Three dimensional Sklyanin algebras and Gröbner bases

We consider a Sklyanin algebra S   with 3 generators, which is the quadratic algebra over a field KK with 3 generators x, y, z   given by 3 relations pxy+qyx+rzz=0pxy+qyx+rzz=0, pyz+qzy+rxx=0pyz+qzy+rxx=0 and pzx+qxz+ryy=0pzx+qxz+ryy=0, where p,q,r∈Kp,q,r∈K. This class of algebras enjoyed much of attention, in particular, using tools from algebraic geometry, Feigin, Odesskii [15], and Artin, Tate and Van den Bergh [3], showed that if at least two of the parameters p, q and r   are non-zero and at least two of three numbers p3p3, q3q3 and r3r3 are distinct, then S is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 variables.It became commonly accepted, that it is impossible to achieve the same objective by purely algebraic and combinatorial means, like the Gröbner basis technique. The main purpose of this paper is to trace the 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. We call them generalized Sklyanin algebras. We classify up to isomorphism all generalized Sklyanin algebras with the same Hilbert series as commutative polynomials on 3 variables. We show that generalized Sklyanin algebras in general position have a Golod–Shafarevich Hilbert series (with exception of the case of field with two elements).