Three dimensional Sklyanin algebras and Groebner bases

Abstract

We consider a Sklyanin algebra S with 3 generators, which is the quadratic algebra over a field k with three generators x,y,z given by three relations pxy+qyx+rzz=0, pyz+qzy+rxx=0 and pzx+qxz+ryy=0, where p,q,r are parameters from the fileld k. This class of algebras enjoyed much of attention, in particular, using tools from algebraic geometry, Feigin & Odesskii, and Artin, Tate & Van den Berg, showed that if at least two of the parameters p, q and r are non-zero and at least two of three numbers p^3,q^3 and r^3 are distinct, then S is Koszul and has the same Hilbert series as the algebra of commutative polynomials in three variables. It became commonly accepted, that it is impossible to achieve the same objective by purely algebraic and combinatorial means, like the Groebner basis technique. The main purpose of this paper is to trace combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van den Bergh. Further, we study a wider class of Sklyanin algebras, namely the situation when all parameters of relations could be different. This class called generalized Sklyanin algebras. We classify up to isomorphism genralized Sklyanin algebras with polynomial Hilbert series. We show that generalized Slyanin algebras in general position have the Golod-Shafarevich Hilbert series (with exception of the case of the field with two elements).