Nichols algebras with many cubic relations

TL;DR

This paper classifies finite-dimensional Nichols algebras of group type with many cubic relations, showing their Hilbert series factor into quantum integers, and links their properties to a conjecture in cellular automata.

Contribution

It provides a classification of Nichols algebras with many cubic relations under a specific assumption, connecting algebraic structures to cellular automata conjectures.

Findings

All such Nichols algebras are finite-dimensional.

Their Hilbert series factor into quantum integers.

Known elementary Nichols algebras of group type have many cubic relations.

Abstract

Nichols algebras of group type with many cubic relations are classified under a technical assumption on the structure of Hurwitz orbits of the third power of the underlying indecomposable rack. All such Nichols algebras are finite-dimensional and their Hilbert series have a factorization into quantum integers. Also, all known finite-dimensional elementary Nichols algebras of group type turn out to have many cubic relations. The technical assumption of our theorem can be removed if a conjecture in the theory of cellular automata can be proven.