Invariant bilinear forms on Leibniz superalgebras

Abstract

In [S. Benayadi, F. Mhamdi and S. Omri, Quadratic (resp. symmetric) Leibniz superalgebras, Commun. Algebra, https://doi.org/10.1080/00927872.2020.1850751 ], the quadratic Leibniz superalgebras, which are (left or right) Leibniz superalgebras provided with even supersymmetric non-degenerate and associative bilinear forms, were investigated. Besides the notion of associativity, there are other kinds of invariance for bilinear forms on Leibniz superalgebras, namely the left invariance and the right invariance. In this paper, we investigate the Leibniz superalgebras endowed with even, supersymmetric, non-degenerate and left invariant bilinear forms and highlight the links between these superalgebras and some other algebraic structures. More precisely, every symmetric Leibniz superalgebra provided with such a bilinear form gives rise to a new type of superalgebra which we call LS-Lie superalgebra. We study LS-Lie superalgebras and we give some interesting informations on the structure of these superalgebras by using certain extensions introduced in [S. Benayadi and F. Mhamdi, Odd-quadratic Leibniz superalgebras, Adv. Pure Appl. Math. 10(4) (2019) 287–298; S. Benayadi, F. Mhamdi and S. Omri, Quadratic (resp. symmetric) Leibniz superalgebras, Commun. Algebra, https://doi.org/10.1080/00927872.2020.1850751 ]. Further, several nontrivial examples of LS-Lie superalgebras are included. Finally, we give similar results for Leibniz superalgebras with right invariant bilinear forms.

Deformations of Symmetric Simple Modular Lie (Super)Algebras

“…If [[c, c]] ∈ Z 3 (g, g) is not a coboundary, the infinitesimal deformation is obstructed and cannot be prolonged. If [[c, α]] = dβ, then −βt 3 gives the third degree prolongation of the deformation. In order to go up to degree 4 then, one has to be able to compensate [[α, α]] + [[c, β]] by a coboundary dγ, and so on.…”

“…The superalgebra satisfying the Jacobi identity, and without any restriction on symmetry is called a Leibniz superalgebra, introduced by Loday, see [3,68].…”

“…3, by (observe thatindices of the cocycles are the degrees in the standard Z-grading)c −9 = y 3 ⊗ x 11 ∧ x 15 + y 3 ⊗ x 13 ∧ x 14 + y 7 ⊗ x 8 ∧ x 15 + y 7 ⊗ x 12 ∧ x 13 + y 8 ⊗ x 7 ∧ x 15 + y 8 ⊗ x 10 ∧ x 14 + y 10 ⊗ x 8 ∧ x 14 + y 10 ⊗ x 11 ∧ x 12 + y 11 ⊗ x 3 ∧ x 15 + y 11 ⊗ x 10 ∧ x 12 + y 12 ⊗ x 7 ∧ x 13 + y 12 ⊗ x 10 ∧ x 11 + y 13 ⊗ x 3 ∧ x 14 + y 13 ⊗ x 7 ∧ x 12 + y 14 ⊗ x 3 ∧ x 13 + y 14 ⊗ x 8 ∧ x 10 + y 15 ⊗ x 3 ∧ x 11 + y 15 ⊗ x 7 ∧ x 8 , c −7 = y 2 ⊗ x 4 ∧ x 15 + y 2 ⊗ x 9 ∧ x 13 + y 4 ⊗ x 2 ∧ x 15 + y 4 ⊗ x 6 ∧ x 14 + y 6 ⊗ x 4 ∧ x 14 + y 6 ⊗ x 14 ∧ y 4 + y 7 ⊗ x 5 ∧ x 6 + y 7 ⊗ x 15 ∧ y 8 + y 10 ⊗ x 2 ∧ x 5 + y 10 ⊗ x 14 ∧ y 8 + y 14 ⊗ x 6 ∧ y 4 + y 14 ⊗ x 10 ∧ y 8 + y 15 ⊗ x 2 ∧ y 4 + y 15 ⊗ x 7 ∧ y 8 + x 4 ⊗ x 2 ∧ x 15 + x 4 ⊗ x 6 ∧ x 14 + x 8 ⊗ x 7 ∧ x 15 + x 8 ⊗ x 10 ∧ x 14 , c 2 −5 = y 1 ⊗ x 4 ∧ x 12 + y 1 ⊗ x 8 ∧ x 9 + y 4 ⊗ x 1 ∧ x 12 + y 4 ⊗x 15 ∧ y 2 + y 8 ⊗ x ∧ x 9 + y 8 ⊗ x 15 ∧ y 7 + y 9 ⊗ x 1 ∧ x 8 + y 9 ⊗ x 13 ∧ y 2 + y 12 ⊗ x 1 ∧ x 4 + y 12 ⊗ x 13 ∧ y 7 + y 13 ⊗ x 9 ∧ y 2 + y 13 ⊗ x 12 ∧ y 7 + y 15 ⊗ x 4 ∧ y 2 + y 15 ⊗ x 8 ∧ y 7 + x 2 ⊗ x 4 ∧ x 15 + x 2 ⊗ x 9 ∧ x 13 + x 7 ⊗ x 8 ∧ x 15 + x 7 ⊗ x 12 ∧ x 13 , c 1 −3 = y 2 ⊗ x 10 ∧ y 5 + y 2 ⊗ x 13 ∧ y 9 + y 6 ⊗ x 7 ∧ y 5 + y 6 ⊗ x 11 ∧ y 9 + y 7 ⊗ x 6 ∧ y 5 + y 7 ⊗ x 13 ∧ y 12 + y 10 ⊗ x 2 ∧ y 5 + y 10 ⊗ x 11 ∧ y 12 + y 11 ⊗ x 6 ∧ y 9 + y 11 ⊗ x 10 ∧ y 12 + y 13 ⊗ x 2 ∧ y 9 + y 13 ⊗ x 7 ∧ y 12 + x 5 ⊗ x 2 ∧ x 10 + x 5 ⊗ x 6 ∧ x 7 + x 9 ⊗ x 2 ∧ x 13 + x 9 ⊗ x 6 ∧ x 11 + x 12 ⊗ x 7 ∧ x 13 + x 12 ⊗ x 10 ∧ x 11 , c 2 −3 = y 4 ⊗ x 12 ∧ y 1 + y 4 ⊗ x 14 ∧ y 6 + y 8 ⊗ x 9 ∧ y 1 + y 8 ⊗ x 14 ∧ y 10 + y 9 ⊗ x 8 ∧ y 1 + y 9 ⊗ x 11 ∧ y 6 + y 11 ⊗ x 9 ∧ y 6 + y 11 ⊗ x 12 ∧ y 10 + y 12 ⊗ x 4 ∧ y 1 + y 12 ⊗ x 11 ∧ y 10 + y 14 ⊗ x 4 ∧ y 6 + y 14 ⊗ x 8 ∧ y 10 + x 1 ⊗ x 4 ∧ x 12…”