[Paper Review] Some Hopf algebras of dimension $72$ without the Chevalley property

Naihong Hu, Rongchuan Xiong|arXiv (Cornell University)|Dec 15, 2016

Algebraic structures and combinatorial models3 references11 citations

TL;DR

This paper constructs new finite-dimensional Hopf algebras of dimension 72 that lack the Chevalley property—meaning their coradical is not a subalgebra—by applying the generalized lifting method to a 12-dimensional Hopf algebra with non-semisimple coradical. It proves that Nichols algebras over non-semisimple modules are infinite-dimensional, and explicitly describes finite-dimensional Nichols algebras via generators and relations, yielding 72-dimensional Hopf algebras without the Chevalley property.

ABSTRACT

In this paper, we consider the Drinfeld double $\D$ of a $12$-dimensional Hopf algebra $\C$ over an algebraically closed field of characteristic zero whose coradical is not a subalgebra and describe its simple modules, projective covers of the simple modules and show that it is of wild representation type. Moreover, we show that the Nichols algebras associated to non-simple indecomposable modules are infinite-dimensional. In particular, for any object $V$ in $\CYD$, if $\BN(V)$ is finite-dimensional, then $V$ must be semisimple. Finally, we describe the Nichols algebras associated to partial simple modules in terms of generators and relations. As a byproduct, we obtain some Hopf algebras of dimension $72$ without the Chevalley property, that is, the coradical is not a subalgebra.

Motivation & Objective

• To classify finite-dimensional Hopf algebras of dimension 72 without the Chevalley property.
• To extend the lifting method beyond the classical setting where the coradical is a Hopf subalgebra.
• To determine when Nichols algebras associated to non-semisimple modules are finite-dimensional.
• To explicitly describe the structure of finite-dimensional Nichols algebras over partial simple modules.
• To construct new examples of Hopf algebras with non-semisimple coradicals using the generalized lifting method.

Proposed method

• Use the generalized lifting method by Andruskiewitsch and Cuadra, replacing the coradical filtration with the standard filtration to handle non-Chevalley property Hopf algebras.
• Construct the Drinfeld double $\mathcal{D}$ of a 12-dimensional Hopf algebra $\mathcal{C}$ whose coradical is not a subalgebra.
• Analyze the representation theory of $\mathcal{D}$, including its simple modules and projective covers, showing it is of wild representation type.
• Compute coproducts and relations in the bosonization $\mathcal{B}(V)\sharp\mathcal{C}$ for specific braided vector spaces $V$ to determine when $\mathcal{B}(V)$ is finite-dimensional.
• Use direct algebraic computation to show that certain commutators like $xy - yx$ vanish in the algebra, leading to relations such as $x^3 = 0$ and $x^2 - \xi^2 y^2 = 0$, implying $A \cong \mathrm{gr}\,A$.
• Verify that the resulting algebras are isomorphic to $\mathcal{B}(V)\sharp\mathcal{C}$ for $V$ isomorphic to $V_{2,2}$, $V_{2,4}$, $V_{3,1}$, or $V_{3,5}$, yielding 72-dimensional Hopf algebras without the Chevalley property.

Experimental results

Research questions

• RQ1 Which finite-dimensional Hopf algebras of dimension 72 fail to satisfy the Chevalley property, and how can they be constructed systematically?
• RQ2 Under what conditions is the Nichols algebra $\mathcal{B}(V)$ finite-dimensional when $V$ is not semisimple?
• RQ3 Can the generalized lifting method be applied to Hopf algebras where the coradical is not a subalgebra?
• RQ4 What are the explicit presentations (generators and relations) of finite-dimensional Nichols algebras over non-semisimple Yetter-Drinfeld modules?
• RQ5 Do the Drinfeld double and standard filtration methods yield new examples of wild representation type Hopf algebras?

Key findings

• The Drinfeld double $\mathcal{D}$ of the 12-dimensional Hopf algebra $\mathcal{C}$ is of wild representation type, as shown by analyzing its simple modules and projective covers.
• For any object $V$ in ${}^{\mathcal{C}}_{\mathcal{C}}\mathcal{YD}$, if $\mathcal{B}(V)$ is finite-dimensional, then $V$ must be semisimple.
• The Nichols algebras associated to partial simple modules $V_{2,2}$, $V_{2,4}$, $V_{3,1}$, and $V_{3,5}$ are finite-dimensional and explicitly described via generators and relations.
• The algebras $\mathcal{B}(V_{2,2})\sharp\mathcal{C}$, $\mathcal{B}(V_{2,4})\sharp\mathcal{C}$, $\mathcal{B}(V_{3,1})\sharp\mathcal{C}$, and $\mathcal{B}(V_{3,5})\sharp\mathcal{C}$ are all of dimension 72 and lack the Chevalley property.
• The relations $x^3 = 0$, $xy - yx = 0$, and $x^2 - \xi^2 y^2 = 0$ hold in the resulting algebras, implying $A \cong \mathrm{gr}\,A$ and confirming the isomorphism to the bosonization.
• The Hopf algebras $\bigwedge\mathds{k}_{\chi^k}\sharp\mathcal{C}$ for $k \in \{1,3,5\}$ are of dimension 24 and also lack the Chevalley property.