由顶点算子代数生成的三类指数增长性李代数
Three Families of Lie Algebras of Exponential Growth from Vertex Operator Algebras
摘要：
研究了由顶点算子代数定义的三类无限维李代数及其性质。对于$N=0$ VOA，我们回顾了由偶格$L$构造Fock空间$V_L$的工作，并利用无鬼定理，通过分解$II_{25,1} = N \oplus II_{1,1}$，从$24$不同的Niemeier格$N$的角度给出了李代数$g_{II_{25,1}}$的代数描述.对于$N=1$ SVOA，我们回顾了Fock空间$V_{NS}$的构造，并为李代数$g_{NS}$的谱生成代数提供了一个明确的基础。对于$N=2$ SVOA，我们将$g^{(2)}_{NS}$的结构明确地描述为一个$\mathbb{Q}$ -分次李代数，并将$II_{2,2}$上的一个左右$SL(2,\mathbb{Z})$作用提升为$g^{(2)}_{NS}$ .
英文摘要：
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 $g^{(2)}_{NS}$.
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