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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2.2. Singular quadratic Lie algebras<br />
Recall that C is not unique (see Remark 2.2.20) and it depends on the choice of V . Let<br />
a = X ⊥ 0 /CX0.<br />
We denote by X the class of an element X ∈ X ⊥ 0 .<br />
Proposition 2.2.23.<br />
Keep the notation above. One has:<br />
(1) the Lie algebra a is Abelian.<br />
(2) define<br />
B(X, Y ) = B(X,Y ), ∀ X,Y ∈ X ⊥ 0 .<br />
Then B is a non-degenerate symmetric bilinear form on a.<br />
(3) define<br />
C(X) = C(X), ∀ X ∈ X ⊥ 0 .<br />
Then C ∈ L (a) is a skew-symmetric map with rank( C) = rank(C) even and rank( C) ≥ 4.<br />
(4) C does not depend on the choice of V . More precisely, if WI = Cα ⊕ φ(V ′ ) and C ′ is the<br />
associated map to V ′ (see Remark 2.2.20), then C ′ = C.<br />
(5) the Lie algebra g is reduced if and only if ker( C) ⊂ Im( C).<br />
Proof.<br />
(1) It follows from Proposition 2.2.22 (1).<br />
(2) It is clear that B is well-defined. Now, since B(X0,Y0) = 1, B(X0,X0) = B(Y0,Y0) = 0, the<br />
restriction of B to span{X0,Y0} is non-degenerate. So<br />
g = span{X0,Y0} ⊥<br />
⊕ span{X0,Y0} ⊥ ,<br />
X ⊥ 0 = CX0 ⊕span{X0,Y0} ⊥ and X ⊥ 0 ⊥ = X ⊥ 0 ∩span{X0,Y0} = CX0. We conclude that B is<br />
non degenerate.<br />
(3) We have C(X ⊥ 0 ) = ad(Y0)(X ⊥ 0 ) ⊂ X ⊥ 0 since X ⊥ 0 is an ideal of g. Moreover, C(X0) = 0, so<br />
C is well-defined. The image of C is contained in X ⊥ 0 and Im(C) ∩ CX0 = {0}, therefore<br />
dim(Im(C)/CX0) = dim(Im( C)) = dim(Im(C)). Now it is enough to apply Proposition<br />
2.2.22.<br />
(4) By Remark 2.2.20, we have C ′ = C + α ⊗ X1 − β ⊗ X0. But α(X0) = 0, so C ′ = C.<br />
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