Constructive Approach to Three Dimensional Sklyanin Algebras

Abstract

A three dimensional Sklyanin is the quadratic algebra over a field $\mathbb{k}$ with 3 generators x; y; z given by 3 relations xy - ayx - szz = 0, yz - azy - sxx = 0 and zx - axz - syy = 0, where a,s $\in \mathbb{k}$. A generalized Sklyanin algebra is the algebra given by relations $\mathrm{xy - a_{1}yx - s_{1}zz = 0, yz - a_{2}zy - s_{2}xx = 0 and zx - a_{3}xz - s_{3}yy = 0}$, where $\mathrm{a_{i}, s_{i}} \in \mathbb{k}$. In this paper we announce the following results; the complete proofs will appear elsewhere. We determine explicitly the parameters for which these algebras has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates as well as when these algebras are Koszul and PBW, using constructive combinatorial methods. These provide new direct proofs of results established first by Artin, Tate, and Van Den Bergh.