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Connected Hopf Algebras that are Not Hopf Ore Extensions of Enveloping Algebras

Mengying Hu, Quanshui Wu

TL;DR

The paper constructs UM(r,2s), a family of connected Hopf algebras with finite GK-dimension that are not IHOEs of the enveloping algebra of their primitive parts, answering Li–Zhou negatively. Despite not arising as IHOEs, these algebras admit an explicit iterated crossed-product presentation by enveloping algebras, realized through a decomposition involving 𝔩, Y, and so(B). The authors compute the Nakayama automorphism, showing UM(r,2s) is Calabi–Yau when r=2s, and establish a Calabi–Yau/Poisson-deformation viewpoint for these algebras. The minimal non-IHOE example appears at UM(4,4) (GK-dimension 19), with UM(2,2) serving as a known IHOE of the base field, highlighting a nuanced landscape between IHOEs and general connected Hopf algebras of finite GK-dimension.

Abstract

We construct a family of connected Hopf algebras with finite Gelfand-Kirillov dimension, none of which is an iterated Hopf Ore extension of the universal enveloping algebra of its primitive part. This provides a negative answer to a question posed by Li and Zhou. It is also demonstrated that these connected Hopf algebras can be formulated as an iterated crossed product of enveloping algebras.

Connected Hopf Algebras that are Not Hopf Ore Extensions of Enveloping Algebras

TL;DR

The paper constructs UM(r,2s), a family of connected Hopf algebras with finite GK-dimension that are not IHOEs of the enveloping algebra of their primitive parts, answering Li–Zhou negatively. Despite not arising as IHOEs, these algebras admit an explicit iterated crossed-product presentation by enveloping algebras, realized through a decomposition involving 𝔩, Y, and so(B). The authors compute the Nakayama automorphism, showing UM(r,2s) is Calabi–Yau when r=2s, and establish a Calabi–Yau/Poisson-deformation viewpoint for these algebras. The minimal non-IHOE example appears at UM(4,4) (GK-dimension 19), with UM(2,2) serving as a known IHOE of the base field, highlighting a nuanced landscape between IHOEs and general connected Hopf algebras of finite GK-dimension.

Abstract

We construct a family of connected Hopf algebras with finite Gelfand-Kirillov dimension, none of which is an iterated Hopf Ore extension of the universal enveloping algebra of its primitive part. This provides a negative answer to a question posed by Li and Zhou. It is also demonstrated that these connected Hopf algebras can be formulated as an iterated crossed product of enveloping algebras.

Paper Structure

This paper contains 8 sections, 17 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. preliminary
  3. Umbrella Hopf Algebras
  4. Umbrella Hopf algebras are not IHOEs of primitive parts
  5. Properties of Umbrella Hopf algebras
  6. Crossed product of Hopf algebras
  7. Umbrella algebras are iterated crossed products
  8. Calabi-Yau property of the umbrella algebras

Key Result

Theorem 1.2

The connected Hopf algebra $\mathop{\mathrm{UM}}\nolimits(r,2s)$ is not an IHOE of the universal enveloping algebra of its primitive part. Furthermore, if $r=2s$ and $s\ge 2$, then $\mathop{\mathrm{UM}}\nolimits(r,2s)$ is not an Ore extension of any Hopf subalgebra of codimension $1$.

Figures (2)

  • Figure 1: Top view
  • Figure 2: Bottom view

Theorems & Definitions (45)

  • Theorem 1.2: Corollary \ref{['CoroMain']}
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 35 more