Related papers: Coxeter categories and quantum groups
Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one-parameter family of flat connections D on h with values in any finite-dimensional h-module V and simple poles on the root hyperplanes. The…
We define the Coxeter cochain complex of a Coxeter group (G,S) with coefficients in a Z[G]-module A. This is closely related to the complex of simplicial cochains on the abstract simplicial complex I(S) of the commuting subsets of S. We…
It is easy to find algebras $\mathbb{T}\in\mathcal{C}$ in a finite tensor category $\mathcal{C}$ that naturally come with a lift to a braided commutative algebra $\mathsf{T}\in Z(\mathcal{C})$ in the Drinfeld center of $\mathcal{C}$. In…
Double-bosonisation associates to a braided group in the category of modules of a quantum group, a new quantum group. We announce the semiclassical version of this inductive construction.
Given a category with a bifunctor and natural isomorphisms for associativity, commutativity and left and right identity we do not assume that extra constraining diagrams hold. We introduce groupoids of coupling trees to describe a version…
A braided category of C*-algebras is constructed. Its objects are C*-algebras endowed with an action of the group R, its morphisms are C*-algebras morphisms intertwining the action of R, the crossed product of its two objects essentially…
A quantum groups of type $A$ is defined in terms of a Hecke symmetry. We show in this paper that the representation category of such a quantum group is uniquely determined as an abelian braided monoidal category by the bi-rank of the Hecke…
The inclusion of the unit in a braided tensor category $\mathcal{V}$ induces a 1-morphism in the Morita 4-category of braided tensor categories $BrTens$. We give criteria for the dualizability of this morphism. When $\mathcal{V}$ is a…
This paper is devoted to further results on the nontrivially associated categories $\mathcal{C}$ and $\mathcal{D}$, which are constructed from a choice of coset representatives for a subgroup of a finite group. We look at the construction…
In recent years, there has been great interest in the study of categorification, specifically as it applies to the theory of quantum groups. In this thesis, we would like to provide a new approach to this problem by looking at Hall…
In Artin-Tits groups attached to Coxeter groups of spherical type, we give a combinatorial formula to express the simple elements of the dual braid monoids in the classical Artin generators. Every simple dual braid is obtained by lifting an…
A linear Gr-category is a category of finite-dimensional vector spaces graded by a finite group together with natural tensor product. We classify the braided monoidal structures of a class of linear Gr-categories via explicit computations…
A braided generalization of the concept of Hopf algebra (quantum group) is presented. The generalization overcomes an inherent geometrical inhomogeneity of quantum groups, in the sense of allowing completely pointless objects. All…
We study the braided monoidal structure that the fusion product induces on the abelian category $\mathcal{W}_p$-mod, the category of representations of the triplet $W$-algebra $\mathcal{W}_p$. The $\mathcal{W}_p$-algebras are a family of…
This paper charts a very direct path between the categorical approach to quantum mechanics, due to Abramsky and Coecke, and the older convex-operational approach based on ordered vector spaces (recently reincarnated as "generalized…
We prove that the quantum unipotent coordinate algebra $A_q(\mathfrak{n}(w))\ $ associated with a symmetric Kac-Moody algebra and its Weyl group element $w$ has a monoidal categorification as a quantum cluster algebra. As an application of…
We define two algebra automorphisms $T_0$ and $T_1$ of the $q$-Onsager algebra $B_c$, which provide an analog of G. Lusztig's braid group action for quantum groups. These automorphisms are used to define root vectors which give rise to a…
We introduce the idea of a geometric categorical Lie algebra action on derived categories of coherent sheaves. The main result is that such an action induces an action of the braid group associated to the Lie algebra. The same proof shows…
We show that if $g_\Gamma$ is the quantum tangent space (or quantum Lie algebra in the sense of Woronowicz) of a bicovariant first order differential calculus over a coquasitriangular Hopf algebra $(A,r)$, then a certain extension of it is…
We define the braided differential algebras which can be interpreted as quantization of the differential operator algebra defined on some algebraic varieties supplied with the action of the group GL(m). The algebra is generated by right…