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Register a new userRemarks on $phi$-coordinated modules for quantum vertex algebras
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Authors
Haisheng Li
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This paper is about $phi$-coordinated modules for weak quantum vertex algebras. Among the main results, several canonical connections among $phi$-coordinated modules for different $phi$ are established. For vertex operator algebras, a reinterpretation of Frenkel-Huang-Lepowskys theorem on contragredient module is given in terms of $phi$-coordinated modules.
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We study $N$-graded $phi$-coordinated modules for a general quantum vertex algebra $V$ of a certain type in terms of an associative algebra $widetilde{A}(V)$ introduced by Y.-Z. Huang. Among the main results, we establish a bijection between the set of equivalence classes of irreducible $N$-graded $phi$-coordinated $V$-modules and the set of isomorphism classes of irreducible $widetilde{A}(V)$-modules. We also show that for a vertex operator algebra, rationality, regularity, and fusion rules are independent of the choice of the conformal vector.
To give a unified treatment on the association of Lie algebras and vertex algebras, we study $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules for vertex algebras, where $G$ is a group with $chi_phi$ a linear character of $G$ and $phi$ is an associate of the one-dimensional additive formal group. The theory of $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules for nonlocal vertex algebra is established in cite{JKLT}. In this paper, we concentrate on the context of vertex algebras. We establish several conceptual results, including a generalized commutator formula and a general construction of vertex algebras and their $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules. Furthermore, for any conformal algebra $mathcal{C}$, we construct a class of Lie algebras $widehat{mathcal{C}}_phi[G]$ and prove that restricted $widehat{mathcal{C}}_phi[G]$-modules are exactly $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules for the universal enveloping vertex algebra of $mathcal{C}$. As an application, we determine the $(G,chi_phi)$-equivariant $phi$-coordinated quasi modules for affine and Virasoro vertex algebras.
We study $phi_epsilon$-coordinated modules for vertex algebras, where $phi_epsilon$ with $epsilon$ an integer parameter is a family of associates of the one-dimensional additive formal group. As the main results, we obtain a Jacobi type identity and a commutator formula for $phi_epsilon$-coordinated modules. We then use these results to study $phi_epsilon$-coordinated modules for vertex algebras associated to Novikov algebras by Primc.
For any nullity $2$ extended affine Lie algebra $mathcal{E}$ of maximal type and $ellinmathbb{C}$, we prove that there exist a vertex algebra $V_{mathcal{E}}(ell)$ and an automorphism group $G$ of $V_{mathcal{E}}(ell)$ equipped with a linear character $chi$, such that the category of restricted $mathcal{E}$-modules of level $ell$ is canonically isomorphic to the category of $(G,chi)$-equivariant $phi$-coordinated quasi $V_{mathcal{E}}(ell)$-modules. Moreover, when $ell$ is a nonnegative integer, there is a quotient vertex algebra $L_{mathcal{E}}(ell)$ of $V_{mathcal{E}}(ell)$ modulo by a $G$-stable ideal, and we prove that the integrable restricted $mathcal{E}$-modules of level $ell$ are exactly the $(G,chi)$-equivariant $phi$-coordinated quasi $L_{mathcal{E}}(ell)$-modules.
We study twisted modules for (weak) quantum vertex algebras and we give a conceptual construction of (weak) quantum vertex algebras and their twisted modules. As an application we construct and classify irreducible twisted modules for a certain family of quantum vertex algebras.
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