Non-semisimple Lie algebras with Levi factor  o (3),   (2,  ) and their invariants

Abstract

We analyze the number N of functionally independent generalized Casimir invariants for non-semisimple Lie algebras s − → ⊕ R r with Levi factors isomorphic to so (3) and sl (2, R) in dependence of the pair (R, r) formed by a representation R of s and a solvable Lie algebra r. We show that for any dimension n ≥ 6 there exist Lie algebras s − → ⊕ Rr with non-trivial Levi decomposition such that N s − → ⊕ Rr = 0.

Intrinsic formulae for the Casimir operators of semidirect products of the exceptional Lie algebraG2and a Heisenberg Lie algebra

“…The difficulty for the general case of real forms s of a simple Lie algebra is to find the appropiate subalgebra to make the reduction and obtain the new variables corresponding to the generators of s. Finally, we can ask whether the semidirect products of simple and Heisenberg Lie algebras is the only case where the Casimir operators can be described using the classical formulae for these invariants, or if other radicals are possible [6,23] . Even if Heisenberg algebras occupy a privileged position within the possible candidates for radicals of semidirect products, due to the deep relation between their compatibility and the quaternionic, complex or real character of representations of simple Lie algebras, examples where the method is still applicable for solvable non-nilpotent radicals exist.…”

“…For the classical algebras, explicit formulae to obtain the Casimir operators and their eigenvalues were succesively developed by Racah, Perelomov and Popov or Gruber and O'Raifeartaigh, among others [1,2,3,4,5]. For nonsemisimple algebras, the problem is far from being solved, although various results have been obtained [6,7]. In [8,9] the Casimir operators of various semidirect products g of classical Lie algebras and Heisenberg algebras were obtained by application of the Perelomov-Popov formulae.…”

Casimir operators induced by the Maurer–Cartan equations

“…The latter constitutes an isolated type of algebras, and it follows from the proof that their Casimir operator depends only on the variables of the radical [13]. The other possibility, dim R < dim s, presents additional features that are of relevance for the Casimir operator.…”

“…This explains naturally why the determinant method developed there holds. In particular, the contact form implies that these algebras contract onto the Heisenberg algebra of the same dimension [13].…”

Determinantal formulae for the Casimir operators of inhomogeneous Lie algebras

“…In all these problems, effective expressions for the Casimir operators of the unitary and inhomogeneous unitary algebras are needed. In contrast to the simple or reductive algebras, the computation of the invariants of inhomogeneous algebras g is not feasible using the classical theory, and various approaches have been developed [3,4,5,6], using either the universal enveloping algebra U(g) explicitly or by means of analytical reductions of differential equations.…”