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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
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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

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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