Non-solvable contractions of semisimple Lie algebras in low dimension

Abstract

Abstract. The problem of non-solvable contractions of Lie algebras is analyzed. By means of a stability theorem, the problem is shown to be deeply related to the embeddings among semisimple Lie algebras and the resulting branching rules for representations. With this procedure, we determine all deformations of indecomposable Lie algebras having a nontrivial Levi decomposition onto semisimple Lie algebras of dimension n ≤ 8, and obtain the non-solvable contractions of the latter class of algebras.

Internal labelling operators and contractions of Lie algebras

“…This bound has an important interpretation, namely, that the number of invariants of a contraction is, in some sense, determined by the number of available missing label operators for the missing label problem with respect to a maximal subalgebra of s that remains invariant by the contraction. This fact establishes a quite strong restriction to semidirect products of semisimple and Abelian Lie algebras to appear as contractions of semisimple Lie algebras [20].…”

Commutativity of missing label operators in terms of Berezin brackets

“…In particular, if h is a Casimir invariant of g, then X l (h) = 0, and by the symmetrization map Λ we obtain that [X l , Λ(h)] = 0. + This fact suggests that (17) can be used to obtain an analytical criterion for the commutativity of labelling operators. is called the Berezin bracket of g and h and the lower order terms correspond to the symmetrization of the polynomial F [14,21].…”

“…If we restrict to the important case where g, h are homogeneous polynomials, then equation(17) can be rewritten as[Λ (g) , Λ (h)] = Λ ({g, h}) + L.O.T.,(19)¶ Expressed in brackets, this condition is given by [Λ (g) , Λ (h)] = Λ(f ). + This identification has already been used in the frame of completely integrable Hamiltonian systems[22].…”

Embeddings of Lie algebras, contractions and the state labelling problem

“…The analytical approach to the missing label problem has the advantage of being formally very similar to the problem of finding the generalized Casimir invariants of Lie algebras. Although in general the missing label operators are neither invariants of the algebra nor any of its subalgebras, they can actually be determined by means of differential equations with the same ansatz as the general invariant problem [9,[14][15][16][17][18][19][20].…”