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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3.4. Quadratic Lie superalgebras with 2-dimensional even part<br />
3.4.1 Double extension of a symplectic vector space<br />
In Chapter 2, the double extension of a quadratic vector space by a skew-symmetric map<br />
is a solvable singular quadratic Lie algebra. Next, we have a similar definition for a symplectic<br />
vector space as follows:<br />
Definition 3.4.6.<br />
(1) Let (q,Bq) be a symplectic vector space equipped with the symplectic bilinear form Bq<br />
and C : q → q be a skew-symmetric map, that is,<br />
Bq(C(X),Y ) = −Bq(X,C(Y )), ∀ X,Y ∈ q.<br />
Let (t = span{X 0 ,Y 0 },Bt) be a 2-dimensional quadratic vector space with Bt defined by<br />
Consider the vector space<br />
Bt(X 0 ,X 0 ) = Bt(Y 0 ,Y 0 ) = 0, Bt(X 0 ,Y 0 ) = 1.<br />
g = t ⊥<br />
⊕ q<br />
equipped with a bilinear form B = Bt + Bq and define a bracket on g by<br />
[λX 0 + µY 0 + X,λ ′ X 0 + µ ′ Y 0 +Y ] = µC(Y ) − µ ′ C(X) + B(C(X),Y )X 0,<br />
for all X,Y ∈ q,λ,µ,λ ′ , µ ′ ∈ C. Then (g,B) is a quadratic solvable Lie superalgebra with<br />
g 0 = t and g 1 = q. We say that g is the double extension of q by C.<br />
(2) Let gi be double extensions of symplectic 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 symplec 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 />
Lemma 3.4.7. We keep the notation above.<br />
(1) Let g be the double extension of q by C. Then<br />
[X,Y ] = B(X 0,X)C(Y ) − B(X 0,Y )C(X) + B(C(X),Y )X 0, ∀ X,Y ∈ g,<br />
where C = ad(Y 0). Moreover, X 0 ∈ Z(g) and C|q = C.<br />
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