Quantization of infinitesimal braidings and pre-Cartier quasi-bialgebras

TL;DR

This paper extends Cartier's deformation theorem to non-symmetric braided categories, introducing pre-Cartier quasi-bialgebras and demonstrating their quantization, with explicit deformations of certain algebraic structures and new concepts like the infinitesimal quantum Yang-Baxter equation.

Contribution

It introduces pre-Cartier quasi-bialgebras extending Drinfeld's quasitriangular quasi-bialgebras and demonstrates their quantization, expanding the theory of braided monoidal categories.

Findings

Quantization of infinitesimal R-matrices is possible for Cartier quasi-bialgebras.

Introduction of the infinitesimal quantum Yang-Baxter equation.

Explicit deformations of gauge deformed quasitriangular quasi-bialgebras E(n).

Abstract

In this paper we extend Cartier's deformation theorem of braided monoidal categories admitting an infinitesimal braiding to the non-symmetric case. The algebraic counterpart of these categories is the notion of a pre-Cartier quasi-bialgebra, which extends the well-known notion of quasitriangular quasi-bialgebra given by Drinfeld. Our result implies that one can quantize the infinitesimal $\mathcal{R}$-matrix of any Cartier quasi-bialgebra. We further discuss the emerging concepts of infinitesimal quantum Yang-Baxter equation and Cartier ring, the latter containing braid groups with additional generators that correspond to infinitesimal braidings. Explicit deformations of the representation categories of the gauge deformed quasitriangular quasi-bialgebras $E(n)$ are provided.