Internal labelling operators and contractions of Lie algebras

Abstract

We analyze under which conditions the missing label problem associated to a reduction chain ${\mathfrak{s}}^{\prime }\subset \mathfrak{s}$ of (simple) Lie algebras can be completely solved by means of an Inönü-Wigner contraction $\mathfrak{g}$ naturally related to the embedding. This provides a new interpretation of the missing label operators in terms of the Casimir operators of the contracted algebra, and shows that the available labeling operators are not completely equivalent. Further, the procedure is used to obtain upper bounds for the number of invariants of affine Lie algebras arising as contractions of semisimple algebras.

20 pages, 2 tables

References in corpus (3)

• Invariants of Lie Algebras with Fixed Structure of Nilradicals
• Invariants of a semi-direct sum of Lie algebras
• Non-solvable contractions of semisimple Lie algebras in low dimension

Cited by in corpus (7)

• Virtual copies of semisimple Lie algebras in enveloping algebras of semidirect products and Casimir operators
• Kinematical superalgebras and Lie algebras of order 3
• Commutativity of missing label operators in terms of Berezin brackets
• The $SU(3)\supset SO(3)$ missing label problem and the analytical Bethe ansatz
• Polynomial algebras from Lie algebra reduction chains $\mathfrak{g}\supset {\mathfrak{g}}^{\prime }$
• Casimir operators induced by Maurer-Cartan equations
• Subalgebra chains and nuclear physics: Commutant approach and construction of polynomial algebras