Exotic elliptic algebras

The 4-dimensional Sklyanin algebras, over C, A(E, τ), are constructed from an elliptic curve E and a translation automorphism τ of E. The Klein vierergruppe Γ acts as graded algebra automorphisms of A(E, τ). There is also an action of Γ as automorphisms of the matrix algebra M ₂ (C) making it isomorphic to the regular representation. The main object of study in this paper is the invariant subalgebra Ã:= (A(E, τ) ⊗M ₂ (C)) ^Γ . Like A(E, τ), à is noetherian, generated by 4 degree-one elements modulo six quadratic relations, Koszul, Artin-Schelter regular of global dimension 4, has the same Hilbert series as the polynomial ring on 4 variables, satisfies the χ condition, and so on. These results are special cases of general results proved for a triple (A, T, H) consisting of a Hopf algebra H, an (often graded) H-comodule algebra A, and an H-torsor T . Those general results involve transferring properties between A, A ⊗ T, and (A ⊗ T) ^coH . We then investigate à from the point of view of non-commutative projective geometry. We examine its point modules, line modules, and a certain quotient ˜B:= Ã/(Θ, Θ ^′ ) where Θ and Θ ^′ are homogeneous central elements of degree two. In doing this we show that à differs from A in interesting ways. For example, the point modules for A are parametrized by E and 4 more points, whereas à has exactly 20 point modules. Although ˜B is not a twisted homogeneous coordinate ring in the sense of Artin and Van den Bergh, a certain quotient of the category of graded ˜B-modules is equivalent to the category of quasi-coherent sheaves on the curve E/E[2] where E[2] is the 2-torsion subgroup. We construct line modules for à that are parametrized by the disjoint union (E/〈ξ ₁ 〉) ⊔ (E/〈ξ ₂ 〉) ⊔ (E/〈ξ ₃ 〉) of the quotients of E by its three subgroups of order 2.