Graded superalgebras and superinvolutions

Gradings by arbitrary groups have been classified for many important algebras in different varieties. In this talk, we will consider gradings on associative superalgebras with superinvolution. There is a version of Wedderburn-Artin theory for graded associative algebras and superalgebras, which allows us to reduce the classification of graded-simple graded-Artinian (super)algebras to that of graded-division (super)algebras. Using a graded version of the Racine’s result that any superinvolution on a simple Artinian superalgebra is given by the superadjoint with respect to a super-Hermitian form, we classify gradings on finite-dimensional superinvolution-simple superalgebras over an algebraically closed field. In characteristic zero, this implies a classification of gradings on non-exceptional classical simple Lie superalgebras except A(1, 1). This is joint work with Caio Hornhardt.