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Register a new userAssociating quantum vertex algebras to Lie algebra $gl_{infty}$
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In this paper, we present a canonical association of quantum vertex algebras and their $phi$-coordinated modules to Lie algebra $gl_{infty}$ and its 1-dimensional central extension. To this end we construct and make use of another closely related infinite-dimensional Lie algebra.
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We study a particular category ${cal{C}}$ of $gl_{infty}$-modules and a subcategory ${cal{C}}_{int}$ of integrable $gl_{infty}$-modules. As the main results, we classify the irreducible modules in these two categories and we show that every module in category ${cal{C}}_{int}$ is semi-simple. Furthermore, we determine the decomposition of the tensor products of irreducible modules in category ${cal{C}}_{int}$.
In this paper, we study nullity-2 toroidal extended affine Lie algebras in the context of vertex algebras and their $phi$-coordinated modules. Among the main results, we introduce a variant of toroidal extended affine Lie algebras, associate vertex algebras to the variant Lie algebras, and establish a canonical connection between modules for toroidal extended affine Lie algebras and $phi$-coordinated modules for these vertex algebras. Furthermore, by employing some results of Billig, we obtain an explicit realization of irreducible modules for the variant Lie algebras.
In this paper, we study a certain deformation $D$ of the Virasoro algebra that was introduced and called $q$-Virasoro algebra by Nigro,in the context of vertex algebras. Among the main results, we prove that for any complex number $ell$, the category of restricted $D$-modules of level $ell$ is canonically isomorphic to the category of quasi modules for a certain vertex algebra of affine type. We also prove that the category of restricted $D$-modules of level $ell$ is canonically isomorphic to the category of $mathbb{Z}$-equivariant $phi$-coordinated quasi modules for the same vertex algebra. In the process, we introduce and employ a certain infinite dimensional Lie algebra which is defined in terms of generators and relations and then identified explicitly with a subalgebra of $mathfrak{gl}_{infty}$.
A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral Z_2-lattice. The irreducible decomposition of the representation is explicitly computed and described. As a by-product, some fundamental representations of affine Kac-Moody Lie algebra of type $A_n^{(2)}$ are recovered by the new method.
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.
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