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

y
Introduction
We realize that with a quadratic Lie superalgebra (g,B), if we define a trilinear form I on g
I(X,Y,Z) = B([X,Y ],Z), ∀ X,Y,Z ∈ g
then I is a super-antisymmetric trilinear form in E (3,0) (g) and therefore, it seems to be natural to
ask the question: does it happen {I,I} = 0? We give an affirmative answer this in the first part
of Chapter 3 (Theorem 3.1.17). Moreover, we obtain that quadratic Lie superalgebra structures
with bilinear form B are in one-to-one correspondence with elements I in E (3,0) (g) satisfying
{I,I} = 0 (Proposition 3.1.18)
As in Chapter 2, we give the notion of dup-number of a non-Abelian quadratic Lie superalgebra
g defined by
dup(g) = dim({α ∈ g ∗ | α ∧ I = 0})
and suggest considering the set of quadratic Lie superalgebras where the dup-number is nonzero.
Thanks to Lemma 3.2.1, we focus on a singular quadratic Lie superalgebra g with
dup(g) = 1. It is called a singular quadratic Lie superalgebra of type S1. Remark that in
this case, the element I may be decomposable.
We detail some particular cases. When the element I is decomposable, we obtain a classification
of reduced corresponding Lie superalgebras as in Proposition 3.3.3 where the even part
g 0 of g is a singular quadratic Lie algebra or 2-dimensional. Actually, we prove in Proposition
3.4.1 that if g is a non-Abelian quadratic Lie superalgebra with 2-dimensional even part then g
is a singular quadratic Lie superalgebra of type S1.
Note that if we replace the quadratic vector space q in the definition of double extension by
a symplectic vector space then we obtain a quadratic Lie superalgebra with 2-dimensional even
part (Definitions 2.2.26 and 3.4.6). Thus, we have the following result (Theorem 3.4.8):
THEOREM 10:
A quadratic Lie superalgebra has the 2-dimensional even part if and only if it is a double
extension.
By a completely similar process as in Chapter 2, a classification of quadratic Lie superalgebras
with 2-dimensional even part is given as follows. Let S(2 + 2n) be the set of such
structures on C 2+n . We call an algebra g ∈ S(2 + 2n) diagonalizable (resp. invertible) if it is
the 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 isomorphic 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. Keeping the other notations then we have the
classification result (Theorems 3.4.13 and 3.4.14):
THEOREM 11:
(1) Let g and g ′ be elements in S(2 + 2n). Then g and g ′ are i-isomorphic if and only if they
are isomorphic.
(2) 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) of
2n in which odd parts occur with even multiplicity.
xiii