Killing form

In particular, all so-called semisimple Lie algebras satisfy this condition. The bilinear form tr(adxudy)(or Re tr adxady) is called the Killing form. Let us denote it simply by (X, Y). Semisimple Lie algebras are characterized by the fact that the form (X, y) is nondegenerate. 

First we consider our model example sl(n,C). Consider an involution a A = A on G, that is, an operation of complex conjugation. The set of fixed points of this involution coincides, obviously, with the subalgebra of real matrices, which is the Lie algebra sl(n,R). It is clear that the Killing form on the algebra sl(n, R) is a... 

The compact form Gc is constructed here as follows. Consider an involution T G G,tA = —ATf where T implies transposition. The fixed points of this involution are skew-Hermitian matrices with a zero trace. It is readily seen that this space is the Lie algebra of the compact group su(n). Indeed, calculating the Killing form on this real form Go, we immediately obtain that it is negative definite. 

Now consider an adjoint action of the Lie algebra Gc on itself, that is, examine the action of transformation of the form ad/ Gc — Gc, where h iTq. Since the element h lies in the Cart an subalgebra, it follows that the transformation adh carries into itself the plane orthogonal to the plane tTo. We make use of the fact that the operators ad are skew-symmetric with respect to the Killing form and therefore preserve the orthogonal complement by carrying it into itself. 

Here grad H X) may be treated as the element of the Lie algebra G identified with G by means of the Killing form. 

Lemma 4.2.1. The operator (pabD is symmetric with respect to the Killing form for any a, 6, D satisfying the above-mentioned conditions. 

Theorem 4.2.6. Let p G G be a linear operator, on a semisimple Lie algebra G, self-conjugate with respect to the Killing form. The Euler equation X = [X, pX] is Hamiltonian simultaneously with respect to both Poisson brackets (the element a is a covector of general position), and, a if and only if p [Pg.217]

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