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 />
(1) g s 4,1 is the double extension of the 2-dimensional symplectic vector space q = C2 by the<br />
map having matrix:<br />
C =<br />
<br />
0 1<br />
0 0<br />
in a canonical basis {E1,E2} of q where B(E1,E2) = 1.<br />
(2) g s 4,2 the double extension of the 2-dimensional symplectic vector space q = C2 by the<br />
map having matrix:<br />
C =<br />
<br />
1 0<br />
0 −1<br />
in a canonical basis {E1,E2} of q where B(E1,E2) = 1.<br />
(3) g s 6 is the double extension of the 4-dimensional symplectic vector space q = C4 by the<br />
map having matrix:<br />
⎛<br />
0<br />
⎜<br />
C = ⎜0<br />
⎝0<br />
1<br />
0<br />
0<br />
0<br />
0<br />
0<br />
⎞<br />
0<br />
0 ⎟<br />
0⎠<br />
0 0 −1 0<br />
in a canonical basis {E1,E2,E3,E4} of q where B(E1,E3) = B(E2,E4) = 1, the other are<br />
zero.<br />
Let (q,B) be a symplectic vector space. We recall that Sp(q) is the isometry group of the<br />
symplectic form B and sp(q) is its Lie algebra, i.e. the Lie algebra of skew-symmetric maps<br />
with respects to B. The adjoint action is the action of Sp(q) on sp(q) by conjugation (see<br />
Chapter 1).<br />
Proposition 3.4.10. Let (q,B) be a symplectic vector space. Let g = (CX0 ⊕ CY0 ) ⊥<br />
⊕ q and<br />
g ′ = (CX ′<br />
⊥<br />
⊕CY ′ ) ⊕ q be double extensions of q, by skew-symmetric maps C and C<br />
0 0 ′ respectively.<br />
Then:<br />
(1) there exists a Lie superalgebra isomorphism between g and g ′ if and only if there exist an<br />
invertible map P ∈ L (q) and a non-zero λ ∈ C such that C ′ = λ PCP −1 and P ∗ PC = C<br />
where P ∗ is the adjoint map of P with respect to B.<br />
(2) there exists an i-isomorphism between g and g ′ if and only if C ′ is in the Sp(q)-adjoint<br />
orbit through λC for some non-zero λ ∈ C.<br />
Proof. The assertions are obvious if C = 0. We assume C = 0.<br />
(1) Let A : g → g ′ be a Lie superalgebra isomorphism then A(CX0 ⊕ CY0) = CX ′ 0<br />
A(q) = q. It is obvious that C ′ = 0. It is easy to see that CX0 = Z(g) ∩ g0 and CX ′ 0 =<br />
Z(g ′ ) ∩ g ′ 0 then one has A(CX0) = CX ′ 0 . It means A(X0) = µX ′ for some non-zero µ ∈ C.<br />
0<br />
Let A|q = Q and assume A(Y0) = βY ′<br />
0 + γX ′ . For all X, Y ∈ q, we have A([X,Y ]) =<br />
0<br />
. Also,<br />
µB(C(X),Y )X ′<br />
0<br />
A([X,Y ]) = [Q(X),Q(Y )] ′ = B(C ′ Q(X),Q(Y ))X ′<br />
0 .<br />
80<br />
⊕ CY ′<br />
0 and