Alexander Sergeev — Page 2

arXiv1998

The invariant polynomials on simple Lie superalgebras

Chevalley's theorem states that for any simple finite dimensional Lie algebra G (1) the restriction homomorphism of the algebra of polynomials on G onto the Cartan subalgebra H induces an isomorphism between the algebra of G-invariant polynomials on G with the algebra of W-invariant polynomals on H, where W is the Weyl group of G, (2) each G-invariant polynomial is a linear combination of the powers of traces tr r(x), where r is a finite dimensional representation of G. None of these facts is necessarily true for simple Lie superalgebras. We reformulate Chevalley's theorem so as to embrace Lie superalgebras. Chevalley's theorem for anti-invariant polynomials is also given.

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arXiv1998

An analog of the classical invariant theory for Lie superlagebras

Let V be a finite-dimensional superspace and G a simple (or a ``close'' to simple) matrix Lie superalgebra, i.e., a Lie subsuperalgebra in GL(V). Under the classical invariant theory for G we mean the description of G-invariant elements of the tensor algebra of V. We give such description for GL(V), SL(V) and OSP(V) and their ``odd'' analogs: Q(V), SQ(V); Pe(V) and SPe(V).

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