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3.4. Quadratic Lie superalgebras with 2-dimensional even part
3.4.1 Double extension of a symplectic vector space
In Chapter 2, the double extension of a quadratic vector space by a skew-symmetric map
is a solvable singular quadratic Lie algebra. Next, we have a similar definition for a symplectic
vector space as follows:
Definition 3.4.6.
(1) Let (q,Bq) be a symplectic vector space equipped with the symplectic bilinear form Bq
and C : q → q be a skew-symmetric map, that is,
Bq(C(X),Y ) = −Bq(X,C(Y )), ∀ X,Y ∈ q.
Let (t = span{X 0 ,Y 0 },Bt) be a 2-dimensional quadratic vector space with Bt defined by
Consider the vector space
Bt(X 0 ,X 0 ) = Bt(Y 0 ,Y 0 ) = 0, Bt(X 0 ,Y 0 ) = 1.
g = t ⊥
⊕ q
equipped with a bilinear form B = Bt + Bq and define a bracket on g by
[λX 0 + µY 0 + X,λ ′ X 0 + µ ′ Y 0 +Y ] = µC(Y ) − µ ′ C(X) + B(C(X),Y )X 0,
for all X,Y ∈ q,λ,µ,λ ′ , µ ′ ∈ C. Then (g,B) is a quadratic solvable Lie superalgebra with
g 0 = t and g 1 = q. We say that g is the double extension of q by C.
(2) Let gi be double extensions of symplectic vector spaces (qi,Bi) by skew-symmetric maps
Ci ∈ L (qi), for 1 ≤ i ≤ k. The amalgamated product
is defined as follows:
g = g1 × a g2 × a ... × a gk
• consider (q,B) be the symplec vector space with q = q1 ⊕ q2 ⊕ ··· ⊕ qk and the
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 ≤
k.
• the skew-symmetric map C ∈ L (q) is defined by C(∑ k i=I Xi) = ∑ k i=I Ci(Xi), for Xi ∈
qi, 1 ≤ i ≤ k.
Then g is the double extension of q by C.
Lemma 3.4.7. We keep the notation above.
(1) Let g be the double extension of q by C. Then
[X,Y ] = B(X 0,X)C(Y ) − B(X 0,Y )C(X) + B(C(X),Y )X 0, ∀ X,Y ∈ g,
where C = ad(Y 0). Moreover, X 0 ∈ Z(g) and C|q = C.
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