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

3.4. Quadratic Lie superalgebras with 2-dimensional even part
Definition 3.4.15. Keep the notation Jp for the Jordan block of size p and define two types of
double extension as follows:
• for p ≥ 2, we consider the symplectic vector space q = C 2p equipped with its canonical
bilinear form B and the map C J
2p having matrix
Jp
0
0 − t Jp
in a canonical basis. Then C J
2p ∈ sp(2p) and we denote by j2p the double extension of q
by C J
2p. So j2p ∈ N(2 + 2p).
• for p ≥ 1, we consider the symplectic vector space q = C2p equipped with its canonical
bilinear form B and the map C J
p+p with matrix
Jp M
0 − t Jp
in a canonical basis where M = (mi j) denotes the p× p-matrix with mp,p = 1 and mi j = 0
otherwise. Then C J
p+p ∈ sp(2p) and we denote by jp+p the double extension of q by
C J
p+p. So jp+p ∈ N(2 + 2p).
Lie superalgebras j2p or jp+p will be called nilpotent Jordan-type Lie superalgebras.
Keep the notations as in Chapter 1. For n ∈ N, n = 0, each [d] ∈ P−1(2n) can be written as
[d] = (p1, p1, p2, p2,..., pk, pk,2q1,...2qℓ),
with all pi odd, p1 ≥ p2 ≥ ··· ≥ pk and q1 ≥ q2 ≥ ··· ≥ qℓ.
We associate the partition [d] with the map C [d] ∈ sp(2n) having matrix
diag k+ℓ (C J
2p1 ,CJ
2p2 ,...,CJ
2pk ,CJ
q1+q1 ,...,CJ
qℓ+qℓ )
in a canonical basis of C 2n and denote by g [d] the double extension of C 2n by C [d]. Then
g [d] ∈ N(2+2n) and g [d] is an amalgamated product of nilpotent Jordan-type Lie superalgebras,
more precisely,
g [d] = j2p1 × a j2p2 × a ... × a j2pk × a jq1+q1 × a ... × a
jqℓ+qℓ .
Proposition 3.4.16. Each g ∈ N(2 + 2n) is i-isomorphic to a unique amalgamated product
g [d], [d] ∈ P−1(2n) of nilpotent Jordan-type Lie superalgebras.
For reduced diagonalizable case, let g s 4 (λ) be the double extension of q = C2 by C =
λ 0
, λ = 0. By Lemma 3.4.7, g
0 −λ
s 4 (λ) is i-isomorphic to gs 4 (1) = gs 4,2 .
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