Three Families of Lie Algebras of Exponential Growth from Vertex Operator Algebras

TL;DR

This paper explores three families of infinite-dimensional Lie algebras derived from Vertex Operator Algebras, revealing their structures, explicit bases, and symmetries, and connecting them to lattice decompositions and modular group actions.

Contribution

It introduces new constructions and explicit descriptions of Lie algebras from VOAs for different N values, highlighting their algebraic structures and symmetries.

Findings

Construction of Lie algebras from lattice VOAs and SVOAs.

Explicit bases for spectrum-generating algebras.

Description of $g^{(2)}_{NS}$ as a graded Lie algebra with $SL(2,bZ)$ symmetry.

Abstract

We study three families of infinite-dimensional Lie algebras defined from Vertex Operator Algebras and their properties. For $N=0$ VOAs, we review the construction of the Fock space $V_L$ from an even lattice $L$ and provide an algebraic description of the Lie algebra $g_{II_{25,1}}$ from the perspective of $24$ different Niemeier lattices $N$ via the decomposition $II_{25,1} = N \oplus II_{1,1}$ using the no-ghost theorem. For $N=1$ SVOAs we review the construction of the Fock space $V_{NS}$ and provide an explicit basis for the spectrum-generating algebra of the Lie algebra $g_{NS}$. For $N=2$ SVOAs, we describe the structure of $g^{(2)}_{NS}$ explicitly as a $\mathbb{Q}$-graded Lie algebra and we lift a left and right $SL(2,\mathbb{Z})$ action on $II_{2,2}$ to $g^{(2)}_{NS}$.