Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지

GROBNER-SHIRSHOV BASES FOR REPRESENTATION THEORY

  • Kang, Seok-Jin (Department of Mathematics Seoul National University) ;
  • Lee, Kyu-Hwan (Department of Mathematics Seoul National University)
  • Published : 2000.01.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we develop the Grobner-Shirshov basis theory for the representations of associative algebras by introducing the notion of Grobner-Shirshov pairs. Our result can be applied to solve the reduction problem in representation theory and to construct monomial bases of representations of associative algebras. As an illustration, we give an explicit construction of Grobner-Shirshov pairs and monomial bases for finite dimensional irreducible representations of the simple tie algebra sl$_3$. Each of these monomial bases is in 1-1 correspondence with the set of semistandard Young tableaux with a given shape.

Keywords

References

  1. Adv. Math. v.29 The diamond lemma for ring theory G. M. Bergman
  2. Algebra and Logic. v.15 Imbedding into simple associative algebras L. A. Bokut
  3. J. Algebra v.217 Grobner-Shirshov bases for Lie superalgebras and their universal enveloping algebras L. A. Bokut;S.-J. Kang;K.-H. Lee;P. Malcolmson
  4. Internat. J. Algebra Comput. v.6 Serre relations Grobner-Shirshov bases for simple Lie algebras Ⅰ, Ⅱ L. A. Bokut;A. A. Klein
  5. Proceedings of ICCAC 97 Grobner-Shirshov bases for exceptional Lie algebras E6-E8
  6. J. Pure Appl. Algebra. v.133 Grobner-Shirshov bases for exceptional Lie algebras Ⅰ
  7. Ph. D. Thesis AN algorithm for finding a basis for the residue class ring of a zero-dimensional ideal B. Buchberger
  8. Infinite dimensional Lie algebras(3th ed.) V. G. Kac
  9. Duke Math. J. v.63 On cystal base of q-analogue of universal enveloping algebras M. Kashiwara
  10. J. Algebra v.165 Crystal graphs for representations of the q-analogue of classical Lie algebras M. Kashiwara;T. Nakashima
  11. M. S. Thesis Grobner-Shirshov bases for Kac-Moody algebras An⑴ E. N. Poroshenko
  12. Siberian Math. J. v.3 Some algorithmic problems for Lie algebras A. I. Shirshov