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 ...
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2.1. Preliminaries<br />
∞<br />
∑ C<br />
k=0<br />
k (g,V ). The coboundary operator δ : C(g,V ) → C(g,V ) is defined by<br />
δ f (X0,...,Xk) =<br />
k<br />
∑<br />
i=0<br />
(−1) i ρ(Xi)( f (X0,..., Xi,...,Xk))<br />
+∑ (−1)<br />
i< j<br />
i+ j f ([Xi,Xj],X0,..., Xi,..., Xj,...,Xk)<br />
for all f ∈ Ck (g,V ), X0,...,Xk ∈ g. It is known that δ 2 = 0. We say that f ∈ Ck (g,V ) is a<br />
k-cocycle if δ f = 0 and f is a k-coboundary if there is g ∈ Ck−1 (g,V ) such that f = δg.<br />
In particular, the dual g∗ is a g-module with respect to the coadjoint representation of g.<br />
Consider a bilinear map θ : g × g → g∗ and define on the vector space T ∗<br />
θ (g) = g ⊕ g∗ the<br />
product as follows:<br />
[X + f ,Y + g] = [X,Y ]g + f ◦ adg(Y ) − g ◦ adg(X) + θ(X,Y ), ∀ X,Y ∈ g, f ,g ∈ g ∗ .<br />
It is easy to check that T ∗<br />
θ (g) is a Lie algebra if and only if θ is a 2-cocycle. In this case,<br />
T ∗<br />
θ (g) is called the T ∗-extension of g by means of θ. Moreover, if θ satisfies θ(X,Y )Z =<br />
θ(Y,Z)X, for all X,Y,Z ∈ g (cyclic condition) then T ∗<br />
θ (g) becomes a quadratic Lie algebra with<br />
the bilinear form B defined by<br />
B(X + f ,Y + g) = f (Y ) + g(X), ∀ X,Y ∈ g, f ,g ∈ g ∗ .<br />
Proposition 2.1.13. [Bor97]<br />
Let (g,B) be an even-dimensional quadratic Lie algebra over C. If g is solvable then g<br />
is i-isomorphic to a T ∗-extension T ∗<br />
θ (h) of h where h is the quotient algebra of g by a totally<br />
isotropic ideal.<br />
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