$Z^N$-graded Lie algebras: Fock representations and reducibility conditions

TL;DR

This paper constructs Fock representations for certain extended $Z^N$-graded algebras on the torus, revealing reducibility conditions similar to Virasoro discrete series, with implications for algebraic structure and representation theory.

Contribution

It introduces a renormalization approach to construct Fock modules for non-central extensions of $Z^N$-graded algebras and establishes reducibility conditions akin to Virasoro discrete series.

Findings

Constructed Fock representations via renormalization.

Identified reducibility conditions for algebra extensions.

Linked results to Virasoro discrete series.

Abstract

Manifestly consistent Fock representations of non-central (but ``core-central'') extensions of the $Z^N$-graded algebras of functions and vector fields on the $N$-dimensional torus $T^N$ are constructed by a kind of renormalization procedure. These modules are of lowest-energy type, but the energy is not a linear function of the momentum. Modulo a technical assumption, reducibility conditions are proved for the extension of $vect(T^N)$, analogous to the discrete series of Virasoro representations.