SIGMA 11 (2015), 019, 16 pages      arXiv:1503.01542      https://doi.org/10.3842/SIGMA.2015.019
Contribution to the Special Issue on New Directions in Lie Theory

Vertex Algebras $\mathcal{W}(p)^{A_m}$ and $\mathcal{W}(p)^{D_m}$ and Constant Term Identities

Dražen Adamović ^a, Xianzu Lin ^b and Antun Milas ^c
a) Department of Mathematics, University of Zagreb, Bijenicka 30, 10000 Zagreb, Croatia
^b) College of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350108, China
^c) Department of Mathematics and Statistics, SUNY-Albany, 1400 Washington Avenue, Albany 12222,USA

Received October 03, 2014, in final form February 25, 2015; Published online March 05, 2015

Abstract
We consider $AD$-type orbifolds of the triplet vertex algebras $\mathcal{W}(p)$ extending the well-known $c=1$ orbifolds of lattice vertex algebras. We study the structure of Zhu's algebras $A(\mathcal{W}(p)^{A_m})$ and $A(\mathcal{W}(p)^{D_m})$, where $A_m$ and $D_m$ are cyclic and dihedral groups, respectively. A combinatorial algorithm for classification of irreducible $\mathcal{W}(p)^\Gamma$-modules is developed, which relies on a family of constant term identities and properties of certain polynomials based on constant terms. All these properties can be checked for small values of $m$ and $p$ with a computer software. As a result, we argue that if certain constant term properties hold, the irreducible modules constructed in [Commun. Contemp. Math. 15 (2013), 1350028, 30 pages; Internat. J. Math. 25 (2014), 1450001, 34 pages] provide a complete list of irreducible $\mathcal{W}(p)^{A_m}$ and $\mathcal{W}(p)^{D_m}$-modules. This paper is a continuation of our previous work on the $ADE$ subalgebras of the triplet vertex algebra $\mathcal{W}(p)$.

Key words: $C_{2}$-cofiniteness, triplet vertex algebra, orbifold subalgebra, constant term identities.

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