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

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