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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1.1. Definitions<br />
and gε(V ) = {C ∈ L (V ) | C is skew-symmetric}.<br />
Then Iε(V ) is the isometry group of the bilinear form Bε and gε(V ) is its Lie algebra. Denote<br />
by A ∗ ∈ L (V ) the adjoint map of an element A ∈ L (V ) with respect to Bε, then A ∈ Iε(V ) if<br />
and only if A −1 = A ∗ and C ∈ gε(V ) if and only if C ∗ = −C. If ε = 1 then Iε(V ) is denoted by<br />
O(V ) and gε(V ) is denoted by o(V ). If ε = −1 then Sp(V ) stands for Iε(V ) and sp(V ) stands<br />
for gε(V ).<br />
Recall that the adjoint action Ad of Iε(V ) on gε(V ) is given by<br />
AdU(C) = UCU −1 , ∀ U ∈ Iε(V ), C ∈ gε(V ).<br />
We denote by OC = Ad Iε(V )(C), the adjoint orbit of an element C ∈ gε(V ) by this action.<br />
If V = C n , we call Bε a canonical bilinear form of C n . And with respect to Bε, we define<br />
a canonical basis B = {E1,...,Em} of C m as follows. If m even, m = 2n, write B =<br />
{E1,...,En,F1,...,Fn}, if m is odd, m = 2n + 1, write B = {E1,...,En,G,F1,...,Fn} and one<br />
has:<br />
• if m = 2n then<br />
where 1 ≤ i, j ≤ n.<br />
B1(Ei,Fj) = B1(Fj,Ei) = δi j, B1(Ei,Ej) = B1(Fi,Fj) = 0,<br />
B−1(Ei,Fj) = −B−1(Fj,Ei) = δi j, B−1(Ei,Ej) = B−1(Fi,Fj) = 0,<br />
• if m = 2n + 1 then ε = 1 and<br />
⎧<br />
⎪⎨ B1(Ei,Fj) = δi j, B1(Ei,Ej) = B1(Fi,Fj) = 0,<br />
B1(Ei,G) = B1(Fj,G) = 0,<br />
⎪⎩<br />
B1(G,G) = 1<br />
where 1 ≤ i, j ≤ n.<br />
Also, in the case V = C m , we denote by GL(m) instead of GL(V ), O(m) stands for O(V )<br />
and o(m) stands for o(V ). If m = 2n then Sp(2n) stands for Sp(V ) and sp(2n) stands for sp(V ).<br />
We will also write Iε = Iε(C m ) and gε = gε(C m ). The goal of this chapter is classifying all of<br />
Iε-adjoint orbits of gε.<br />
Finally, let V is an m-dimensional vector space. If V is quadratic then V is isometrically<br />
isomorphic to the quadratic space (C m ,B1) and if V is symplectic then V is isometrically isomorphic<br />
to the symplectic space (C m ,B−1) [Bou59].<br />
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