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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1.2. Nilpotent orbits<br />
where µi = i(r + 1 − i) for 1 ≤ i ≤ r.<br />
By Corollary 1.2.5, ρr defines an irreducible representation of sl(2) of highest weight r and<br />
dimension r + 1. Moreover, every irreducible finite-dimensional representation of sl(2) arises<br />
in this way.<br />
Conveniently, for a partition [d] = [d1,...,dk] of n, denote by O [d] the orbit through X [d].<br />
Now, let O be a non-zero nilpotent orbit in sl(n) then there exists a partition [d1,...,dk] of n<br />
such that O = O [d1,...,dk]. We define the homomorphism φO : sl(2) → sl(n) by<br />
φO = ⊕ ρdi−1.<br />
1≤i≤k<br />
<br />
0 1<br />
Then φO(X) = X [d1,...,dk] where X = ∈ sl(2). Note that if O is a non-zero nilpotent<br />
0 0<br />
orbit then the partition [d1,...,dk] does not coincide with [1n ], the image φO is not trivial and<br />
it is isomorphic to sl(2). Therefore, to each non-zero nilpotent orbit O = OX in sl(n), we can<br />
attach a sl(2)-triple {H,X,Y } such that the nilpositive element is X. More precisely, choose X<br />
exactly having the Jordan form and set<br />
<br />
<br />
1 0<br />
0 1<br />
0 0<br />
H := φO<br />
, X := φO<br />
and Y := φO<br />
.<br />
0 −1<br />
0 0<br />
1 0<br />
This can be done for an arbitrary complex semisimple algebra g, not necessarily sl(n), by<br />
the theorem of Jacobson-Morozov as follows.<br />
Proposition 1.2.6 (Jacobson-Morozov).<br />
Let g be a complex semisimple Lie algebra. If X is a non-zero nilpotent element of g then<br />
it is the nilpositive element of a sl(2)-triple. Equivalently, for any nilpotent element X, there<br />
0 1<br />
exists a homomorphism φ : sl(2) → g such that: φ<br />
= X.<br />
0 0<br />
Proof. We follow the proof given in [CM93]. First, we prove the following lemma.<br />
Lemma 1.2.7. Let g be a complex semisimple Lie algebra and X be a nilpotent element in g.<br />
Then one has:<br />
(1) κ(X,g X ) = 0 where κ is the Killing form and g X is the centralizer of X in g defined by<br />
g X = {Y ∈ g | [X,Y ] = 0}.<br />
(2) [g,X] = (g X ) ⊥ where the notation (g X ) ⊥ denotes the orthogonal subspace of g X by κ.<br />
Proof.<br />
(1) Let Z be an element of g X then by the Jacobi identity, one has ad(X) ◦ ad(Z) = ad(Z) ◦<br />
ad(X). As a consequence, (ad(X) ◦ ad(Z)) k = ad k (X) ◦ ad k (Z) for any k in N. Since X<br />
is nilpotent then ad(X) is nilpotent. It implies that ad k (X) = 0 for some k and therefore<br />
ad(X) ◦ ad(Z) is nilpotent. That means trace(ad(X) ◦ ad(Z)) = κ(X,Z) = 0.<br />
(2) By the invariance of κ, one has κ([g,X],g X ) = κ(g,[X,g X ]) = 0. Hence, [g,X] ⊂ (g X ) ⊥ .<br />
Since dim(g) = dim(ker(ad(X)))+dim(Im(ad(X))) then dim(g) = dim(g X )+dim([g,X]).<br />
By the non-degeneracy of κ, we obtain [g,X] = (g X ) ⊥ .<br />
8