An extension based determinantal method to compute Casimir operators of Lie algebras

Abstract

We present a method based on degree one extensions of Lie algebras by a derivation to compute the Casimir operator of perfect Lie algebras having only one invariant for the coadjoint representation and an Abelian radical. In particular, the Casimir operator of the special affine Lie algebras sa(n, R) results from the determinant of the commutator matrix of an extension. Examples are given for the case of non-Abelian radicals, and the corresponding generalization of the method for this case is formulated.