Analogue of the classical invariant theory for Lie superalgebras. (English. Russian original) Zbl 0838.17036

Sergeev, A. N.

Funct. Anal. Appl. 26, No. 3, 223-225 (1992); translation from Funkts. Anal. Prilozh. 26, No. 3, 88-90 (1992).

Let \(V\) be a finite-dimensional superspace over \(\mathbb{C}\), and let \({\mathfrak g}\) be a Lie superalgebra lying in \({\mathfrak gl} (V)\). By the classical invariant theory of a Lie superalgebra \({\mathfrak g}\) we mean a description of \({\mathfrak g}\)-invariant elements of the algebra \({\mathfrak A}^{p,q}_{k,l} = S(V^k \oplus \pi (V)^l \oplus V^{*p} \oplus \pi (V)^{*q})\). It is easy to see that \({\mathfrak A}^{p,q}_{k,l} = S(U \otimes V \oplus V^* \otimes W)\), where \(\dim U = (k,l)\) and \(\dim W = (p,q)\). Thus, Lie superalgebras \({\mathfrak gl} (U)\) and \({\mathfrak gl} (W)\) and, hence, their universal enveloping algebra act on \({\mathfrak A}^{p,q}_{k,l}\). Elements of the universal enveloping algebra are called polarization operators. They commute with the natural action of \({\mathfrak gl} (V)\). A set \({\mathfrak M}\) of invariants of a Lie superalgebra \({\mathfrak g}\) is called basic if the algebra of \({\mathfrak g}\)-invariants coincides with the least subalgebra that contains \({\mathfrak M}\) and is invariant under polarization operators. We describe such a set \({\mathfrak M}\) for each series of classical superalgebras and their central extensions.

MSC:

17B70	Graded Lie (super)algebras
17A70	Superalgebras
17B35	Universal enveloping (super)algebras

Keywords:

invariant theory; Lie superalgebra; universal enveloping algebra; polarization operators; central extensions

References:

[1]	H. Weyl, Classical Groups, their Invariants and Representations, Princeton University Press, Princeton (1946). · Zbl 1024.20502
[2]	D. A. Leites, Supermanifold Theory [in Russian], Petrozavodsk (1983).
[3]	A. N. Sergeev, Mat. Sb.,123, 422-430 (1984).