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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Introduction<br />
structures can be determined completely up to isometrical isomorphisms (or i-isomorphisms, for<br />
short) [PU07]. Such Lie algebras appear also in the classification of Lie algebras whose coadjoint<br />
orbits of dimension at most 2 (done by D. Arnal, M. Cahen and J. Ludwig [ACL95]). A<br />
remarkable point is that the dup-number is invariant by i-isomorphism, that is, two i-isomorphic<br />
quadratic Lie algebras have the same dup-number.<br />
The first goal of our study is to determine quadratic Lie algebra structures in the case<br />
dup(g) = 1. The classification of such structures is one of the aims of this thesis. Here, we<br />
want to emphasize that our classification is considered in two senses: up to i-isomorphisms and<br />
more strongly, up to isomorphisms. This study is interesting by itself. It allows us to regard two<br />
distinguished kinds of classes: quadratic Lie algebras whose dup-number is non-zero and those<br />
whose dup-number is zero.<br />
We say that a non-Abelian quadratic Lie algebra g is ordinary if dup(g) = 0. Otherwise, g<br />
is called singular. By a technical requirement, we separate singular quadratic Lie algebras into<br />
two classes: those of type S1 if their dup-number is 1 and of type S3 if their dup-number is 3.<br />
For n ≥ 1, let O(n) be the set of ordinary, S(n) be the set of singular and Q(n) be the set of<br />
non-Abelian quadratic Lie algebra structures on C n . The distinction of two sets O(n) and S(n)<br />
is shown in Theorem 2.2.13 as follows:<br />
THEOREM 1:<br />
(1) O(n) is Zariski-open and S(n) is Zariski-closed in Q(n).<br />
(2) Q(n) = /0 if and only if n ≥ 3.<br />
(3) O(n) = /0 if and only if n ≥ 6.<br />
Next, we shall give a complete classification of singular quadratic Lie algebras, up to iisomorphisms<br />
and up to isomorphisms. It is done mainly on a solvable framework by the<br />
reason below. There are four main steps to reach this goal:<br />
(1) Using the identity {I,I} = 0, we determine the Lie bracket on a solvable singular quadratic<br />
Lie algebra (Proposition 2.2.22).<br />
(2) We describe a solvable singular quadratic Lie algebra as a double extension of a quadratic<br />
vector space by a skew-symmetric map (or double extension, for short) (this notion is<br />
initiated by V. Kac [Kac85] and generally developed by A. Medina and P. Revoy [MR85]).<br />
As a consequence of (1), a quadratic Lie algebra is singular and solvable if and only if it<br />
is a double extension (Proposition 2.2.28 and Proposition 2.2.29).<br />
(3) We find the i-isomorphic and isomorphic conditions for two solvable singular quadratic<br />
Lie algebras (Theorem 2.2.30 and Corollary 2.2.31). These conditions allow us to establish<br />
a one-to-one correspondence between the set of i-isomorphic class of solvable<br />
singular quadratic Lie algebras and the set P1 (o(n)) of O(n)-orbits in P1 (o(n)), where<br />
P1 (o(n)) denotes the projective space of the Lie algebra o(n).<br />
(4) Finally, we prove that the i-isomorphic and isomorphic notions coincide for solvable singular<br />
quadratic Lie algebras.<br />
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