Quadratic Lie Superalgebras with the Completely Reducible Action of the Even Part on the Odd Part

Abstract

A quadratic Lie superalgebra is a Lie superalgebra g=g0⊕g1 with a non-degenerate supersymmetric consistent (i.e., even) g-invariant bilinear form B; B is called an invariant scalar product of g. We obtain an inductive classification of quadratic Lie superalgebras g=g0⊕g1 such that the action of g0 on g1 is completely reducible and g0 is a reductive Lie algebra. In the case of quadratic Lie superalgebras g=g0⊕g1 such that the action of g0 on g1 is completely reducible, we give an affirmative answer to the following open question (H. Benamor and S. Benayadi, Comm. Algebra27, No. 1 (1999), 67–88): can every B-irreducible non-simple quadratic Lie superalgebra be obtained by double extension? Next, we get an inductive classification of solvable quadratic Lie superalgebras g=g0⊕g1 such that the action of g0 on g1 is completely reducible. Finally, we give the classification of quadratic semisimple Lie superalgebras with the completely reducible action of the even part on the odd part.