  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).
