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Bosonization of curved Lie bialgebras

I. Heckenberger, L. Vendramin

Abstract

We use Cartier's preadditive symmetric monoidal categories to study Lie bialgebras. We prove that bosonization can be done consistently in this framework. In the last part of the paper we present explicit examples and indicate a deep relationship between certain curved Lie bialgebras and Nichols algebras over abelian groups.

Bosonization of curved Lie bialgebras

Abstract

We use Cartier's preadditive symmetric monoidal categories to study Lie bialgebras. We prove that bosonization can be done consistently in this framework. In the last part of the paper we present explicit examples and indicate a deep relationship between certain curved Lie bialgebras and Nichols algebras over abelian groups.
Paper Structure (10 sections, 10 theorems, 86 equations)

This paper contains 10 sections, 10 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Symmetric monoidal categories with additional structure
  3. Lie algebras in symmetric categories
  4. Lie coalgebras in symmetric categories
  5. Lie bialgebras in Cartier categories
  6. Bosonization of Lie bialgebras in Cartier categories
  7. Examples
  8. The Jordan plane
  9. The super Jordan plane
  10. The Laistrygonian Lie bialgebras

Key Result

Lemma 5.2

The category $\prescript{\mathfrak{f}}{\mathfrak{f}}\mathcal{C}$ is preadditive symmetric monoidal, where the identity is the identity of $\mathcal{C}$ with zero action and coaction, the action and the coaction of $\mathfrak{f}$ on tensor products are diagonal, and the braiding is the braiding of $\

Theorems & Definitions (25)

  • Definition 2.1
  • Remark 2.2
  • Remark 5.1
  • Lemma 5.2
  • Remark 5.3
  • Lemma 5.4
  • proof
  • Lemma 5.5
  • proof
  • Proposition 5.6
  • ...and 15 more