[Paper Review] Hopf algebras of GK-dimension two with vanishing Ext-group

D.-G. Wang, J.J. Zhang+1 more|arXiv (Cornell University)|Apr 30, 2011

Algebraic structures and combinatorial models16 references3 citations

TL;DR

This paper constructs and classifies a new family of finitely generated, noetherian Hopf domains of Gelfand-Kirillov dimension two with vanishing first Ext-group, $ext^1_H(k,k)=0$, thereby answering negatively a question by Goodearl and Zhang regarding whether all such Hopf domains must satisfy the non-vanishing Ext condition. The authors introduce a new class $B(n,\{p_i\}_{1}^{s},q,\{\alpha_i\}_{1}^{s})$ and prove that under a root-of-unity exclusion hypothesis, these are the only such algebras satisfying the vanishing Ext condition.

ABSTRACT

We construct and study a family of finitely generated Hopf algebra domains $H$ of Gelfand-Kirillov dimension two such that $\Ext^1_H(k,k)=0$. Consequently, we answer a question of Goodearl and the second-named author.

Motivation & Objective

• To construct finitely generated noetherian Hopf domains of GK-dimension two with $\operatorname{Ext}^1_H(k,k) = 0$, answering a question posed by Goodearl and Zhang.
• To classify such Hopf domains under the hypothesis $\Omega'$, which excludes $A(1,q)$ as a Hopf subalgebra when $q$ is a primitive 5th or 7th root of unity.
• To provide evidence that the new family $B(n,\{p_i\}_{1}^{s},q,\{\alpha_i\}_{1}^{s})$ exhausts all such Hopf domains of GK-dimension two with vanishing Ext-group under the given restriction.
• To show that the new algebras are finitely generated over their affine centers and have finite global dimension if and only if $s=2$ and $\alpha_1 \neq \alpha_2$.

Proposed method

• Construct a new family of Hopf algebras $B(n,\{p_i\}_{1}^{s},q,\{\alpha_i\}_{1}^{s})$ as a modification of the algebra $B(n,p_0,\dots,p_s,q)$ from Goodearl-Zhang.
• Use the structure of skew primitive and grouplike elements to define relations: $x^{-1}y_i x = q_i y_i$, $y_i^{p_i} = y_1^{p_1} + \alpha_i(x^M - 1)$, and $q_i^{n_j} = q_j^{n_i} = 1$.
• Apply the coradical filtration and GK-dimension arguments to analyze the structure of Hopf subalgebras and classify them via the hypothesis $\Omega'$.
• Use the vanishing of $\operatorname{Ext}^1_H(k,k)$ to eliminate cases where all $\alpha_i = 0$, which would otherwise force $H$ to be isomorphic to one of the seven known families with non-vanishing Ext.
• Leverage results from Goodearl-Zhang's classification under condition ($\natural$) and extend them to the vanishing Ext case via a new classification theorem.
• Prove that the new family $B(n,\{p_i\}_{1}^{s},q,\{\alpha_i\}_{1}^{s})$ is the only possible Hopf domain of GK-dimension two with $\operatorname{Ext}^1_H(k,k) = 0$ under $\Omega'$, using properties of skew primitive elements and their weights.

Experimental results

Research questions

• RQ1 Does there exist a Hopf domain of GK-dimension two with $\operatorname{Ext}^1_H(k,k) = 0$, contradicting the non-vanishing condition ($\natural$) assumed in prior classifications?
• RQ2 Can all finitely generated Hopf domains of GK-dimension two with vanishing $\operatorname{Ext}^1_H(k,k)$ be classified under a suitable restriction?
• RQ3 Is the new family $B(n,\{p_i\}_{1}^{s},q,\{\alpha_i\}_{1}^{s})$ the only such family of Hopf domains of GK-dimension two with $\operatorname{Ext}^1_H(k,k) = 0$ when excluding certain roots of unity?
• RQ4 What are the homological properties (e.g., injective dimension, global dimension) of the new Hopf algebras $B(n,\{p_i\}_{1}^{s},q,\{\alpha_i\}_{1}^{s})$?
• RQ5 Under what conditions is the global dimension of $B(n,\{p_i\}_{1}^{s},q,\{\alpha_i\}_{1}^{s})$ finite, and when is it exactly 2?

Key findings

• The paper constructs a new family of Hopf algebras $B(n,\{p_i\}_{1}^{s},q,\{\alpha_i\}_{1}^{s})$ of GK-dimension two with $\operatorname{Ext}^1_H(k,k) = 0$, providing the first known examples where the Ext condition ($\natural$) fails.
• Under the hypothesis $\Omega'$, which excludes $A(1,q)$ as a Hopf subalgebra for $q$ a primitive 5th or 7th root of unity, any finitely generated Hopf domain of GK-dimension two with $\operatorname{Ext}^1_H(k,k) = 0$ must be isomorphic to $B(n,\{p_i\}_{1}^{s},q,\{\alpha_i\}_{1}^{s})$ with $\alpha_i \neq \alpha_j$ for some $i \neq j$.
• The algebra $B(n,\{p_i\}_{1}^{s},q,\{\alpha_i\}_{1}^{s})$ is finitely generated over its affine center, and its injective dimension is exactly 2.
• The global dimension of $B(n,\{p_i\}_{1}^{s},q,\{\alpha_i\}_{1}^{s})$ is finite if and only if $s = 2$ and $\alpha_1 \neq \alpha_2$, in which case it equals 2.
• The classification is complete under $\Omega'$: any such Hopf domain is either isomorphic to one of the seven families with non-vanishing $\operatorname{Ext}^1_H(k,k)$ or to the new family $B(n,\{p_i\}_{1}^{s},q,\{\alpha_i\}_{1}^{s})$ with non-zero $\alpha_i$'s.
• The construction shows that the vanishing of $\operatorname{Ext}^1_H(k,k)$ is possible in GK-dimension two, thus answering [GZ, Question 0.3] in the negative.