ADESubalgebras of the Triplet Vertex Algebra 𝒲(p):E6,E7

Abstract

It is the forth paper in a series devoted to the investigation of the |$ADE$| subalgebras of the triplet vertex algebra |$\mathcal {W}(p)$|⁠. In this paper, we prove |$C_{2}$|-cofiniteness of the invariant subalgebras |$\mathcal {W}(p)^{\Gamma }$| and study their representations when |$\Gamma$| is the tetrahedral group |$T$| (of |$E_6$|-type), or the octahedral group |$O$| (of |$E_7$|-type). We give a list of |$21p$| irreducible |$\mathcal {W}(p)^{T}$|-modules and a list of |$28p$| irreducible |$\mathcal {W}(p)^{O}$|-modules. The completeness of these two lists can be reduced to some conjectures about constant term identities which have been verified in the case |$p=1$|⁠. For the proof of |$C_{2}$|-cofiniteness of |$\mathcal {W}(p)^{\Gamma }$|⁠, and the calculations of the Zhu's algebras |$A(\mathcal {W}(p)^{\Gamma })$|⁠, we develop several quite universal formulas (Theorem 4.3 and its corollaries). In Appendix, we further apply them to derive a useful tool of structure coefficients for the triplet vertex algebra |$\mathcal {W}(p)$| and its irreducible modules, which unifies almost all the calculations in this paper; as an unexpected byproduct, we give a new proof of |$\mathfrak {sl}_{2}$|-action on |$\mathcal {W}(p)$|⁠.