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Hopf bimodules and Yetter-Drinfeld modules of Hopf algebroids

Xiao Han

TL;DR

This work extends the theory of Hopf bimodules and Yetter-Drinfeld modules from Hopf algebras to Hopf algebroids, establishing (pre-)braided and braided monoidal structures on their categories. It proves an equivalence between the categories of Hopf bimodules and Yetter-Drinfeld modules over a Hopf algebroid, and develops a duality theory for finitely generated projective Yetter-Drinfeld modules via dual pairings. The results provide foundational categorical tools for quantum groupoids and potential differential-geometric constructions (e.g., quantum Atiyah Lie algebroids) within the Hopf algebroid setting. Together, these contributions offer a robust framework for studying quantum symmetries and their modules in a generalized algebroid context, with implications for quantum geometry and cyclic (co)homology theories.

Abstract

We construct Hopf bimodules and Yetter-Drinfeld modules of Hopf algebroids as a generalization of the theory for Hopf algebras. More precisely, we show that the categories of Hopf bimodules and Yetter-Drinfeld modules over a Hopf algebroid are equivalent (pre-)braided monoidal categories. Moreover, we also study the duality between finitely generated projective Yetter-Drinfeld modules.

Hopf bimodules and Yetter-Drinfeld modules of Hopf algebroids

TL;DR

This work extends the theory of Hopf bimodules and Yetter-Drinfeld modules from Hopf algebras to Hopf algebroids, establishing (pre-)braided and braided monoidal structures on their categories. It proves an equivalence between the categories of Hopf bimodules and Yetter-Drinfeld modules over a Hopf algebroid, and develops a duality theory for finitely generated projective Yetter-Drinfeld modules via dual pairings. The results provide foundational categorical tools for quantum groupoids and potential differential-geometric constructions (e.g., quantum Atiyah Lie algebroids) within the Hopf algebroid setting. Together, these contributions offer a robust framework for studying quantum symmetries and their modules in a generalized algebroid context, with implications for quantum geometry and cyclic (co)homology theories.

Abstract

We construct Hopf bimodules and Yetter-Drinfeld modules of Hopf algebroids as a generalization of the theory for Hopf algebras. More precisely, we show that the categories of Hopf bimodules and Yetter-Drinfeld modules over a Hopf algebroid are equivalent (pre-)braided monoidal categories. Moreover, we also study the duality between finitely generated projective Yetter-Drinfeld modules.
Paper Structure (10 sections, 22 theorems, 195 equations)

This paper contains 10 sections, 22 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Basic algebraic preliminaries
  3. Left and right bialgebroids and its modules
  4. Hopf algebroids
  5. Hopf bimodules of Hopf algebroids
  6. (Pre-)braided monoidal category of Yetter-Drinfeld modules and Hopf bimodules
  7. (Pre-)braided monoidal category of Hopf bimodules
  8. (Pre-)braided monoidal category of Yetter-Drinfeld modules
  9. Dual pairings between Yetter-Drinfeld modules
  10. Properties of Hopf algebroids

Key Result

Proposition 2.21

Let $\Gamma$ be a right comodule of $(\mathcal{L}, \mathcal{R}, S)$, then $a^{.}\rho^{.} a'=a^{*}\rho^{*}a'$ and $\Gamma^{co\mathcal{R}}=\Gamma^{co\mathcal{L}}$, where $\Gamma^{co\mathcal{L}}:=\{\ \eta\in\Gamma \ |\ \delta_{L}(\eta)=\eta\otimes_{B}1 \}$ and $\Gamma^{co\mathcal{R}}:=\{\ \eta\in\Gamma

Theorems & Definitions (70)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 60 more