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

3.2. The dup-number of a quadratic Lie superalgebra
(3) Let g ′ be a quadratic Lie superalgebra and A : g → g ′ be a Lie superalgebra isomorphism.
Then
g ′ ′
⊥
= z ⊕ l ′
where z ′ = A(z) is central, l ′ = A(z) ⊥ , Z(l ′ ) is totally isotropic and l and l ′ are isomorphic.
Moreover if A is an i-isomorphism, then l and l ′ are i-isomorphic.
Proof. The proof is exactly as Proposition 2.1.5 where z is a complementary subspace of Z(g)∩
[g,g] in Z(g) and l = z ⊥ .
Clearly, if z = {0} then Z(g) is totally isotropic. Moreover, one has
Lemma 3.2.4. Let g be a non-Abelian quadratic Lie superalgebra. Write g = z ⊥
⊕ l as in Proposition
3.2.3 then dup(g) = dup(l).
Proof. Since [z,g] = {0} then I ∈ E (3,0) (l). Let α ∈ g ∗ such that α ∧I = 0, we show that α ∈ l ∗ .
Assume that α = α1 + α2, where α1 ∈ z ∗ and α2 ∈ l ∗ . Since α ∧ I = 0, α1 ∧ I ∈ E(z) ⊗ E(l) and
α2 ∧ I ∈ E(l) then one has α1 ∧ I = 0. Therefore, α1 = 0 since I is nonzero in E (3,0) (l). That
mean α ∈ l ∗ and then dup(g) = dup(l).
Definition 3.2.5. A quadratic Lie superalgebra g is reduced if:
(1) g = {0}
(2) Z(g) is totally isotropic.
Notice that a reduced quadratic Lie superalgebra is necessarily non-Abelian.
Definition 3.2.6. Let g be a non-Abelian quadratic Lie superalgebra.
(1) g is an ordinary quadratic Lie superalgebra if dup(g) = 0.
(2) g is a singular quadratic Lie superalgebra if dup(g) ≥ 1.
(i) g is a singular quadratic Lie superalgebra of type S1 if dup(g) = 1.
(ii) g is a singular quadratic Lie superalgebra of type S3 if dup(g) = 3.
By Lemma 3.2.1, if g is a singular quadratic Lie superalgebra of type S3 then g is an or-
⊥
thogonal direct sum g = g0 ⊕ V 2n where g1 = V 2n , [g1,g 1] = {0}, g0 is a singular quadratic Lie
algebra of type S3 and the classification is known (more details in Proposition 3.3.3). Therefore,
we are interested in singular quadratic Lie superalgebras of type S1.
Before studying completely the structure of singular quadratic Lie superalgebras of type S1,
we begin with some simple properties as follows:
Proposition 3.2.7. Let (g,B) be a singular quadratic Lie superalgebra of type S1. If g 0 is
non-Abelian then g 0 is a singular quadratic Lie algebra.
Proof. By the proof of Lemma 3.2.1, one has VI = VI0 ∩VI1 . Therefore, dim(VI0 ) ≥ 1. It means
that g0 is a singular quadratic Lie algebra.
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