Application of the Gel'fand Matrix Method to the Missing Label Problem in Classical Kinematical Lie Algebras

Abstract

Abstract. We briefly review a matrix based method to compute the Casimir operators of Lie algebras, mainly certain type of contractions of simple Lie algebras. The versatility of the method is illustrated by constructing matrices whose characteristic polynomials provide the invariants of the kinematical algebras in (3+1)-dimensions. Moreover it is shown, also for kinematical algebras, how some reductions on these matrices are useful for determining the missing operators in the missing label problem (MLP).

A comment concerning cohomology and invariants of Lie algebras with respect to contractions and deformations

“…This proves the first assertion. By (15), g α has no invariants for α = −1, thus the deformation decreases the rank of A(g) and generates two invariants for the coadjoint representation. 8 2…”

“…This means geometrically that the generic rank of an exterior form in the space spanned by the Maurer-Cartan forms is reduced by the deformation. This fact is of interest for representations, since it indicates the possibility that deformations introduce additional internal labels to describe basis states of a representation [13,14,15]. This result implies the general falseness of the intuitive idea that a contraction of Lie algebras "abelianizes" it.…”

Invariants of triangular Lie algebras

“…The actual problem is the investigation of generalized Casimir operators for classes of solvable Lie algebras or non-solvable Lie algebras with non-trivial radicals of arbitrary finite dimension. There are a number of papers on the partial classification of such algebras and the subsequent calculation of their invariants [1,6,7,14,15,16,20,21,22,23]. In particular, Tremblay and Winternitz [22] classified all the solvable Lie algebras with the nilradicals isomorphic to the nilpotent algebra t 0 (n) of strictly upper triangular matrices for any fixed dimension n. Then in [23] invariants of these algebras were considered.…”

Internal labelling operators and contractions of Lie algebras

“…The analytical approach to the missing label problem has the advantage of pointing out its close relation to the problem of finding the invariants of the coadjoint representation of a Lie algebra. Although in general the missing label operators do not constitute invariants of the algebra or subalgebra, they can actually be determined with the same Ansatz [5,8,9]. Given the Lie algebra g with structure tensor C k ij over a basis {X 1 , .., X n }, we realize the algebra in the space C ∞ (g * ) by means of the differential operators defined by:…”