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.
Appendix A<br />
Lemma A.4. Assume that C is semisimple and invertible. Then there is a basis {e1,...,ep,<br />
f1,..., fp} of V such that Bε(ei,ej) = Bε( fi, f j) = 0, Bε(ei, f j) = δi j, 1 ≤ i, j ≤ p. For 1 ≤ i ≤ p,<br />
there exist non-zero λi ∈ C such that C(ei) = λiei and C( fi) = −λi fi.<br />
Moreover, if Λ denotes the spectrum of C, then λ ∈ Λ if and only if −λ ∈ Λ, λ and −λ have<br />
the same multiplicity.<br />
Proof. We prove the result by induction on dim(V ). Assume that dim(V ) = 2. Let {e1,e2}<br />
be an eigenvector basis of V corresponding to eigenvalues λ1 and λ2. We have Bε(C(v),v ′ ) =<br />
−Bε(v,C(v ′ )) and C is invertible, so Bε(e1,e1) = Bε(e2,e2) = 0, Bε(e1,e2) = 0 and λ2 = −λ1.<br />
1<br />
Let f1 =<br />
Bε(e1,e2) e2, then the basis {e1, f1} is a convenient basis.<br />
Assume that the result is true for vector spaces of dimension n with n ≤ 2(p − 1). Assume<br />
dim(V ) = 2p. Let {e1,...,e2p} be an eigenvector basis with corresponding eigenvalues<br />
λ1,...,λ2p. As before, Bε(ei,ei) = 0, 1 ≤ i ≤ 2p, so there exists j such that Bε(e1,ej) =<br />
1<br />
0. Then λ j = −λ1. Let f1 =<br />
Bε(e1,ej) e j. Then Bε| span{e1, f1} is non-degenerate, so V =<br />
span{e1, f1} ⊥<br />
⊕ V1 where V1 = span{e1, f1} ⊥ . But C maps V1 into itself, so we can apply the<br />
induction assumption and the result follows.<br />
As a consequence, we have this classical result, used in Chapter 1:<br />
Lemma A.5.<br />
(1) Let C be a semisimple element of o(n). Then C belongs to the SO(n)-adjoint orbit<br />
of an element of the standard Cartan subalgebra of o(n) (i.e., an element with matrix<br />
diag 2p (λ1,...,λp,−λ1,...,−λp) if n = 2p and diag 2p+1 (λ1,...,λp,0,−λ1,...,−λp) if<br />
n = 2p + 1 in a canonical basis of C n ).<br />
(2) Let C be a semisimple element of sp(2p). Then C belongs to the Sp(2p)-adjoint orbit<br />
of an element of the standard Cartan subalgebra of sp(2p) (i.e., an element with matrix<br />
diag 2p (λ1,...,λp,−λ1,...,−λp)).<br />
(3) Let C and C ′ be semisimple elements of gε. Then C and C ′ are in the same Iε-adjoint orbit<br />
if and only if they have the same spectrum, with same multiplicities.<br />
Proof.<br />
(1) We have C n = ker(C) ⊥<br />
⊕ Im(C) and rank(C) is even. So dim(ker(C)) is even if n = 2p<br />
and odd, if n = 2p + 1. Then apply Lemma A.4 to C| Im(C) to obtain the result.<br />
(2) The proof is similar to (1) with n even.<br />
(3) If C and C ′ have the same spectrum and their eigenvalues having same multiplicities, they<br />
are Iε-conjugate to the same element of the standard Cartan subalgebra.<br />
131