TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...

2.2. Singular quadratic Lie algebras
2.2 Singular quadratic Lie algebras
2.2.1 Super-Poisson bracket and quadratic Lie algebras
Let (V,B) be a quadratic vector space. Denote by A (V ) the (Z-graded) Grassmann algebra
of alternating multilinear forms on V . For X ∈ V , we recall the derivation ιX of A (V ) defined
by:
ιX(Ω)(Y1,...,Yk) = Ω(X,Y1,...,Yk), ∀ Ω ∈ A k+1 (V ), X,Y1,...,Yk ∈ V (k ≥ 0),
and ιX(1) = 0. Then the super-Poisson bracket on A (V ) is defined as follows (see [PU07] for
details): fix an orthonormal basis {v1,...,vn} of V , then one has
{Ω,Ω ′ n
k+1
} = (−1) ∑ ιvj
j=1
(Ω) ∧ ιvj (Ω′ ), ∀ Ω ∈ A k (V ), Ω ′ ∈ A (V ). (I)
For instance, if α ∈ V ∗ , one has
{α,Ω} = ι φ −1 (α) (Ω), ∀ Ω ∈ A (V ),
and if α ′ ∈ V ∗ , {α,α ′ } = B(φ −1 (α),φ −1 (α ′ )). This definition does not depend on the choice
of the basis.
For any Ω ∈ A k (V ), define adP(Ω) by
adP(Ω) Ω ′ = {Ω,Ω ′ }, ∀ Ω ′ ∈ A (V ).
Then adP(Ω) is a super-derivation of degree k − 2 of the algebra A (V ). One has:
adP(Ω) {Ω ′ ,Ω ′′ } = {adP(Ω)(Ω ′ ),Ω ′′ } + (−1) kk′
{Ω ′ ,adP(Ω)(Ω ′′ )},
for all Ω ′ ∈ A k′ (V ), Ω ′′ ∈ A (V ). That implies that A (V ) is a graded Lie algebra for the
super-Poisson bracket.
Proposition 2.2.1. [PU07]
Let (g,B) be a quadratic Lie algebra. We define a 3-form I on g as follows:
Then one has:
(1) I ∈ A 3 (g).
(2) {I,I} = 0.
I(X,Y,Z) = B([X,Y ],Z), ∀ X,Y,Z ∈ g.
Conversely, assume that g is a finite-dimensional quadratic vector space. Let I ∈ A 3 (g) and
define
[X,Y ] = φ −1 (ιX∧Y (I)), ∀ X,Y ∈ g.
This bracket satisfies the Jacobi identity if and only if {I,I} = 0 [PU07]. In this case, g becomes
a quadratic Lie algebra with the 3-form I.
Definition 2.2.2. The 3-form I in the previous proposition is called the 3-form associated to g.
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