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Quasideterminants and Casimir elements for the general linear Lie superalgebra. (English) Zbl 1079.17006

The authors construct new families of Casimir elements for the general linear Lie superalgebra \({\mathfrak g}{\mathfrak l}(m| n)\). These families are given explicitly in terms of an oriented graph associated with \({\mathfrak g}{\mathfrak l}(m| n)\). Then the authors calculate the images under the Harish-Chandra isomorphism which turn out to be the elementary, complete, and power sums supersymmetric functions, respectively.
The authors start with the quantum Berezinian of Nazarov and obtain its quasideterminant factorization. Then they use it to give the graph presentation of the Casimir elements. The main results of the paper are superanalogues of some of the results in [I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh and J.-Y. Thibon, Adv. Math. 112, No. 2, 218–348 (1995; Zbl 0831.05063)].

MSC:

17B35 Universal enveloping (super)algebras
05E05 Symmetric functions and generalizations
16S30 Universal enveloping algebras of Lie algebras

Citations:

Zbl 0831.05063
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