Sklyanin algebra

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 $$k$$ be a field with a primitive cube root of unity. Let $$\mathfrak{D}$$ be the following subset of the projective plane $$\textbf{P}_k^2$$:

$$\mathfrak{D} = \{ [1:0:0], [0:1:0], [0:0:1] \} \sqcup \{ [a:b:c ] \big| a^3=b^3=c^3\}.$$

Each point $$[a:b:c] \in \textbf{P}_k^2$$ gives rise to a (quadratic 3-dimensional) Sklyanin algebra,

$$S_{a,b,c} = k \langle x,y,z \rangle / (f_1, f_2, f_3),$$

where,

$$f_1 = ayz + bzy + cx^2, \quad f_2 = azx + bxz + cy^2, \quad f_3 = axy + b yx + cz^2.$$

Whenever $$[a:b:c ] \in \mathfrak{D}$$ we call $$S_{a,b,c}$$ a degenerate Sklyanin algebra and whenever $$[a:b:c] \in \textbf{P}^2 \setminus \mathfrak{D}$$ we say the algebra is non-degenerate.

Properties
The non-degenerate case shares many properties with the commutative polynomial ring $$k[x,y,z]$$, 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 $$S_{\text{deg}}$$ be a degenerate Sklyanin algebra.


 * $$S_{\text{deg}}$$ contains non-zero zero divisors.
 * The Hilbert series of $$S_{\text{deg}}$$ is $$H_{S_{\text{deg}}} = \frac{1+t}{1-2t}$$.
 * Degenerate Sklyanin algebras have infinite Gelfand–Kirillov dimension.
 * $$S_{\text{deg}}$$ is neither left nor right Noetherian.
 * $$S_{\text{deg}}$$ is a Koszul algebra.
 * Degenerate Sklyanin algebras have infinite global dimension.

Properties of non-degenerate Sklyanin algebras
Let $$S$$ be a non-degenerate Sklyanin algebra.


 * $$S$$ contains no non-zero zero divisors.
 * The hilbert series of $$S$$ is $$H_{S} = \frac{1}{(1-t)^3}$$.
 * Non-degenerate Sklyanin algebras are Noetherian.
 * $$S$$ is Koszul.
 * Non-degenerate Sklyanin algebras are Artin-Schelter regular. Therefore, they have global dimension 3 and Gelfand–Kirillov dimension 3.
 * There exists a normal central element in every non-degenerate Sklyanin algebra.

Degenerate Sklyanin algebras
The subset $$\mathfrak{D}$$ consists of 12 points on the projective plane, which give rise to 12 expressions of degenerate Sklyanin algebras. However, some of these are isomorphic and there exists a classification of degenerate Sklyanin algebras into two different cases. Let $$S_{\text{deg}} = S_{a,b,c}$$ be a degenerate Sklyanin algebra.


 * If $$a=b$$ then $$S_{\text{deg}}$$ is isomorphic to $$k \langle x,y,z \rangle /(x^2,y^2,z^2)$$, which is the Sklyanin algebra corresponding to the point $$[0:0:1] \in \mathfrak{D}$$.
 * If $$a \neq b$$ then $$S_{\text{deg}}$$ is isomorphic to $$k \langle x,y,z \rangle /(xy,yx,zx)$$, which is the Sklyanin algebra corresponding to the point $$[1:0:0] \in \mathfrak{D}$$.

These two cases are Zhang twists of each other and therefore have many properties in common.

Non-degenerate Sklyanin algebras
The commutative polynomial ring $$k[x,y,z]$$ is isomorphic to the non-degenerate Sklyanin algebra $$S_{1,-1,0} = k \langle x,y,z \rangle /( xy-yx, yz-zy, zx- xz)$$ and is therefore an example of a non-degenerate Sklyanin algebra.

Point modules
The study of point modules is a useful tool which can be used much more widely than just for Sklyanin algebras. Point modules are a way of finding projective geometry in the underlying structure of noncommutative graded rings. Originally, the study of point modules was applied to show some of the properties of non-degenerate Sklyanin algebras. For example to find their Hilbert series and determine that non-degenerate Sklyanin algebras do not contain zero divisors.

Non-degenerate Sklyanin algebras
Whenever $$abc \neq 0$$ and $$\left( \frac{a^3+b^3+c^3}{3abc} \right) ^3 \neq 1$$ in the definition of a non-degenerate Sklyanin algebra $$S=S_{a,b,c}$$, the point modules of $$S$$ are parametrised by an elliptic curve. If the parameters $$a,b,c$$ do not satisfy those constraints, the point modules of any non-degenerate Sklyanin algebra are still parametrised by a closed projective variety on the projective plane. If $$S$$ is a Sklyanin algebra whose point modules are parametrised by an elliptic curve, then there exists an element $$g \in S$$ which annihilates all point modules i.e. $$Mg = 0$$ for all point modules $$M $$ of $$S$$.

Degenerate Sklyanin algebras
The point modules of degenerate Sklyanin algebras are not parametrised by a projective variety.