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Quantization of infinitesimal braidings and pre-Cartier quasi-bialgebras

Chiara Esposito, Andrea Rivezzi, Jonas Schnitzer, Thomas Weber

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.

Quantization of infinitesimal braidings and pre-Cartier quasi-bialgebras

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 -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 are provided.

Paper Structure

This paper contains 16 sections, 22 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Braided monoidal categories
  4. Quasitriangular quasi-bialgebras
  5. Infinitesimal braid relations and Drinfeld associators
  6. Infinitesimal braidings and pre-Cartier quasi-bialgebras
  7. Pre-Cartier categories
  8. Pre-Cartier quasi-bialgebras
  9. Infinitesimal braid relations in pre-Cartier quasi-bialgebras
  10. The quasi counterpart of the infinitesimal quantum Yang-Baxter equation
  11. Cartier rings and their representations
  12. Deforming braided monoidal categories
  13. Infinitesimal braid relations in pre-Cartier categories
  14. Deformation of Cartier categories
  15. Deformation of pre-Cartier categories
  16. ...and 1 more sections

Key Result

Theorem 2.5

Let $\mathcal{C}$ be a braided monoidal category. Then there exists a strict braided monoidal category $\mathcal{C}^{\mathrm{str}}$ such that $\mathcal{C} \cong \mathcal{C}^{\mathrm{str}}$.

Theorems & Definitions (68)

  • Definition 2.1: MacLan63JS93
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8: Doikou22
  • Proposition 2.9: Doikou22
  • Corollary 2.10
  • ...and 58 more