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Register a new userQuantum continuous $gl_infty$: Tensor products of Fock modules and $W_n$ characters
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We construct a family of irreducible representations of the quantum continuous $gl_infty$ whose characters coincide with the characters of representations in the minimal models of the $W_n$ algebras of $gl_n$ type. In particular, we obtain a simple combinatorial model for all representations of the $W_n$-algebras appearing in the minimal models in terms of $n$ interrelating partitions.
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Every irreducible finite-dimensional representation of the quantized enveloping algebra U_q(gl_n) can be extended to the corresponding quantum affine algebra via the evaluation homomorphism. We give in explicit form the necessary and sufficient conditions for irreducibility of tensor products of such evaluation modules.
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.
Given formal differential operators $F_i$ on polynomial algebra in several variables $x_1,...,x_n$, we discuss finding expressions $K_l$ determined by the equation $exp(sum_i x_i F_i)(exp(sum_j q_j x_j)) = exp(sum_l K_l x_l)$ and their applications. The expressions for $K_l$ are related to the coproducts for deformed momenta for the noncommutative space-times of Lie algebra type and also appear in the computations with a class of star products. We find combinatorial recursions and derive formal differential equations for finding $K_l$. We elaborate an example for a Lie algebra $su(2)$, related to a quantum gravity application from the literature.
We study the meromorphic open-string vertex algebras and their modules over the two-dimensional Riemannian manifolds that are complete, connected, orientable, and of constant sectional curvature $K eq 0$. Using the parallel tensors, we explicitly determine a basis for the meromorphic open-string vertex algebra, its modules generated by eigenfunctions of the Laplace-Beltrami operator, and their irreducible quotients. We also study the modules generated by lowest weight subspace satisfying a geometrically interesting condition. It is showed that every irreducible module of this type is generated by some (local) eigenfunction on the manifold. A classification is given for modules of this type admitting a composition series of finite length. In particular and remarkably, if every composition factor is generated by eigenfunctions of eigenvalue $p(p-1)K$ for some $pin mathbb{Z}_+$, then the module is completely reducible.
This is the first part of a series of two papers aiming to construct a categorification of the braiding on tensor products of Verma modules, and in particular of the Lawrence--Krammer--Bigelow representations. In this part, we categorify all tensor products of Verma modules and integrable modules for quantum $mathfrak{sl_2}$. The categorification is given by derived categories of
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