Twisted Hamiltonian Lie Algebras and Their Multiplicity-Free Representations

TL;DR

This paper introduces a new class of Lie algebras derived from quadratic conformal algebras and Poisson structures, and constructs irreducible, multiplicity-free representations for them.

Contribution

It generalizes Hamiltonian Lie algebras via twisting and provides a complete simplicity classification and explicit irreducible representations.

Findings

Complete classification of simplicity for the new Lie algebras

Construction of a family of irreducible, multiplicity-free representations

Extension of Hamiltonian Lie algebra theory to twisted structures

Abstract

We construct a class of new Lie algebras by generalizing the one-variable Lie algebras generated by the quadratic conformal algebras (or corresponding Hamiltonian operators) associated to Poisson algebras and a quasi-derivation found by Xu. These algebras can be viewed as certain twists of Xu's generalized Hamiltonian Lie algebras. The simplicity of these algebras is completely determined. Moreover, we construct a family of multiplicity-free representations of these Lie algebras and prove their irreducibility.