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

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.