TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
TH`ESE A NEW INVARIANT OF QUADRATIC LIE ALGEBRAS AND ...
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Chapter 2<br />
Quadratic Lie algebras<br />
In the first part of this chapter, we recall some preliminary definitions and results of quadratic<br />
Lie algebras. Next, we study a new invariant under isomorphisms of quadratic Lie algebras<br />
that we call dup-number. Moreover, we give a classification of singular quadratic Lie algebras,<br />
i.e. those for which the invariant does not vanish. The classification is closely related to<br />
O(m)-adjoint orbits of o(m) mentioned in the Chapter 1. To prove these results, we need to<br />
fully describe the structure of singular quadratic Lie algebras by properties of super-Poisson<br />
bracket defined on the (Z-graded) Grassmann algebra of alternating multilinear forms of an ndimensional<br />
quadratic vector space [PU07] and in terms of double extensions ([Kac85], [FS87]<br />
and [MR85]). The invariance of dup-number is a consequence of calculating the quadratic<br />
dimension of singular quadratic Lie algebras in the reduced case.<br />
Another effective method called T ∗ -extension is introduced by M. Bordemann to descibe<br />
solvable quadratic Lie algebras [Bor97] and we use it to study the 2-step nilpotent case. Finally,<br />
we also obtain a familiar result: the classification of 2-step nilpotent quadratic Lie algebras up<br />
to isometrical isomorphisms is equivalent to the classification all of associated 3-forms.<br />
2.1 Preliminaries<br />
Definition 2.1.1. A quadratic Lie algebra (g,B) is a vector space g equipped with a nondegenerate<br />
symmetric bilinear form B and a Lie algebra structure on g such that B is invariant<br />
(that means, B([X,Y ],Z) = B(X,[Y,Z]), for all X, Y , Z ∈ g).<br />
Let (g,B) be a quadratic Lie algebra. Since B is non-degenerate and invariant, we have some<br />
simple properties of g as follows:<br />
Proposition 2.1.2.<br />
(1) If I is an ideal of g then I⊥ is also an ideal of g. Moreover, if I is non-degenerate then so<br />
is I⊥ and g = I ⊕ I⊥ . Conveniently, in this case we use the notation g = I ⊥<br />
⊕ I⊥ .<br />
(2) Z(g) = [g,g] ⊥ where Z(g) is the center of g. And then<br />
dim(Z(g)) + dim([g,g]) = dim(g).<br />
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