# Some algebras having relations like those for the 4-dimensional Sklyanin algebras

## TL;DR

This paper investigates algebras similar to 4-dimensional Sklyanin algebras, disproves a conjecture about a related family of algebras, and explores their structural features and classifications.

## Contribution

It refutes a conjecture linking certain algebras to Sklyanin algebras and introduces a new classification approach using projective space graph structures.

## Key findings

- Most Cho-Hong-Lau algebras are characterized by a bijection graph in P^3
- The studied algebras share features with Sklyanin algebras but also differ in key ways
- A new family of 4-generator 6-relator algebras includes Sklyanin and Cho-Hong-Lau algebras

## Abstract

The 4-dimensional Sklyanin algebras are a well-studied 2-parameter family of non-commutative graded algebras, often denoted A(E,tau), that depend on a quartic elliptic curve E in P^3 and a translation automorphism tau of E. They are graded algebras generated by four degree-one elements subject to six quadratic relations and in many important ways they behave like the polynomial ring on four indeterminates apart from the minor difference that they are not commutative. They are elliptic analogues of the enveloping algebra of sl(2,C) and the quantized enveloping algebras U_q(gl_2). Recently, Cho, Hong, and Lau, conjectured that a certain 2-parameter family of algebras arising in their work on homological mirror symmetry consists of 4-dimensional Sklyanin algebras. This paper shows their conjecture is false in the generality they make it. On the positive side, we show their algebras exhibit features that are similar to, and differ from, analogous features of the 4-dimensional Sklyanin algebras in interesting ways. We show that most of the Cho-Hong-Lau algebras determine, and are determined by the graph of a bijection between two 20-point subsets of the projective space P^3. The paper also examines a 3-parameter family of 4-generator 6-relator algebras admitting presentations analogous to those of the 4-dimensional Sklyanin algebras. This class includes the 4-dimensional Sklyanin algebras and most of the Cho-Hong-Lau algebras.