Potential algebras with few generators

TL;DR

This paper classifies quadratic and cubic potential algebras with few generators over algebraically closed fields, providing conditions for finite-dimensionality and clarifying the structure for arbitrary generators.

Contribution

It offers a complete classification of potential algebras with 2 or 3 generators and establishes a necessary condition for finite-dimensionality in two-generator cases.

Findings

Complete classification of quadratic and cubic potential algebras with few generators.

Finite-dimensionality requires the potential to contain degree three terms in two-generator cases.

Clarification of potential algebra structures for arbitrary numbers of generators.

Abstract

We give a complete description of quadratic potential and twisted potential algebras on 3 generators as well as cubic potential and twisted potential algebras on 2 generators up to graded algebra isomorphisms under the assumption that the ground field is algebraically closed and has characteristic different from 2 or 3. We also prove that for two generated potential algebra necessary condition of finite-dimensionality is that potential contains terms of degree three, this answers a question of Agata Smoktunowicz and the first named author, formulated in [AN]. We clarify situation in case of arbitrary number of generators as well.