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 $\mathfrak$ be the following subset of the projective plane $\textbf_k^2$:

$\mathfrak = \ \sqcup \.$

Each point :b:c\in \textbf_k^2 gives rise to a (quadratic 3-dimensional) Sklyanin algebra,

$S_ = k \langle x,y,z \rangle / (f_1, f_2, f_3),$

where,

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

Whenever :b:c \in \mathfrak we call $S_$ a degenerate Sklyanin algebra and whenever :b:c\in \textbf^2 \setminus \mathfrak we say the algebra is non-degenerate.

Properties

The non-degenerate case shares many properties with the commutative polynomial ring k ,y,z/math>, 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 $S_$ be a degenerate Sklyanin algebra.

* $S_$ contains non-zero zero divisors.
* The Hilbert series of $S_$ is $H_ = \frac$.
* Degenerate Sklyanin algebras have infinite Gelfand–Kirillov dimension.
* $S_$ is neither left nor right Noetherian.
* $S_$ 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 $H_ = \frac$.
* 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.

Examples

Degenerate Sklyanin algebras

The subset $\mathfrak$ 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 $S_ = S_$ be a degenerate Sklyanin algebra.

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

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 ,y,z/math> is isomorphic to the non-degenerate Sklyanin algebra $S_ = k \langle x,y,z \rangle /( xy-yx, yz-zy, zx- xz)$ 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 $\left( \frac \right) ^3 \neq 1$ in the definition of a non-degenerate Sklyanin algebra $S=S_$, 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.

References

{{reflist

Algebra