[Paper Review] $Z^N$-graded Lie algebras: Fock representations and reducibility conditions

T. A. Larsson|ArXiv.org|Dec 9, 1992

Algebraic structures and combinatorial models2 references3 citations

TL;DR

This paper constructs consistent Fock representations for Z^N-graded Lie algebras of vector fields and functions on the N-torus via a renormalization procedure, demonstrating that these modules are of lowest-energy type with non-linear energy-momentum dependence. It establishes reducibility conditions for the extended vect(T^N) algebra analogous to the discrete series in Virasoro representations, under a technical assumption.

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.

Motivation & Objective

• To develop manifestly consistent Fock representations for non-central but 'core-central' extensions of Z^N-graded Lie algebras on the N-torus.
• To analyze the structure of lowest-energy representations where energy is not linear in momentum.
• To establish reducibility conditions for the extended algebra vect(T^N), analogous to the discrete series in Virasoro representations.
• To extend the understanding of infinite-dimensional Lie algebras in higher-dimensional toroidal settings.

Proposed method

• A renormalization procedure is employed to construct Fock representations that are consistent despite non-central extensions.
• The representations are of lowest-energy type, with energy defined as a non-linear function of momentum.
• The construction relies on a technical assumption to ensure consistency and closure of the algebraic structure.
• The method draws parallels to the Virasoro algebra's discrete series, adapting the reducibility analysis to the Z^N-graded setting.
• The approach uses graded commutators and Fock space formalism to define the module structure.
• The analysis focuses on the extended vect(T^N) algebra, treating it as a central extension with core-central structure.

Experimental results

Research questions

• RQ1 How can consistent Fock representations be constructed for non-central extensions of Z^N-graded Lie algebras on the N-torus?
• RQ2 What is the nature of the energy-momentum relation in these lowest-energy representations when the energy is not linear in momentum?
• RQ3 Under what conditions does the extended vect(T^N) algebra admit reducible representations similar to the discrete series in the Virasoro algebra?
• RQ4 How does the renormalization procedure ensure consistency in the representation space?
• RQ5 What role does the technical assumption play in enabling the reducibility analysis?

Key findings

• Fock representations for the Z^N-graded Lie algebras are constructed via a renormalization procedure, ensuring consistency despite non-central extensions.
• The resulting modules are of lowest-energy type, but the energy operator is not a linear function of the momentum quantum numbers.
• The paper proves reducibility conditions for the extended vect(T^N) algebra under a technical assumption, mirroring the discrete series classification in the Virasoro algebra.
• The structure of the representations suggests a generalization of the Virasoro discrete series to higher-dimensional toroidal algebras.
• The results indicate a non-trivial interplay between grading, central extensions, and representation theory in infinite-dimensional Lie algebras.
• The analysis provides a framework for studying similar algebras in higher-dimensional conformal field theories and string theory.