv., 1973, 28: 17–32.MATHMathSciNetGoogle Scholar
  2. 2.
    Dlab, V., Ringel, C. M., Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc., 1976, 173: 1–57.MathSciNetGoogle Scholar
  3. 3.
    Ringel, C. M., Hall polynomials for the representation-finite hereditary algebras, Adv. Math., 1990, 84: 137–178.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ringel, C. M., Hall algebras and quantum groups, Inventions Mathematicae, 1990, 101: 583–592.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Sevenhant, B., Van den Bergh, M., On the double of the Hall algebra of a quiver, J. Algebra, 1999, 221: 135–160.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Xiao, J., Yang, S. L., BGP-reflection functors and Lusztig’s symmetries: A Ringel-Hall algebra approach to quantum groups, J. Algebra, 2001, 241: 204–246.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Frenkel, I., Malkin, A., Vybornov, M., Affine Lie algebras and tame quivers, Selcta Mathematica, New Series, 2001, 7: 1–56.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Lusztig, G., Introduction to quantum groups, Progress in Math. 110, Boston: Birkhauser, 1993.Google Scholar
  9. 9.
    Peng, L. G., Xiao, J., Triangulated categories and Kac-Moody algebras, Inventions Mathematicae, 2000, 140: 563–603.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Peng, L. G., Xiao, J., Root categories and simple Lie algebras, J. Algebra, 1997, 198: 19–56.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Happel, D., Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, LMN119, London: Mathematical Society, 1988.Google Scholar
  12. 12.
    Auslander, M., Platzeck, M., Reiten, I., Coxeter functors without diagrams, Trans. Amer. Math. Soc., 1979, 250: 1–46.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kac, V., Infinite Dimensional Lie Algebras, 3rd ed., Cambridge: Cambridge University Press, 1988.Google Scholar

Copyright information

© Science in China Press 2005

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  2. 2.Department of MathematicsYancheng Teachers CollegeYanchengChina

Personalised recommendations