Related papers: Coxeter categories and quantum groups

Let g be a symmetrizable Kac-Moody algebra and U_h(g) its quantized enveloping algebra. The quantum Weyl group operators of U_h(g) and the universal R-matrices of its Levi subalgebras endow U_h(g) with a natural quasi-Coxeter…

Let g be a symmetrisable Kac-Moody algebra, and U_h(g) the corresponding quantum group. We showed in arXiv:1610.09744 and arXiv:1610.09741 that the braided quasi-Coxeter structure on integrable, category O representations of U_h(g) which…

We construct a categorification of the braid groups associated with Coxeter groups inside the homotopy category of Soergel's bimodules. Classical actions of braid groups on triangulated categories should come from an action of this monoidal…

The author, and independently De Concini, conjectured that the monodromy of the Casimir connection of a simple Lie algebra g is described by the quantum Weyl group operators of the quantum group U_h(g). The aim of this paper, and of its…

Let g be a symmetrisable Kac-Moody algebra and U_h(g) its quantised enveloping algebra. Answering a question of P. Etingof, we prove that the quantum Weyl group operators of U_h(g) give rise to a canonical action of the pure braid group of…

Braid groups may be defined for every Coxeter diagram. Artin's braid group is of type A. Analogs of Temperley-Lieb, Hecke and Birman-Wenzl algebras exist for B-type. Our general hypothethis is that the braid group of B-type replaces Artin's…

In this expository paper, we discuss and compare the notions of braided and coboundary monoidal categories. Coboundary monoidal categories are analogues of braided monoidal categories in which the role of the braid group is replaced by the…

Within the framework of braided or quasisymmetric monoidal categories braided Q-supersymmetry is investigated, where Q is a certain functorial isomorphism in a braided symmetric monoidal category. For an ordinary (co-)quasitriangular Hopf…

We give an alternative presentation of braided monoidal categories. Instead of the usual associativity and braiding we have just one constraint (the b-structure). In the unital case, the coherence conditions for a b-structure are shown to…

Let ${\mathfrak g}$ be a finite dimensional complex semisimple Lie algebra. The finite dimensional representations of the quantized enveloping algebra $U_q({\mathfrak g})$ form a braided monoidal category $O_{int}$. We show that the…

We define 2-functors on the categorified quantum group of a simply-laced Kac-Moody algebra that induce Lusztig's internal braid group action at the level of the Grothendieck group.

Let g be a symmetrisable Kac-Moody algebra and V an integrable g-module in category O. We show that the monodromy of the (normally ordered) rational Casimir connection on V can be made equivariant with respect to the Weyl group W of g, and…

In the present work we describe the category $\mathsf{WC}_2$ of weighted 2-complexes and its subcategory $\mathsf{WC}_1$ of weighted graphs. Since a Coxeter group is defined by its Coxeter graph, the construction of Coxeter groups defines a…

For a quasitriangular C*-quantum group, we enrich the twisted tensor product constructed in the first part of this series to a monoidal structure on the category of its continuous coactions on C*-algebras. We define braided C*-quantum…

Quantum matrices $A(R)$ are known for every $R$ matrix obeying the Quantum Yang-Baxter Equations. It is also known that these act on `vectors' given by the corresponding Zamalodchikov algebra. We develop this interpretation in detail,…

For each braided category $\mathcal{C}$ we show that, under mild hypotheses, there is an associated category of "half braided algebras" and their bimodules internal to $\mathcal{C}$ which is not only monoidal but even braided and balanced.…

In this paper, we construct the Lusztig symmetries for quantum Borcherds-Bozec algebra $U_q(\mathscr g)$ and its weight module $M\in \mathcal O$, on which the generators with real indices of $U_q(\mathscr g)$ act nilpotently. We show that…

Let $V$ be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the…

We consider two families of categories. The first is the family of semisimple quotients of H. Andersen's tilting module categories for quantum groups of Lie type $B$ specialized at odd roots of unity. The second consists of categories…

We present a general theory of braided quantum groups in the C*-algebraic framework using the language of multiplicative unitaries. Starting with a manageable multiplicative unitary in the representation category of the quantum codouble of…