Manin triples and non-degenerate anti-symmetric bilinear forms on Lie superalgebras in characteristic 2

Abstract

In this paper, we introduce and develop the notion of a Manin triple for a Lie superalgebra g defined over a field of characteristic p=2. We find cohomological necessary conditions for the pair (g,g⁎) to form a Manin triple. We introduce the concept of Lie bi-superalgebras for p=2 and establish a link between Manin triples and Lie bi-superalgebras. In particular, we study Manin triples defined by a classical r-matrix with an extra condition (called an admissible classical r-matrix). A particular case is examined where g has an even invariant non-degenerate bilinear form. In this case, admissible r-matrices can be obtained inductively through the process of double extensions. In addition, we introduce the notion of double extensions of Manin triples, and show how to get a new Manin triple from an existing one.