TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...

3.4. Quadratic Lie superalgebras with 2-dimensional even part
Proof. By Theorem 3.4.8, g is a double extension. Let us write g = (CX0 ⊕ CY0 ) ⊥
C = ad(Y0 )|q. Define A : g → V by A(X0 ) = X ′ ) = Y ′
⊕ q and
0 , A(Y0 0 and A = A|q any isometry from
q → q ′ . It is clear that A is an isometry from g to V . Now, define the Lie super-bracket on V by:
[X,Y ] = A [A −1 (X),A −1 (Y )] , ∀ X,Y ∈ V.
Then V is a quadratic Lie superalgebra, that is i-isomorphic to g. Moreover, V is obviously the
double extension of q ′ by C ′ = A C A −1 .
Theorem 3.4.8, Proposition 3.4.10, Corollary 3.4.11 and Lemma 3.4.12 are enough for us to
apply the method of classification in Chapter 2 for the set S(2 + 2n) of quadratic Lie superalgebra
structures on the quadratic Z2-graded vector space C2 ⊕ C
Z2
2n by only replacing the isometry
group O(m) by Sp(2n) and o(m) by sp(2n) to obtain completely similar results. One has the
first characterization of the set S(2 + 2n):
Theorem 3.4.13. Let g and g ′ be elements in S(2 + 2n). Then g and g ′ are i-isomorphic if and
only if they are isomorphic.
By using the notion of double extension, we call the Lie superalgebra g ∈ S(2 + 2n) diagonalizable
(resp. invertible) if it is a double extension by a diagonalizable (resp. invertible)
map. Denote the subsets of nilpotent elements, diagonalizable elements and invertible elements
in S(2 + 2n), respectively by N(2 + 2n), D(2 + 2n) and by Sinv(2 + 2n). Denote by N(2 + 2n),
D(2 + 2n), Sinv(2 + 2n) the sets of isomorphism classes in N(2 + 2n), D(2 + 2n), Sinv(2 + 2n),
respectively and Dred(2 + 2n) the subset of D(2 + 2n) including reduced ones. Keep the notations
as in Chapter 1 and Chapter 2. Then we have the classification result of these sets as
follows:
Theorem 3.4.14.
(1) There is a bijection between N(2 + 2n) and the set of nilpotent Sp(2n)-adjoint orbits of
sp(2n) that induces a bijection between N(2 + 2n) and the set of partitions P−1(2n).
(2) There is a bijection between D(2 + 2n) and the set of semisimple Sp(2n)-orbits of
P 1 (sp(2n)) that induces a bijection between D(2 + 2n) and Λn/Hn. In the reduced case,
Dred(2 + 2n) is bijective to Λ + n /Hn.
(3) There is a bijection between Sinv(2 + 2n) and the set of invertible Sp(2n)-orbits of
P 1 (sp(2n)) that induces a bijection between Sinv(2 + 2n) and Jn/C ∗ .
(4) There is a bijection between S(2 + 2n) and the set of Sp(2n)-orbits of P 1 (sp(2n)) that
induces a bijection between S(2 + 2n) and D(2n)/C ∗ .
Next, we will describe the sets N(2 + 2n), Dred(2 + 2n) the subset of D(2 + 2n) including
reduced ones, and Sinv(2+2n) in term of amalgamated product in Definition 3.4.6. Remark that
except for the nilpotent case, the amalgamated product may have a bad behavior with respect to
isomorphisms.
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