### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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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