Normalized invariant form on a Kac-Moody Algebra

Question

For a symmetrizable Kac-Moody Algebra, we can define a normalized invariant form that performs the same role as the Killing form in the finite dimensional case. My question is, do these forms coincide, or are they related in the finite dimensional case (on a f.d. semisimple Lie algebra)?

Answer 1

There are textbook discussions which deal with such questions in detail: see for example Chapter 16 in the book Lie Algebras of Finite and Affine Type by Roger Carter (Cambridge Univ. Press, 2005). As Carter points out on page 370, the standard construction for a symmetrizable Kac-Moody algebra gives back a nonzero scalar multiple of the traditional Killing form when the Lie algebra is finite dimensional. (This much uniqueness is all you can expect.)

For an infinite dimensional Lie algebra there is of course no obvious "trace" form. One feature that makes the construction possible here is the fact that the analogue of the traditional Cartan subalgebra is still finite dimensional and admits a suitable nondegenerate form. But the symmetrizability of the generalized Cartan matrix also comes into play.