Quadratic symplectic Lie superalgebras and Lie bi-superalgebras

Abstract

We establish some relations between quadratic symplectic Lie superalgebras and Manin superalgebras. Next, we introduce some concepts of double extensions of quadratic symplectic Lie superalgebras and of Manin superalgebras in order to give inductive descriptions of quadratic symplectic Lie superalgebras.

Lie Superalgebras With Some Homogeneous Structures

“…This study complete the inductive descriptions of homogeneous quadratic symplectic Lie superalgebras started in [6]. Finally, we generalize to the case of homogeneous quadratic symplectic Lie superargebras some relations between even-quadratic even-symplectic Lie superalgebras and Manin superalgebras established in [6].…”

“…The second description was obtained by means of the concept of double extension of Manin algebras. Recently in [6], a generalization to the case of Lie superalgebra of the quadratic symplectic double extension was done and an inductive description of quadratic symplectic Lie superalgebras was obtained by using this notion. Moreover, in the same paper, an other inductive description of quadratic symplectic Lie superalgebras was also obtained by introducing the generalized double extension of Manin superalgebras.…”

“…Moreover, in the same paper, an other inductive description of quadratic symplectic Lie superalgebras was also obtained by introducing the generalized double extension of Manin superalgebras. The work in [6] lead us to ask two questions.…”

“…We use this concept to give an inductive description of nilpotent homogeneoussymplectic Lie superalgebras. Several examples are included to show the existence of homogeneous quadratic symplectic Lie superalgebras other than evenquadratic even-symplectic considered in [6]. We study the structures of even (resp.…”

Omni-Lie superalgebras and Lie 2-superalgebras

“…In [3], quadratic Lie superalgebras are studied for a given inner product B on V . In this case, one has the orthogonal Lie superalgebra o(V ) ⊂ gl(V ) and it is easy to see that (V, ω, B) is a quadratic Lie superalgebra if and only if ω satisfies the two conditions in Proposition 3.3 above as well as ad ω x ∈ o(V ), ∀x ∈ V .…”