Casimir operators induced by the Maurer–Cartan equations

Abstract

It is shown that for inhomogeneous Lie algebras having only one Casimir operator, the latter can be explicitly constructed from the Maurer-Cartan equations by means of wedge products. It is further proved that this constraint imposes sharp bounds for the dimension of the representation R defining the semidirect product. The procedure is generalized to compute also the rational invariant of some Lie algebras.

Generalized conformal pseudo-Galilean algebras and their Casimir operators

“…The reformulation of condition (4) in terms of differential forms (see e.g. [30]) allows to compute N (g) quite efficiently and even to obtain the Casimir operators under special circumstances [31,32]. In terms of the Maurer-Cartan equations, the Lie algebra g is described as follows: If C k ij denotes the structure tensor over the basis {X 1 , .., X n }, the identification of the dual space g * with the left-invariant 1-forms on the simply connected Lie group the Lie algebra of which is isomorphic to g allows to define an exterior differential d on g * by dω…”

Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators

“…Although polynomial invariants of an arbitrary Lie algebra always belong to the center of U(g), for non-semisimple algebras this direct approach is quite complicated in practice, due to the absence of structural properties such as the Killing form and the difficulties of their representation theory. For these types of algebras, an analytic approach has been shown to be more effective [7][8][9][10][11][12][13].…”