On real forms of complex Lie superalgebras and complex algebraic supergroups

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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 such that its complexification is the original complex Lie superalgebra. The second is related to considering A-points of a Lie superalgebra over a commutative complex superalgebra A equipped with superconjugation. It is not difficult to see that the first real form can be obtained as the set of fixed points of an antilinear involutive automorphism and the second is related to an automorphism ϕ such that ϕ² is identity on even part and negative identity on the odd part. The generalized notion of the real form is subsequently introduced also for complex algebraic supergroups.