Self-adjoint representations of braid groups

$U_q(sl_2)$ 的辫子余积、对极与伴随作用

Braided coproduct, antipode and adjoint action for $U_q(sl_2)$

本文为量子代数 $U_q(\mathfrak{sl}_2)$ 显式给出辫子余积、对极及伴随作用，并证明其辫子伴随作用与普通量子伴随作用相同。

Categorical Chern character and braid groups

To a braid $β\in Br_n$ we associate a complex of sheaves $S_β$ on $Hilb_n(C^2)$ such that the previously defined triply graded link homology of the closure $L(β)$ is isomorphic to the homology of $S_β$. The construction of $S_β$ relies on the Chern functor $CH: MF_n^{st}\to D^{per}_{C^*\times C^*}(Hilb_n(C^2))$ defined in the paper together with its adjoint functor $HC$. We prove a formula for the closure of sufficiently positive elements of the Jucys-Murphy algebra previously conjectured by Gorsky, Negut and Rasmussen.

A category for the adjoint representation

We construct an abelian category C and exact functors in C which on the Grothendieck group descend to the action of a simply-laced quantum group in its adjoint representation. The braid group action in the adjoint representation lifts to an action in the derived category of C. The category C is the direct sum of a semisimple category and the category of modules over a certain algebra A, associated to a Dynkin diagram. In the second half of the paper we show how these algebras appear in the modular representation theory and in the McKay correspondence and explore their relationship with root sy…

Symmetric Categories as a Means of Virtualization for Braid Groups and Racks

In this paper we propose, firstly, a categorification of virtual braid groups and groupoids in terms of "locally" braided objects in a symmetric category (SC), and, secondly, a definition of self-distributive structures (SDS) in an arbitrary SC. SDS are shown to produce braided objects in a SC. As for examples, we interpret associativity and the Jacobi identity in a SC as generalized self-distributivity, thus endowing associative and Leibniz algebras with a "local" braiding. Our "double braiding" approach provides a natural interpretation for virtual racks and twisted Burau representation. Hom…

Symmetric self-adjoint Hopf categories and a categorical Heisenberg double

Motivated by the work of of A. Zelevinsky on positive self-adjoint Hopf algebras, we define what we call a symmetric self-adjoint Hopf structure for a certain kind of semisimple abelian categories. It is known that every positive self-adjoint Hopf algebra admits a natural action of the associated Heisenberg double. We construct canonical morphisms lifting the relations that define this action on the algebra level and define an object that we call a categorical Heisenberg double that is a natural setting for considering these morphisms. As examples, we exhibit the symmetric self-adjoint Hopf st…