Exotic elliptic algebras

Abstract

The 4-dimensional Sklyanin algebras, over $\mathbb {C}$, $A(E,\tau )$, are constructed from an elliptic curve $E$ and a translation automorphism $\tau$ of $E$. The Klein vierergruppe $\Gamma$ acts as graded algebra automorphisms of $A(E,\tau )$. There is also an action of $\Gamma$ as automorphisms of the matrix algebra $M_2(\mathbb {C})$ making it isomorphic to the regular representation. The main object of study in this paper is the invariant subalgebra $\widetilde {A}:=\big (A(E,\tau ) \otimes M_2(\mathbb {C})\big )^{\Gamma }$. Like $A(E,\tau )$, $\widetilde {A}$ 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 $\chi$ 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 $A$, $A \otimes T$, and $(A \otimes T)^\textrm {{co} H}$. We then investigate $\widetilde {A}$ from the point of view of non-commutative projective geometry. We examine its point modules, line modules, and a certain quotient $\widetilde {B}:=\widetilde {A}/(\Theta ,\Theta ’)$ where $\Theta$ and $\Theta ’$ are homogeneous central elements of degree two. In doing this we show that $\widetilde {A}$ differs from $A$ in interesting ways. For example, the point modules for $A$ are parametrized by $E$ and 4 more points, whereas $\widetilde {A}$ has exactly 20 point modules. Although $\widetilde {B}$ is not a twisted homogeneous coordinate ring in the sense of Artin and Van den Bergh, a certain quotient of the category of graded $\widetilde {B}$-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 $\widetilde {A}$ that are parametrized by the disjoint union $(E/\langle \xi _1\rangle ) \sqcup (E/\langle \xi _2\rangle ) \sqcup (E/\langle \xi _3\rangle )$ of the quotients of $E$ by its three subgroups of order 2.