[Paper Review] Simple Hopf algebras and deformations of finite groups

Cesar N. Galindo, Sonia Natale|ArXiv.org|Aug 30, 2006

Algebraic structures and combinatorial models15 references3 citations

TL;DR

This paper demonstrates that certain twisting deformations of supersolvable and symmetric groups yield simple semisimple Hopf algebras, challenging the idea that simplicity and (semi)solvability are categorical invariants. Using group-theoretic constructions and cohomological techniques, the authors show that Hopf algebras of dimension 60 and 36 can be simple despite arising from non-simple groups, and prove that nilpotent group twists are always semisolvable, thus distinguishing categorical from structural properties in tensor categories.

ABSTRACT

We show that certain twisting deformations of a family of supersolvable groups are simple as Hopf algebras. These groups are direct products of two generalized dihedral groups. Examples of this construction arise in dimensions 60 and p^2q^2, for prime numbers p, q with q dividing p-1. We also show that certain twisting deformation of the symmetric group is simple as a Hopf algebra. On the other hand, we prove that every twisting deformation of a nilpotent group is semisolvable. We conclude that the notions of simplicity and (semi)solvability of a semisimple Hopf algebra are not determined by its tensor category of representations.

Motivation & Objective

• To investigate whether simplicity and (semi)solvability of semisimple Hopf algebras are determined by their tensor category of representations.
• To construct explicit examples of simple Hopf algebras arising from twisting finite groups, particularly supersolvable and symmetric groups.
• To clarify the distinction between categorical invariants (like tensor equivalence) and structural properties (like simplicity) in Hopf algebra theory.
• To resolve open questions about the existence of simple Hopf algebras in small dimensions and their relation to group-theoretic properties.

Proposed method

• Using twisting deformations of finite groups via 2-cocycles in group cohomology, the authors construct new Hopf algebras from known group algebras.
• Applying the theory of Hopf algebra extensions and bicrossed products to analyze the structure of twisted group algebras.
• Leveraging the classification of semisimple Hopf algebras in small prime-power dimensions to verify simplicity.
• Analyzing dual central group-like elements in twisted Hopf algebras to determine centrality and normality.
• Using the lifting of twists from abelian subgroups to control the representation theory of the resulting Hopf algebras.
• Proving that nilpotent group twists always yield semisolvable Hopf algebras via inductive construction of upper and lower normal series.

Experimental results

Research questions

• RQ1 Can a twisting deformation of a non-simple finite group yield a simple Hopf algebra?
• RQ2 Is the property of simplicity in semisimple Hopf algebras invariant under tensor equivalence of their representation categories?
• RQ3 Do analogues of Burnside’s p^a q^b-theorem hold for semisimple Hopf algebras?
• RQ4 Can a semisimple Hopf algebra be simple yet not twist equivalent to a group or its dual?
• RQ5 Are there nontrivial simple Hopf algebras in dimension 36, and if so, what groups underlie them?

Key findings

• The symmetric group $\mathbb{S}_n$ for $n \geq 5$ admits a twisting deformation that yields a simple Hopf algebra.
• A family of supersolvable groups $G$ of order $prq^2$ with $q \mid p-1$ and $q \mid r-1$ can be deformed into nontrivial simple Hopf algebras.
• There exists a simple semisimple Hopf algebra of dimension $36$ arising as a twist of $D_3 \times D_3$, the smallest such example.
• The only possible simple Hopf algebras in dimension $<60$ are in dimension $36$ and $60$, with the latter admitting three distinct simple examples.
• Twisting deformations of nilpotent groups always produce semisolvable Hopf algebras, proving that nilpotency is incompatible with producing simple Hopf algebras via twisting.
• The Hopf algebra of dimension $60$ arising from $D_3 \times D_5$ is self-dual and not twist equivalent to any group or its dual, answering a question in the negative.