## 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.

Original language | English (US) |
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Pages (from-to) | 199-250 |

Number of pages | 52 |

Journal | Journal of Algebra |

Volume | 614 |

DOIs | |

State | Published - Jan 15 2023 |

## Keywords

- Admissible classical r-matrices
- Characteristic 2
- Left-alternating
- Left-symmetric
- Manin triple
- Modular Lie superalgebra

## ASJC Scopus subject areas

- Algebra and Number Theory