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

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 $g^{(2)}_{NS}$.