Exotic Elliptic Algebras

TL;DR

This paper introduces a method for twisting comodule algebras using torsors, with a focus on a specific 4-dimensional Sklyanin algebra, resulting in a new class of exotic elliptic algebras with unique properties.

Contribution

It develops a general framework for algebra twisting via torsors and explores a novel example involving Sklyanin algebras leading to exotic elliptic algebras.

Findings

Twisted algebra retains many properties of the original Sklyanin algebra.

The twisted algebra exhibits new, unusual properties.

The method applies broadly to comodule algebras and torsors.

Abstract

This paper examines a general method for producing twists of a comodule algebra by tensoring it with a torsor then taking co-invariants. We examine the properties that pass from the original algebra to the twisted algebra and vice versa. We then examine the special case where the algebra is a 4-dimensional Sklyanin algebra viewed as a comodule algebra over the Hopf algebra of functions on the non-cyclic group of order 4 with the torsor being the 2x2 matrix algebra. The twisted algebra is an "exotic elliptic algebra". We show that the twisted algebra has many of the good properties that the Sklyanin algebra has, and that it has some new properties that make it quite unusual by comparison.