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Register a new userCategorification of infinite-dimensional $mathfrak{sl}_2$-modules and braid group 2-actions I: tensor products
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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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In this paper we study an approximation of tensor product of irreducible integrable $hat{mathfrak{sl}_2}$ representations by infinite fusion products. This gives an approximation of the corresponding coset theories. As an application we represent characters of spaces of these theories as limits of certain restricted Kostka polynomials. This leads to the bosonic (which is known) and fermionic (which is new) formulas for the $hat{mathfrak{sl}_2}$ branching functions.
In this paper we explore the possibility of endowing simple infinite-dimensional ${mathfrak{sl}_2(mathbb{C})}$-modules by the structure of the graded module. The gradings on finite-dimensional simple module over simple Lie algebras has been studied in [arXiv:1308.6089] and [arXiv:1601.03008].
We give a description of the centralizer algebras for tensor powers of spin objects in the pre-modular categories $SO(N)_2$ (for $N$ odd) and $O(N)_2$ (for $N$ even) in terms of quantum $(n-1)$-tori, via non-standard deformations of $Umathfrak{so}_N$. As a consequence we show that the corresponding braid group representations are Gaussian representations, the images of which are finite groups. This verifies special cases of a conjecture that braid group representations coming from weakly integral braided fusion categories have finite image.
The Witt group of nondegenerate braided fusion categories $mathcal{W}$ contains a subgroup $mathcal{W}_text{un}$ consisting of Witt equivalence classes of pseudo-unitary nondegenerate braided fusion categories. For each finite-dimensional simple Lie algebra $mathfrak{g}$ and positive integer $k$ there exists a pseudo-unitary category $mathcal{C}(mathfrak{g},k)$ consisting of highest weight integerable $hat{g}$-modules of level $k$ where $hat{mathfrak{g}}$ is the corresponding affine Lie algebra. Relations between the classes $[mathcal{C}(mathfrak{sl}_2,k)]$, $kgeq1$ have been completely described in the work of Davydov, Nikshych, and Ostrik. Here we give a complete classification of relations between the classes $[mathcal{C}(mathfrak{sl}_3,k)]$, $kgeq1$ with a view toward extending these methods to arbitrary simple finite dimensional Lie algebras $mathfrak{g}$ and positive integer levels $k$.
Let $V$ be the two-dimensional simple module and $M$ be a projective Verma module for the quantum group of $mathfrak{sl}_2$ at generic $q$. We show that for any $rge 1$, the endomorphism algebra of $Motimes V^{otimes r}$ is isomorphic to the type $B$ Temperley-Lieb algebra $rm{TLB}_r(q, Q)$ for an appropriate parameter $Q$ depending on $M$. The parameter $Q$ is determined explicitly. We also use the cellular structure to determine precisely for which values of $r$ the endomorphism algebra is semisimple. A key element of our method is to identify the algebras $rm{TLB}_r(q,Q)$ as the endomorphism algebras of the objects in a quotient category of the category of coloured ribbon graphs of Freyd-Yetter or the tangle diagrams of Turaev and Reshitikhin.
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