Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T05:23:00.491Z Has data issue: false hasContentIssue false
  • English
  • Français

Kac modules of sl (2/2) and the extended Kac-Weyl formulas

Part of: Jordan algebras

Published online by Cambridge University Press:  26 February 2010

H. Shakibi
H. Shakibi
Affiliation:
School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London. El 4NS
Get access

Extract

Abstract. We consider the structure of the Kac modules V(Λ) for dominant integral doubly atypical weights Λ of the Lie superalgebra s1(2/2). Primitive vectors of V(Λ) are constructed and it is shown that the number of composition factors of V(Λ) for such Λ is in exact agreement with the conjectures of [HKV]. These results are used to show that the extended Kac-Weyl character formula which was proved in [VHKTl] for singly atypical simple modules of s1(m/n), and conjectured to be valid for all finite dimensional irreducible representations of sl(m/n) in [VHKT2] is in fact valid for all finite-dimensional doubly atypical simple modules of s1(2/2).

MSC classification

Secondary: 17C70: Super structures
Type
Research Article
Information
Mathematika , Volume 39 , Issue 1 , June 1992 , pp. 37 - 48
Copyright
Copyright © University College London 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[Kl]Kac, V. G., Representations of Classical Lie Superalgebras, Lecture Notes in Mathematics 676, 579–626, eds Bleuler, K., Petry, H. and Reetz, A. (Springer, Berlin, 1977).Google Scholar
[K2]Kac, V. G., Lie superalgebras, Adv. Math., 26 (1977) 8–96.CrossRefGoogle Scholar
[VHKT1]Jeugt, J. Van der, Hughes, J. W. B., King, R. C. and Thierry-Mieg, J., A character formula for singly atypical modules of the Lie superalgebra s1 (m/n), Commun. Alg., 18(10) (1990) 3453–3480.CrossRefGoogle Scholar
[VHKT2]Jeugt, J. Van der, Hughes, J. W. B., King, R. C. and Thierry-Mieg, J., Character formulae for irreducible modules of the Lie superalgebra s1 (m/n), J. Math. Phys., 31 (1990) 2278.CrossRefGoogle Scholar
[PS]Palev, T. D. and Stoilova, N. I., Finite-dimensional representations of the Lie superalgebra gl (2/2) in a gl (2) ⊕ gl (2) basis. II. Nontypical representations, J. Math. Phys., 31 (1990) 953.CrossRefGoogle Scholar
[HKV]Hughes, J. W. B., King, R. C., Jeugt, J. Van der, On the composition factors of the Kac module V(Λ) for the Lie superalgebra s1(m/n), J. Math. Phys., 33 (1992), 470–491.CrossRefGoogle Scholar
[BL]Bernstein, I. N. and Leites, D. A., A formula for the characters of the irreducible finite-dimensional representations of Lie superalgebras of series gl and s1. C.R. Acad. Bulg. Sci., 33 (1980) 1049–1051 (in Russian).Google Scholar
[S]Shakibi, H., A study of Kac modules for Lie superalgebras of type s1 (m/n), Ph.D. Thesis (Queen Mary and Westfield College, 1992).Google Scholar