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Register a new userRectification of algebras and modules
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V. Hinich
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Let O be a topological (colored) operad. The Lurie infinity-category of O-algebras with values in (infinity-category of) complexes is compared to the infinity-category underlying the model category of (classical) dg O-algebras. This can be interpreted as a rectification result for Lurie operad algebras. A similar result is obtained for modules over operad algebras, as well as for algebras over topological PROPs.
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We develop a theory of toroidal vertex algebras and their modules, and we give a conceptual construction of toroidal vertex algebras and their modules. As an application, we associate toroidal vertex algebras and their modules to toroidal Lie algebras.
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.
In this work various symbol spaces with values in a sequentially complete locally convex vector space are introduced and discussed. They are used to define vector-valued oscillatory integrals which allow to extend Rieffels strict deformation quantization to the framework of sequentially complete locally convex algebras and modules with separately continuous products and module structures, making use of polynomially bounded actions of $mathbb{R}^n$. Several well-known integral formulas for star products are shown to fit into this general setting, and a new class of examples involving compactly supported $mathbb{R}^n$-actions on $mathbb{R}^n$ is constructed.
In this paper, we continue the study on toroidal vertex algebras initiated in cite{LTW}, to study concrete toroidal vertex algebras associated to toroidal Lie algebra $L_{r}(hat{frak{g}})=hat{frak{g}}otimes L_r$, where $hat{frak{g}}$ is an untwisted affine Lie algebra and $L_r=$mathbb{C}[t_{1}^{pm 1},ldots,t_{r}^{pm 1}]$. We first construct an $(r+1)$-toroidal vertex algebra $V(T,0)$ and show that the category of restricted $L_{r}(hat{frak{g}})$-modules is canonically isomorphic to that of $V(T,0)$-modules.Let $c$ denote the standard central element of $hat{frak{g}}$ and set $S_c=U(L_r(mathbb{C}c))$. We furthermore study a distinguished subalgebra of $V(T,0)$, denoted by $V(S_c,0)$. We show that (graded) simple quotient toroidal vertex algebras of $V(S_c,0)$ are parametrized by a $mathbb{Z}^r$-graded ring homomorphism $psi:S_crightarrow L_r$ such that Im$psi$ is a $mathbb{Z}^r$-graded simple $S_c$-module. Denote by $L(psi,0}$ the simple $(r+1)$-toroidal vertex algebra of $V(S_c,0)$ associated to $psi$. We determine for which $psi$, $L(psi,0)$ is an integrable $L_{r}(hat{frak{g}})$-module and we then classify irreducible $L(psi,0)$-modules for such a $psi$. For our need, we also obtain various general results.
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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