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Hopf actions of some quantum groups on path algebras

Ryan Kinser, Amrei Oswald

Abstract

Our first collection of results parametrize (filtered) actions of a quantum Borel $U_q(\mathfrak{b}) \subset U_q(\mathfrak{sl}_2)$ on the path algebra of an arbitrary (finite) quiver. When $q$ is a root of unity, we give necessary and sufficient conditions for these actions to factor through corresponding finite-dimensional quotients, generalized Taft algebras $T(r,n)$ and small quantum groups $U_q(\mathfrak{sl}_2)$. In the second part of the paper, we shift to the language of tensor categories. Here we consider a quiver path algebra equipped with an action of a Hopf algebra $H$ to be a tensor algebra in the tensor category of representations $H$. Such a tensor algebra is generated by an algebra and bimodule in this tensor category. Our second collection of results describe the corresponding bimodule categories via an equivalence with categories of representations of certain explicitly described quivers with relations.

Hopf actions of some quantum groups on path algebras

Abstract

Our first collection of results parametrize (filtered) actions of a quantum Borel on the path algebra of an arbitrary (finite) quiver. When is a root of unity, we give necessary and sufficient conditions for these actions to factor through corresponding finite-dimensional quotients, generalized Taft algebras and small quantum groups . In the second part of the paper, we shift to the language of tensor categories. Here we consider a quiver path algebra equipped with an action of a Hopf algebra to be a tensor algebra in the tensor category of representations . Such a tensor algebra is generated by an algebra and bimodule in this tensor category. Our second collection of results describe the corresponding bimodule categories via an equivalence with categories of representations of certain explicitly described quivers with relations.

Paper Structure

This paper contains 16 sections, 18 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Context and motivation
  3. Summary of main results
  4. Background and preliminaries
  5. Hopf algebras and their actions
  6. Generalized Taft algebras and related quantum groups
  7. Skew-primitives and $q$-integers
  8. Quivers, path algebras, and their representations
  9. Hopf actions on path algebras
  10. Parametrizing actions by linear algebraic data
  11. Actions of $U_q(\mathfrak{b})$ and Taft algebras on products of $\Bbbk$
  12. Actions of $U_q(\mathfrak{b})$ and Taft algebras on path algebras
  13. Actions of $U_q(\mathfrak{sl}_2)$ and $u_q(\mathfrak{sl}_2)$ on path algebras
  14. Tensor categorical viewpoint
  15. Hopf actions and tensor categories
  16. ...and 1 more sections

Key Result

Theorem 1

The following data determines a (filtered) Hopf action of $U_q(\mathfrak{b})$ (resp., $U_q(\mathfrak{sl}_2)$) on a path algebra $\Bbbk Q$, and all such actions are of this form:

Theorems & Definitions (37)

  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.4
  • Lemma 2.17
  • proof
  • Proposition 2.20
  • proof
  • Lemma 2.27
  • proof
  • ...and 27 more