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 ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Introduction<br />
and Di red (n + 2) the corresponding sets of isomorphism and i-isomorphism classes of elements<br />
in D(n + 2) and Dred(n + 2). It is clear by Theorem 2 that Di (n + 2) is in bijection with the<br />
well-known set of semisimple O(n)-orbits in P1 (o(n)). More precisely, we have the following<br />
result (Proposition 2.2.40, Corollary 2.2.43 and Proposition 2.2.44):<br />
THEOREM 4:<br />
(1) There is a bijection between D i (n+2) and the set of semisimple O(n)-orbits in P 1 (o(n)).<br />
The same result holds for D i red (n + 2) and semisimple invertible orbits in P1 (o(n)).<br />
(2) Let g and g ′ be inDred(n+2). Then n must be even and g i g ′ if and only if g g ′ . Thus,<br />
Dred(2p + 2) = Di red (2p + 2), for all p ≥ 1.<br />
(3) Let (g,B) be a diagonalizable reduced singular quadratic Lie algebra. Consider g4 the<br />
double extension of C2 <br />
1 0<br />
by C = . Then g is an amalgamated product of singular<br />
0 −1<br />
quadratic Lie algebras all i-isomorphic to g4.<br />
Combined with the classification of semisimple O(n)-adjoint orbits of o(n) in Chapter 1,<br />
the sets D(2p + 2) and Dred(2p + 2) can be parametrized as in Theorem 2.2.41 where the set<br />
D(2p + 2) is in bijection with Λp/Hp and the set Dred(2p + 2) is in bijection with Λ + p /Hp (see<br />
Section 1.3 and Subsection 2.2.5 for the respective definitions).<br />
Clearly, the parametrization of i-isomorphic classes of nilpotent or diagonalizable singular<br />
quadratic Lie algebras can be regarded as a direct corollary of the classification of nilpotent or<br />
semisimple O(n)-adjoint orbits of o(n). However, we do not find any reference that shows how<br />
to parametrize O(n)-adjoint orbits of o(n) in the general case. Therefore, our next objective is<br />
to determine such a parametrization. This solution allows us to go further in the classification<br />
of singular quadratic Lie algebras.<br />
We continue with the notion of an invertible singular quadratic Lie algebra (i.e. C is<br />
invertible). Let Sinv(2p+2) be the set of such structures on C2p+2 and Sinv(2p+2) be the set of<br />
isomorphism classes of elements in Sinv(2p + 2). The isomorphic and i-isomorphic notions coincide<br />
in the invertible case as we show in Lemma 2.2.42. Moreover, a description of invertible<br />
singular quadratic Lie algebras in term of amalgamated product can be found in Proposition<br />
2.2.49. The classification of the set Sinv(2p + 2) is deduced from Theorem 6 below.<br />
We turn our attention to the general case. Given a solvable singular quadratic Lie algebra<br />
g, realized as the double extension of Cn by C ∈ o(n), we consider the Fitting components CI<br />
and CN of C and the corresponding double extensions gI = g CI and gN = g CN<br />
that we call the<br />
Fitting components of g. We have gI invertible, gN nilpotent and g is the amalgamated product<br />
of gI and gN. We prove that (Theorem 2.2.52):<br />
THEOREM 5:<br />
Let g and g ′ be solvable singular quadratic Lie algebras and let gN, gI, g ′ N , g′ I<br />
Fitting components, respectively. Then:<br />
ix<br />
be their