Odd-quadratic Lie superalgebras with a weak filiform module as an odd part

Quadratic symplectic Lie superalgebras with a filiform module as an odd part

“…Lemma 2.3. [6] Let (g = g0 ⊕ g1, B) be a quadratic Lie superalgebra with dim(g1) = m > 0 such that g1 has the structure of filiform g0-module concerning to the flag g1 =…”

“…By Lemma 5.1 we know that V 1 = z(g) ∩ g1. By Lemma 2.13 of [6], we have B(g0 ⊕ V m−1 , e 1 ) = {0} and as B is nondegenerate, then there exists e m ∈ V m \V m−1 such that B(e 1 , e m ) = 0. We may assume that B(e 1 , e m ) = 1 (we recall that as B | g1×g1 is skew-symmetric then B(e m , e m ) = 0 and B(e 1 , e 1 ) = 0).…”

“…where a 1 , a 2 are parameters in C, with respect to the basis {X 0 , X 1 , Y 1 , Y 2 , e, e * }, being {X 0 , X 1 } a basis of g0 and {Y 1 , Y 2 , e, e * } a basis of g1, and having B(X 0 , X 1 ) = B(Y 2 , Y 1 ) = B(e * , e) = 1. If a 1 = 0 all the quadratic superalgebras of the above family admit a symplectic structure which can be defined by ω(X, Y ) := B(δ(X), Y ) with δ an even invertible skew-supersymmetric superderivation of (g, B) defined by the (6 × 6)matrix 2cE 1,1 − 2cE 2,2 + cE 3,3 − cE 4,4 + cE 5,5 − cE 6,6 .…”

“…Let us recall now the concept of elementary odd double extensions for quadratic Lie algebras and superalgebras [6]. Thus, let (g, B) be a quadratic Lie algebra with non-null center.…”

“…In [1] were presented some non-trivial examples of quadratic Lie superalgebras such that the even part is a reductive Lie algebra and the action of the even part on the odd part is not completely reducible, and was given an inductive description of this class of quadratic Lie superalgebras. More recently, in [6], the authors characterized in any dimension, via double extensions, a very special family of quadratic Lie superalgebras g = g0 ⊕ g1 such that g0 is a reductive Lie algebra and g1 is not a completely reducible g0-module. In particular, these superalgebras verify that g1 is a filiform g0-module (filiform type).…”