Group gradings on the superalgebras M(m,n), A(m,n) and P(n)

Abstract

We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras P(n), n≥2, and on the simple associative superalgebras M(m,n), m,n≥1, over an algebraically closed field: fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism. As a corollary, we also classify up to isomorphism the G-gradings on the classical Lie superalgebra A(m,n) that are induced from G-gradings on M(m+1,n+1). In the case of Lie superalgebras, the characteristic is assumed to be 0.