Abstract
This paper extends existing Lie algebra representation theory related to Lie algebra gradings. The notion of a representation compatible with a given grading is applied to finite-dimensional representations of sl(n,C) in relation to its Z2-gradings. For representation theory of sl(n,C) the Gel’fand-Tseitlin method turned out very efficient. We show that it is not generally true that every irreducible representation can be compatibly graded.
Highlights
Contractions of Lie algebras, of interest in connecting physical theories, are traditionally understood as limit procedures through which Lie algebras are modified into different, non-isomorphic Lie algebras [5, 8]
Compatibly graded finite-dimensional representations were assumed throughout the paper [13], Eqs. (2.10) and (2.11) which is a valid assumption if the grading is induced by an inner automorphism
One should mention a short note [15] on the subject, but up to now nobody has gone ahead with a further study of representations of Lie algebras related to their gradings, especially when the gradings are induced by outer automorphisms
Summary
Contractions of Lie algebras, of interest in connecting physical theories, are traditionally understood as limit procedures through which Lie algebras are modified into different, non-isomorphic Lie algebras [5, 8]. By considering along with graded Lie algebras their compatibly graded finite-dimensional representations, they obtained a theory of contractions of representations that contains the Lie algebra contractions as a special case for adjoint representation. Z2-gradings are closely related to involutive (second order) automorphisms of Lie algebras In physical applications they are especially useful as generalized parity transformations. In this connection our earlier paper [12] dealt with the well-known space-time parity transformations — space inversion and time reversal — and the associated graded contractions for the de Sitter Lie algebras (type B). Our concrete results are illustrated on the simple Lie algebra sl(3, C)
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