On the triplet vertex algebra W(p)

Details On the triplet vertex algebra W(p) On the triplet vertex algebra W(p)

2007-07-12

arxiv.org/abs/0707.1857v2

Advances in Mathematics, 217 (2008) 2664-2699.

Mathematics

Quantum Algebra

32 pages; v2: a few minor changes, to appear in Advances in Mathematics

Scientific paper

We study the triplet vertex operator algebra $\mathcal{W}(p)$ of central charge $1-\frac{6(p-1)^2}{p}$, $p \geq 2$. We show that $\trip$ is $C_2$-cofinite but irrational since it admits indecomposable and logarithmic modules. Furthermore, we prove that $\trip$ is of finite-representation type and we provide an explicit construction and classification of all irreducible $\mathcal{W}(p)$-modules and describe block decomposition of the category of ordinary $\trip$-modules. All this is done through an extensive use of Zhu's associative algebra together with explicit methods based on vertex operators and the theory of automorphic forms. Moreover, we obtain an upper bound for ${\rm dim}(A(\mathcal{W}(p)))$. Finally, for $p$ prime, we completely describe the structure of $A(\trip)$. The methods of this paper are easily extendable to other $\mathcal{W}$-algebras and superalgebras.

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