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,

$f_{1}=ayz+bzy+cx^{2},\quad f_{2}=azx+bxz+cy^{2},\quad f_{3}=axy+byx+cz^{2}.$

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.

Examples

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.