TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
2.2. Singular quadratic Lie algebras<br />
2.2 Singular quadratic Lie algebras<br />
2.2.1 Super-Poisson bracket and quadratic Lie algebras<br />
Let (V,B) be a quadratic vector space. Denote by A (V ) the (Z-graded) Grassmann algebra<br />
of alternating multilinear forms on V . For X ∈ V , we recall the derivation ιX of A (V ) defined<br />
by:<br />
ιX(Ω)(Y1,...,Yk) = Ω(X,Y1,...,Yk), ∀ Ω ∈ A k+1 (V ), X,Y1,...,Yk ∈ V (k ≥ 0),<br />
and ιX(1) = 0. Then the super-Poisson bracket on A (V ) is defined as follows (see [PU07] for<br />
details): fix an orthonormal basis {v1,...,vn} of V , then one has<br />
{Ω,Ω ′ n<br />
k+1<br />
} = (−1) ∑ ιvj<br />
j=1<br />
(Ω) ∧ ιvj (Ω′ ), ∀ Ω ∈ A k (V ), Ω ′ ∈ A (V ). (I)<br />
For instance, if α ∈ V ∗ , one has<br />
{α,Ω} = ι φ −1 (α) (Ω), ∀ Ω ∈ A (V ),<br />
and if α ′ ∈ V ∗ , {α,α ′ } = B(φ −1 (α),φ −1 (α ′ )). This definition does not depend on the choice<br />
of the basis.<br />
For any Ω ∈ A k (V ), define adP(Ω) by<br />
adP(Ω) Ω ′ = {Ω,Ω ′ }, ∀ Ω ′ ∈ A (V ).<br />
Then adP(Ω) is a super-derivation of degree k − 2 of the algebra A (V ). One has:<br />
adP(Ω) {Ω ′ ,Ω ′′ } = {adP(Ω)(Ω ′ ),Ω ′′ } + (−1) kk′<br />
{Ω ′ ,adP(Ω)(Ω ′′ )},<br />
for all Ω ′ ∈ A k′ (V ), Ω ′′ ∈ A (V ). That implies that A (V ) is a graded Lie algebra for the<br />
super-Poisson bracket.<br />
Proposition 2.2.1. [PU07]<br />
Let (g,B) be a quadratic Lie algebra. We define a 3-form I on g as follows:<br />
Then one has:<br />
(1) I ∈ A 3 (g).<br />
(2) {I,I} = 0.<br />
I(X,Y,Z) = B([X,Y ],Z), ∀ X,Y,Z ∈ g.<br />
Conversely, assume that g is a finite-dimensional quadratic vector space. Let I ∈ A 3 (g) and<br />
define<br />
[X,Y ] = φ −1 (ιX∧Y (I)), ∀ X,Y ∈ g.<br />
This bracket satisfies the Jacobi identity if and only if {I,I} = 0 [PU07]. In this case, g becomes<br />
a quadratic Lie algebra with the 3-form I.<br />
Definition 2.2.2. The 3-form I in the previous proposition is called the 3-form associated to g.<br />
22