Defining relations for classical Lie superalgebras without Cartan matrices

Abstract

The analogs of Chevalley generators are offered for simple (and close to them) Z-graded complex Lie algebras and Lie superalgebras of polynomial growth without Cartan matrix. We show how to derive the defining relations between these generators and explicitly write them for a "most natural" ("distinguished" in terms of Penkov and Serganova) system of simple roots. The results are given mainly for Lie superalgebras whose component of degree zero is a Lie algebra (other cases being left to the reader). Observe presentations of exceptional Lie superalgebras and Lie superalgebras of hamiltonian vector fields. Now we can, at last, q-quantize the Lie Lie superalgebras of hamiltonian vector fields and Poisson superalgebras.