Group gradings on classical lie superalgebras

Abstract

Assuming the base field is algebraically closed, we classify, up to isomorphism, gradings by arbitrary groups on non-exceptional classical simple Lie superalgebras, excluding those of type A(1, 1), and on finite dimensional superinvolution-simple associative superalgebras. We assume the characteristic to be 0 in the Lie case, and different from 2 in the associative case. Our approach is based on a version of Wedderburn Theorem for graded-simple associative superalgebras satisfying a descending chain condition, which allows us to classify superinvolutions using nondegenerate supersymmetric sesquilinear forms on graded modules over a graded-division superalgebra. To transfer the results from the associative case to the Lie case, we use the duality between G-gradings and b G-actions for finite dimensional universal algebras.