We con
sider a Sklyanin algebra
S with 3 generator
s,
which i
s the quadratic algebra over a field <
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with 3 generator
s x,
y,
z given by 3 relation
s <
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span>, <
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where <
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sin;K
span><
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span>. Thi
s cla
ss of algebra
s enjoyed much of attention, in particular, u
sing tool
s from algebraic geometry, Feigin, Ode
sskii <
span id="bbr0150">[15]
span>, and Artin, Tate and Van den Bergh <
span id="bbr0030">[3]
span>,
sho
wed that if at lea
st t
wo of the parameter
s p,
q and
r are non-zero and at lea
st t
wo of three number
s <
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sup>3
sup>
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span>, <
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sup>3
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sup>3
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span> are di
stinct, then
S i
s Ko
szul and ha
s the
same Hilbert
serie
s a
s the algebra of commutative polynomial
s in 3 variable
s.
sp0020">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.
sp0030">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).