A new proof of the ${\mathfrak{sl}}_2$ action on the triplet vertex algebra

Abstract

Let $\mathcal{W}(p)$ be the triplet vertex algebra of central charge $c_p=1-\frac{6(p-1{)}^2}{p}$, $p\ge 2$. As a Virasoro module, we have $$\mathcal{W}(p)=\underset{n=0}{\overset{\infty }{⨁}}(2n+1)L(c_p,n^{2}p+np-n).$$ It was pointed out in \cite{am1} that $\mathcal{W}(p)$ admits an action of ${\mathfrak{sl}}_2$. In this paper we give a combinatorics description of $\mathcal{W}(p)$, from which the action of ${\mathfrak{sl}}_2$ follows quite directly. In the end of this paper we give similar descriptions of the invariant subalgebra $\mathcal{W}(p{)}^{\Gamma }$, these will be useful for the characterizations of the exceptional vertex operator algebras of central charge $1$ in forthcoming papers. We also hope to extend the method of this paper to subalgebra of lattice vertex operator algebras of higher rank.

This paper has been withdrawn, contain a small error

References in corpus (2)

• ADE subalgebras of the triplet vertex algebra W(p): A-series
• Fusion rules of Virasoro Vertex Operator Algebras