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 />
equipped with a bilinear form B = Bq + Bt and define a bracket on g by<br />
[X + λX1 + µY1,Y + λ ′ X1 + µ ′ Y1] = µC(Y ) − µ ′ C(X) + B(C(X),Y )X1,<br />
for all X,Y ∈ q, λ,µ,λ ′ , µ ′ ∈ C. Then (g,B) is a quadratic solvable Lie algebra. We say<br />
that g is the double extension of q by C.<br />
(2) Let gi be double extensions of quadratic vector spaces (qi,Bi) by skew-symmetric maps<br />
Ci ∈ L (qi), for 1 ≤ i ≤ k. The amalgamated product<br />
is defined as follows:<br />
g = g1 × a g2 × a ... × a gk<br />
• consider (q,B) be the quadratic vector space with q = q1 ⊕ q2 ⊕ ··· ⊕ qk and the<br />
bilinear form B such that B(∑ k i=I Xi,∑ k i=I Yi) = ∑ k i=I Bi(Xi,Yi), for Xi,Yi ∈ qi, 1 ≤ i ≤<br />
k.<br />
• the skew-symmetric map C ∈ L (q) is defined by C(∑ k i=I Xi) = ∑ k i=I Ci(Xi), for Xi ∈<br />
qi, 1 ≤ i ≤ k.<br />
Then g is the double extension of q by C.<br />
Next, we will show that double extensions are highly related to solvable singular quadratic<br />
Lie algebras and amalgamated products can be used to decompose double extensions that are<br />
useful in the nilpotent case. However, we notice here that generally this decomposition is a bad<br />
behavior with respect to i-isomorphisms as follows: if g1 g ′ 1 and g2 g ′ 2 , it may happen that<br />
g1 × g2 and g<br />
a ′ 1 × a g′ 2 are not even isomorphic. An example will be given in Remark 2.2.45.<br />
Lemma 2.2.27. We keep the notation above.<br />
(1) Let g be the double extension of q by C. Then<br />
[X,Y ] = B(X1,X)C(Y ) − B(X1,Y )C(X) + B(C(X),Y )X1, ∀ X,Y ∈ g,<br />
where C = ad(Y1). Moreover, X1 ∈ Z(g) and C|q = C.<br />
(2) Let g ′ be the double extension of q by C ′ = λC, λ ∈ C, λ = 0. Then g and g ′ are iisomorphic.<br />
Proof.<br />
(1) This is a straightforward computation from the previous definition.<br />
(2) Write g = q ⊥<br />
⊕ t = g ′ . Denote by [·,·] ′ the Lie bracket on g ′ . Define A : g → g ′ by<br />
A(X1) = λX1, A(Y1) = 1<br />
λ Y1 and A|q = Idq. Then A([Y1,X]) = C(X) = [A(Y1),A(X)] ′ and<br />
A([X,Y ]) = [A(X),A(Y )] ′ , for all X,Y ∈ q. So A is an i-isomorphism.<br />
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