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Iterated Hopf Ore Extensions over Group Rings

Can Hatipoğlu, Christian Lomp

TL;DR

This work defines a broad family of Hopf algebras $H(G, \chi, \eta, b, c, \beta)$ as two-step Hopf-Ore extensions of the group algebra $\mathbb{K}[G]$ and unifies several known Hopf algebra families. A key dichotomy is established: the zero-derivation case yields a skew group ring and enables module construction by induction from one-dimensional subalgebras, while the nonzero-derivation case enforces $\eta=\chi^{-1}$ and yields a noncommutative differential-operator–like regime with new simple modules. The paper provides a complete classification of finite-dimensional simple $H$-modules in both regimes, with explicit isomorphism criteria and constructions, and situates the results within Takeuchi, generalized Taft, Wang–Wu–Tan, and Fantino–García frameworks. Structural properties such as Noetherianity, affine PI-conditions, and Artin–Schelter–Gorenstein aspects are established in appropriate settings, and several Hopf quotients are shown to be finite over $\mathbb{K}[G]$, highlighting the framework's unifying power for pointed Hopf algebras.

Abstract

We introduce and study a class of Hopf algebras $H(G, χ, η, b, c, β)$ which are two-step Ore extensions of a group algebra $\mathbb{K}[G]$. This construction unifies and generalizes some known families of Hopf algebras such as generalized Taft algebras and Hopf algebras related to $\mathfrak{sl}_2$ constructed by Wang, Wu, and Tan. We analyze the ring theoretical properties of these algebras and classify all finite dimensional simple modules over them.

Iterated Hopf Ore Extensions over Group Rings

TL;DR

This work defines a broad family of Hopf algebras as two-step Hopf-Ore extensions of the group algebra and unifies several known Hopf algebra families. A key dichotomy is established: the zero-derivation case yields a skew group ring and enables module construction by induction from one-dimensional subalgebras, while the nonzero-derivation case enforces and yields a noncommutative differential-operator–like regime with new simple modules. The paper provides a complete classification of finite-dimensional simple -modules in both regimes, with explicit isomorphism criteria and constructions, and situates the results within Takeuchi, generalized Taft, Wang–Wu–Tan, and Fantino–García frameworks. Structural properties such as Noetherianity, affine PI-conditions, and Artin–Schelter–Gorenstein aspects are established in appropriate settings, and several Hopf quotients are shown to be finite over , highlighting the framework's unifying power for pointed Hopf algebras.

Abstract

We introduce and study a class of Hopf algebras which are two-step Ore extensions of a group algebra . This construction unifies and generalizes some known families of Hopf algebras such as generalized Taft algebras and Hopf algebras related to constructed by Wang, Wu, and Tan. We analyze the ring theoretical properties of these algebras and classify all finite dimensional simple modules over them.
Paper Structure (7 sections, 20 theorems, 99 equations)

This paper contains 7 sections, 20 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. The Construction
  3. Examples
  4. Basic Properties
  5. Finite Dimensional Simple $H$-modules
  6. The Skew Group Ring Case
  7. The Differential Operator Case

Key Result

Theorem 2.1

For a group $G$ and pairs $(\chi, b), (\eta, c) \in \widehat{G} \times Z(G)$ such that $\eta(b) = \chi(c)^{-1}$ and $\beta \in \mathbb{K}$, there exists a $\tau_\eta$-derivation $\delta$ such that the iterated Ore extension $H = \mathbb{K}[G][x; \tau_\chi][y; \tau_\eta, \delta]$ is a Hopf algebra su In particular, $H$ is generated as an algebra by the elements $g$ of $G$ and $x$ and $y$ such that

Theorems & Definitions (44)

  • Theorem 2.1: The Hopf algebras $H(G, \chi, \eta, b, c, \beta)$
  • proof
  • Remark 2.2
  • Example 3.1: Takeuchi's Hopf Algebra $U(1)$
  • Example 3.2: Generalized Taft Algebras
  • Example 3.3
  • Example 3.4: Fantino & García's pointed Hopf algebras over dihedral groups
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • ...and 34 more