W-realization of Lie algebras: application to so(4,2) and Poincare algebras

TL;DR

This paper introduces a novel method for realizing Lie algebras, specifically applying it to the conformal algebra so(4,2) and the Poincare algebra, connecting algebraic properties with representation theory.

Contribution

It develops a new approach using finite W-algebras as commutants to generate G-realizations from differential realizations, linking to induced representations.

Findings

New realizations of so(4,2) and Poincare algebras obtained.

Unitary irreducible representations identified within the framework.

Method bridges algebraic properties with representation theory techniques.

Abstract

The property of some finite W-algebras to appear as the commutant of a particular subalgebra in a simple Lie algebra G is exploited for the obtention of new G-realizations from a "canonical" differential one. The method is applied to the conformal algebra so(4,2) and therefore yields also results for its Poincare subalgebra. Unitary irreducible representations of these algebras are recognized in this approach, which is naturally compared -or associated- to the induced representation technic.