Real forms of complex Lie superalgebras and complex algebraic supergroups

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 Apoints 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 φ such that φ 2 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.