Sklyanin algebras and a cubic root of 1

TL;DR

This paper presents a new algebraic approach to analyze Sklyanin algebras with three generators, enabling explicit determination of their structure and isomorphisms without relying on algebraic geometry, except in characteristic 3.

Contribution

The authors develop a linear substitution technique to determine Gr"obner bases of Sklyanin algebras directly, simplifying previous geometric methods and explicitly classifying isomorphisms.

Findings

Identified conditions for Sklyanin algebras to be Artin--Schelter regular.

Explicitly determined leading monomials of Gr"obner bases for these algebras.

Classified isomorphisms among Sklyanin algebras based on parameters.

Abstract

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. This class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry Artin, Tate and Van Den Berg \cite{ATV2} showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates. It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Gr\"obner basis technique. The authors have previously dispelled this belief. However our previous proof was no less complicated than the one based on algebraic geometry. It used a construcion of a Gr\"obner basis in a suitable one-sided module over $S$ and had quite a number of cases to consider. In this paper we exhibit a linear substitution after which it becomes possible to determine the leading monomials of a reduced Gr\"obner basis for the ideal of relations of $S$ itself (without passing to a module). We also find out explicitly (in terms of parameters) which Sklyanin algebras are isomorphic. The only drawback of the new technique is that it fails if the characteristic of the ground field equals 3.