Related papers: Topological description of U_q(sl_2)-representatio…

Vladimir Turaev discovered in the early years of quantum topology that the notion of modular category was an appropriate structure for building 3-dimensional Topological Quantum Field Theories (TQFTs for short) containing invariants of…

A complete list of Uq(sl2)-module algebra structures on the quantum plane is produced and the (uncountable family of) isomorphism classes of these structures are described. The composition series of representations in question are computed.…

The goal of this paper is to classify ``finite subgroups in U_q sl(2)'' where $q=e^{\pi\i/l}$ is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of U_q sl(2); we show that this…

We give a diagrammatic presentation of the category of $\textbf{U}_q(\mathfrak{sl}_2)$-tilting modules $\mathfrak{T}$ for $q$ being a root of unity and introduce a grading on $\mathfrak{T}$. This grading is a "root of unity phenomenon" and…

We give a complete list of indecomposable exact module categories over the finite tensor category $\mathrm{Rep}(u_q(\mathfrak{sl}_2))$ of representations of the small quantum group $u_q(\mathfrak{sl}_2)$, where $q$ is a root of unity of odd…

A braided tensor category $FM_{\kappa}$ of `factorizable D-modules' over configuration spaces is introduced, analogous to the category $FS_q$ of factorizable sheaves from q-alg/9604001. This category is equivalent to the category of finite…

A certain class of unitary representations of U_q(sl(2,R)) has the property of being simultanenously a representation of U_{tilde{q}}(sl(2,R)) for a particular choice of tilde{q}(q). Faddeev has proposed to unify the quantum groups…

We study geometric representation theory of Lie algebroids. A new equivalence relation for integrable Lie algebroids is introduced and investigated. It is shown that two equivalent Lie algebroids have equivalent categories of infinitesimal…

We study the unitarity and modularity of ribbon tensor categories derived from simple affine Lie algebras, via their associated quantum groups. Based on numerical calculations, and assuming two conjectures, we provide the complete picture…

Let $\mathfrak u$ be a compact semisimple Lie algebra, and $\sigma$ be a Lie algebra involution of $\mathfrak u$. Let Rep$_q(\mathfrak u)$ be the ribbon braided tensor C*-category of $U_q(\mathfrak u)$-representations for $0<q<1$. We…

For a quasi-compact quasi-separated scheme X and an arbitrary scheme Y we show that the pullback construction implements an equivalence between the discrete category of morphisms Y --> X and the category of cocontinuous tensor functors…

We give a construction of sl_2-categorifications (in the sense of Chuang-Rouquier) for representations of U_q(gl_n), for generic q and for q a root of unity.

The equitable presentation of $U_q(\mathfrak{sl}_2)$ was introduced in 2006 by Ito, Terwilliger, and Weng. This presentation involves some generators $x, y, y^{-1}, z$. It is known that $\{x^r y^s z^t : r, t \in \mathbb{N}, s \in…

We introduce the notion of a relative spherical category. We prove that such a category gives rise to the generalized Kashaev and Turaev-Viro-type 3-manifold invariants defined in arXiv:1008.3103 and arXiv:0910.1624, respectively. In this…

Unitary Ribbon Fusion Categories (URFC) formalize anyonic theories. It has been widely assumed that the same category formalizes a topological quantum computing model. However, in previous work, we addressed and resolved this confusion and…

We show that the mirabolic quantum group $MU(n)$ is a comodule algebra over the quantized enveloping algebra $U_v(\mathfrak{sl}_n)$, and use this structure to give a complete classification of its finite dimensional representations. In…

We categorify the R-matrix isomorphism between tensor products of minuscule representations of U_q(sl(n)) by constructing an equivalence between the derived categories of coherent sheaves on the corresponding convolution products in the…

We classify module categories over the category of representations of quantum $SL(2)$ in a case when $q$ is not a root of unity. In a case when $q$ is a root of unity we classify module categories over the semisimple subquotient of the same…

We prove two results about $\text{SLF}(\bar U_q)$, the algebra of symmetric linear forms on the restricted quantum group $\bar U_q = \bar U_q(\mathfrak{sl}(2))$. First, we express any trace on finite dimensional projective $\bar…

In this paper, we study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. Our aim is to construct knot homologies categorifying…