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 />
• if As = X1 ∧ ··· ∧ Xr ∧ Xjr+1 ∧ ··· ∧ Xjr+s ∧ ··· ∧ Xjk then ιAs I = ±ξJα jr+s ∈ WI(1 ≤<br />
s ≤ k − r).<br />
Therefore, dim(WI) ≥ k.<br />
(2) The first statement of (2) follows Lemma 2.2.4. By the proof of (1), α1,...,αr,αjr+1 ,<br />
...,αjr+s ∈ WI then I ∈ k (WI).<br />
(3) Lemma 2.2.4 shows that I is decomposable if and only if dim(VI) = k. We only prove<br />
that I is decomposable if and only if dim(WI) = k. The last assertion follows. Indeed,<br />
if I is decomposable, then I = α1 ∧ ··· ∧ αk. Therefore, WI = span{α1,...,αk}, i.e.<br />
dim(WI) = k.<br />
Conversely, if dim(WI) = k, we assume that I is not decomposable. Let {α1,...,αr} be a<br />
basis of VI and complete it to get a basis {α1,...,αn} of V ∗ . By Lemma 2.2.4, there exists<br />
a (k − r)-form β such that<br />
I = α1 ··· ∧ αr ∧ β.<br />
We can write β as follows:<br />
β = ∑ ξ jr+1,... jk<br />
r+1≤ jr+1