In mathematics, specifically the field of algebra, Sklyanin algebras are a class of noncommutative algebra named after Evgeny Sklyanin. This class of algebras was first studied in the classification of Artin-Schelter regular algebras of global dimension 3 in the 1980s. Sklyanin algebras can be grouped into two different types, the non-degenerate Sklyanin algebras and the degenerate Sklyanin algebras, which have very different properties. A need to understand the non-degenerate Sklyanin algebras better has led to the development of the study of point modules in noncommutative geometry.
Formal definition Let k {\displaystyle k} be a field with a primitive cube root of unity. Let D {\displaystyle {\mathfrak {D}}} be the following subset of the projective plane P k 2 {\displaystyle {\textbf {P}}_{k}^{2}} :
D = { [ 1 : 0 : 0 ] , [ 0 : 1 : 0 ] , [ 0 : 0 : 1 ] } ⊔ { [ a : b : c ] | a 3 = b 3 = c 3 } . {\displaystyle {\mathfrak {D}}=\{[1:0:0],[0:1:0],[0:0:1]\}\sqcup \{[a:b:c]{\big |}a^{3}=b^{3}=c^{3}\}.}
Each point [ a : b : c ] ∈ P k 2 {\displaystyle [a:b:c]\in {\textbf {P}}_{k}^{2}} gives rise to a (quadratic 3-dimensional) Sklyanin algebra,
S a , b , c = k ⟨ x , y , z ⟩ / ( f 1 , f 2 , f 3 ) , {\displaystyle S_{a,b,c}=k\langle x,y,z\rangle /(f_{1},f_{2},f_{3}),}
where,
f 1 = a y z + b z y + c x 2 , f 2 = a z x + b x z + c y 2 , f 3 = a x y + b y x + c z 2 . {\displaystyle f_{1}=ayz+bzy+cx^{2},\quad f_{2}=azx+bxz+cy^{2},\quad f_{3}=axy+byx+cz^{2}.}
Whenever [ a : b : c ] ∈ D {\displaystyle [a:b:c]\in {\mathfrak {D}}} we call S a , b , c {\displaystyle S_{a,b,c}} a degenerate Sklyanin algebra and whenever [ a : b : c ] ∈ P 2 ∖ D {\displaystyle [a:b:c]\in {\textbf {P}}^{2}\setminus {\mathfrak {D}}} we say the algebra is non-degenerate.
Properties The non-degenerate case shares many properties with the commutative polynomial ring k [ x , y , z ] {\displaystyle k[x,y,z]} , whereas the degenerate case enjoys almost none of these properties. Generally the non-degenerate Sklyanin algebras are more challenging to understand than their degenerate counterparts.
Properties of degenerate Sklyanin algebras Let S deg {\displaystyle S_{\text{deg}}} be a degenerate Sklyanin algebra.
S deg {\displaystyle S_{\text{deg}}} contains non-zero zero divisors. The Hilbert series of S deg {\displaystyle S_{\text{deg}}} is H S deg = 1 + t 1 − 2 t {\displaystyle H_{S_{\text{deg}}}={\frac {1+t}{1-2t}}} . Degenerate Sklyanin algebras have infinite Gelfand–Kirillov dimension.
S deg {\displaystyle S_{\text{deg}}} is neither left nor right Noetherian.
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