Exotic Elliptic Algebras of dimension 4 (with an Appendix by Derek Tomlin)

TL;DR

This paper studies a class of non-commutative 4-dimensional algebras linked to elliptic curves, classifying their modules and geometric properties, revealing intricate symmetries and structures related to elliptic curves and quantum groups.

Contribution

It provides a detailed classification of modules and geometric structures of elliptic algebras of dimension 4, extending previous work with new insights into quantum symmetries.

Findings

Classification of point, fat point, and line modules.

Parametrization of line modules by a degree 20 curve.

Identification of quantum group symmetries acting on the algebra.

Abstract

This is a continuation of our previous paper 1502.01744. We examine a class of non-commutative algebras A that depend on an elliptic curve and a translation automorphism of it. They may be defined in terms of the 4-dimensional Sklyanin algebra S that is associated to the same data. The algebra A has the same Hilbert series as the polynomial ring in 4 variables, and there is an associated non-commutative variety, Proj(A), that is a non-commutative analogue of P^3. The structure and representation theory of A, and the geometric properties of Proj(A) are closely related to the geometric properties of E sitting as a quartic curve in P^3. Our main results concern the classification of point modules, fat point modules, line modules, and the incidence relations between them. The line modules are parametrized by a degree 20 curve in the Grassmannian G(1,3) that is a union of 4 disjoint plane conics and 3 disjoint quartic elliptic curves that are isomorphic to E/(t) where t runs over the three 2-torsion points. A finite quantum group related to the Heisenberg group of size 4^3 acts as auto-equivalences of the category of graded A-modules and those quantum symmetries of A play a central role in our analysis.