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

Chapter 3
Singular quadratic Lie superalgebras
This chapter is a natural adaptation of Chapter 2 for the quadratic Lie superalgebras: Lie
superalgebras endowed with an even invariant non-degenerate bilinear form. In this context,
we also have a trilinear form I. We will recall the construction of the super-exterior algebra
([Sch79], [Gié04]) and the super Z × Z2-Poisson bracket { , } on it [MPU09] to get the same
formula {I,I} = 0 for quadratic Lie superalgebras. These guide us to define a dup-number and
a subclass of quadratic Lie superalgebras having dup-number non-zero which can be characterized
up to isomorphisms. Finally, we show that the dup-number is also an invariant of quadratic
Lie superalgebras.
3.1 Application of Z×Z2-graded Lie superalgebras to quadratic
Lie superalgebras
We begin from a Z2-graded vector space V = V 0 ⊕ V 1 over C. The subspaces V 0 and V 1
are respectively called the even part and the odd part of V . Keep the notation A = A (V 0)
for the Grassmann algebra of alternating multilinear forms on V 0 as in Chapter 2 and denote
by S = S (V 1) the (Z-graded) algebra of symmetric multilinear forms on V 1, i.e. S = S(V ∗
1 )
where S(V ∗
1
) is the symmetric algebra of V ∗
1 . We define a Z × Z2-gradation on A and on S by
A (i,0) = A i , A (i,1) = {0}
and S (i,i) = S i , S (i, j) = {0} if i = j,
where i, j ∈ Z and i, j are the residue classes modulo 2 of i and j, respectively.
Set a gradation:
E(V ) = A ⊗
Z×Z2
S .
More particularly, in terms of the Z-gradations of A and S
E n (V ) =
n m n−m
A ⊗ S ,
m=0
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