Algebras associated to elliptic curves

Abstract

Abstract. This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one.In particular, we study the elliptic algebras A(+), A(−), and A(a), where a ∈ P 2 , which were first defined in an earlier paper. We omit a set S ⊂ P 2 consisting of 11 specified points where the algebras A(a) become too degenerate to be regular. A(a), where a ∈ P 2 \ S. Then A is an Artin-Schelter regular algebra of global dimension three. Moreover, A is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables. Theorem. Let A represent A(+), A(−) orThis, combined with our earlier results, completes the classification.

Noncommutative homological mirror functor

“…Theorem 5.14 ([ATdB90,Ste97]). Let W be a non-zero central element of the Sklyanin algebra Sky 3 (a, b, c).…”

“…It is known by [ATdB90,Ste97] that the quotient algebra of Sklyanin algebra by its central element can be written as a homogeneous coordinate ring. In our setting, it can be written as follows.…”

Untitled

“…Remark 1.8. We believe that by adapting the techniques in this work, one could establish a version of Theorem 1.7 for semisimple Hopf actions on other elliptic algebras, such as in [Sklyanin 1982] (or, see [Smith and Stafford 1992]) and in [Etingof and Ginzburg 2010;Odesskiȋ 2002;Stephenson 1997]. Further, we believe that under an appropriate nondegeneracy condition, there are no finite quantum symmetry results for such elliptic algebras and for twisted homogeneous coordinate rings B(X, σ, L) as above; compare to Theorem 5.2.…”

Regular algebras of dimension 4 and their A∞-Ext-algebras

“…The notation (l) in (AS2) is the degree l shift operation on graded modules. AS regular algebras have been studied in many recent papers, and in particular, AS regular algebras of global dimension 3 have been classified [ASc,ATV1,ATV2,Ste1,Ste2] and their geometry has been studied extensively [ATV2,Ste3]. If A is an AS regular algebra of global dimension 3, then it is generated by two or three elements.…”