Real forms of complex Lie superalgebras and complex algebraic supergroups

Abstract

The paper concerns two versions of the notion of real forms of Lie superalgebras. One is the standard approach, where a real form of a complex Lie superalgebra is a real Lie superalgebra whose complexification is the original complex Lie superalgebra. The second arises from considering A-points of a Lie superalgebra over a commutative complex superalgebra A equipped with superconjugation. The first kind of real form can be obtained as the set of fixed points of an antilinear involutive automorphism; the second is related to an automorphism f such that f2 is the identity on the even part and the negative identity on the odd part. The generalized notion of real forms is then introduced for complex algebraic supergroups.