The tensor algebra of the identity representation as a module over the Lie superalgebras \({\mathfrak Gl}(n,m)\) and Q(n). (English. Russian original) Zbl 0573.17002

Sergeev, A. N.

Math. USSR, Sb. 51, 419-427 (1985); translation from Mat. Sb., Nov. Ser. 123(165), No. 3, 422-430 (1984).

The representation \(g\to T(g)\equiv g\) of a matrix Lie superalgebra is called an identity one. The author decomposes into irreducible representations the tensor powers of the identity representations of the superalgebras \({\mathfrak Gl}(n,m)\) and Q(n). The set \(\Omega\) of irreducible representations which are contained in these tensor powers is described. The character formula for irreducible representations of this set for the Lie superalgebra Q(n) is derived. The results of the paper make it possible to utilize Young tableaux for the representations of \(\Omega\). The author uses the method which was exploited by H. Weyl for the proof of similar results for the representations of the groups GL(n,\({\mathbb{C}})\).

Reviewer: A.Klimyk

MSC:

17A70	Superalgebras
17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

Keywords:

irreducible representations; tensor powers; identity representations; character formula; Lie superalgebra; Young tableaux