The Centers of 3-Dimensional Sklyanin Algebras

Abstract

This chapter describes the centers of three-dimensional Sklyanin algebras. In noncommutative situations, the problem of showing that rings are noetherian is much more difficult than in the commutative case. There is no Hilbert Basis Theorem, even in the case of algebras that are deformations of commutative polynomial rings. Of the seven generic types of these algebras, three come from reducible cubics and four from smooth irreducible ones. The four cases correspond to the nature of the automorphism, which can be a translation, a reflection, or a complex multiplication of order 3 or 4. Such algebra is called a Sklyanin algebra, because Sklyanin gave a construction of a four-dimensional analog. The definition in dimension 4 is in fact more subtle than in dimension 3, but the generalization of the construction to higher dimensions, obtaining from a triple is shown. There are a few other nontrivial examples of regular algebras of dimension known, but a complete classification analogous to that in dimension 3, even for quadratic ones seems far away, even in dimension 4.