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

Notations
We use the notations N, Z, R and C for the set of natural numbers, the set of integers, the
set of real numbers and the set of complex numbers, respectively. The ring of residue classes
modulo 2 of integers is denoted by Z2 which contains two elements 0 and 1. If p ∈ Z, the
notation p indicates its residue classes modulo 2. For a set of numbers K, we denote by K ∗ the
set of non-zero numbers in K. Let Λ be a finite set then we use the notation ♯Λ for the number
of elements of Λ.
If V is a finite-dimensional vector space over a field K of characteristic zero, the notation
End(V ) signs the set of endomorphisms of V . The space End(V ) is also an algebra over K and
it is denoted by L (V ). We denote by V ∗ the dual vector space of V , that is the set of linear
maps from V into K. For each f ∈ V ∗ and each X ∈ V , there is a natural bilinear form 〈 , 〉 is
defined by
〈 f ,X〉 = f (X).
Let W be a subset of V , we denote by W ⊥∗ an orthogonal complement of W in V ∗ by the bilinear
form 〈 , 〉. In addition, if V has a non-degenerate bilinear form B : V ×V → K then W ⊥B (or
W ⊥ , for short) also denotes the set {X ∈ V | B(X,W) = 0}.
The Grassmann algebra of V , that is the algebra of alternating multilinear forms on V ,
with the wedge product is denoted by A (V ). We have A (V ) = (V ∗ ), where (V ∗ ) denotes
the exterior algebra of the dual space V ∗ . We also use the notation S (V ) for the algebra of
symmetric multilinear forms on V , i.e. S (V ) = S(V ∗ ) where S(V ∗ ) denotes the symmetric
algebra of V ∗ . The algebras A (V ) and S (V ) are Z-graded, we denote their homogeneous
subspaces of degee n by A n (V ) and S n (V ), respectively. Thus one has
A (V ) =
A n (V ) and S (V ) =
S n (V )
n∈Z
where A n (V ) = {0} if n /∈ {0,1,...,dim(V )} and S n (V ) = {0} if n is negative.
For n ∈ N ∗ , the Lie algebra of complex square matrices of size n is denoted by gl(n,C) or
gl(n) for short. The subalgebras sl(n,C) of zero trace matrices and o(n,C) of skew-symmetric
matrices of gl(n) are defined as follows:
n∈Z
sl(n,C) = {M ∈ gl(n) | tr(M) = 0},
o(n,C) = {M ∈ gl(n) | t M = −M}
where t M denotes the transpose matrix of the matrix M. If n = 2k then the subalgebra sp(2k,C)
of symplectic matrices of gl(n) is defined by:
sp(2k,C) = {M ∈ gl(2k) | t MJ + JM = 0}
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