Analogue of the classical invariant theory for Lie superalgebras (Q1324684)

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.