Generalization of the Gell-Mann formula for sl(n,R) and su(n) algebras

TL;DR

This paper extends the Gell-Mann formula to be valid for all representations of sl(n,R) and su(n) algebras, providing a universal algebraic tool within a group manifold framework.

Contribution

The authors generalize the Gell-Mann formula for sl(n,R) and su(n) algebras, making it applicable to all tensorial, spinorial, and nonunitary representations.

Findings

Derived a simple, concise generalized Gell-Mann formula.

Obtained closed-form matrix elements for SL(n,R)/Spin(n) and SU(n)/SO(n) generators.

Validated the formula's applicability across all irreducible representations.

Abstract

The so called Gell-Mann or decontraction formula is proposed as an algebraic expression inverse to the Inonu-Wigner Lie algebra contraction. It is tailored to express the Lie algebra elements in terms of the corresponding contracted ones. In the case of sl(n,R) and su(n) algebras, contracted w.r.t. so(n) subalgebras, this formula is generally not valid, and applies only in the cases of some algebra representations. A generalization of the Gell-Mann formula for sl(n,R) and su(n) algebras, that is valid for all tensorial, spinorial, (non)unitary representations, is obtained in a group manifold framework of the SO(n) and/or Spin(n) group. The generalized formula is simple, concise and of ample application potentiality. The matrix elements of the SL(n,R)/Spin(n), i.e. SU(n)/SO(n), generators are determined, by making use of the generalized formula, in a closed form for all irreducible representations.