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 lead to the development of the study of point modules in noncommutative geometry. 
Formal definition.
Let formula_1 be a field with a primitive cube root of unity. Let formula_2 be the following subset of the projective plane formula_3:
formula_4
Each point formula_5 gives rise to a (quadratic 3-dimensional) Sklyanin algebra,
formula_6
where,
formula_7
Whenever formula_8 we call formula_9 a degenerate Sklyanin algebra and whenever formula_10 we say the algebra is non-degenerate.
Properties.
The non-degenerate case shares many properties with the commutative polynomial ring formula_11, whereas the degenerate case enjoys almost none of these properties. Generally the non-degenerate Sklyanin algebras are more challenging to understand than their non-degenerate counterparts.
Properties of degenerate Sklyanin algebras.
Let formula_12 be a degenerate Sklyanin algebra.
Properties of non-degenerate Sklyanin algebras.
Let formula_18 be a non-degenerate Sklyanin algebra.
Examples.
Degenerate Sklyanin algebras.
The subset formula_2 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 formula_24 be a degenerate Sklyanin algebra.
These two cases are Zhang twists of each other and therefore have many properties in common. 
Non-degenerate Sklyanin algebras.
The commutative polynomial ring formula_11 is isomorphic to the non-degenerate Sklyanin algebra formula_34 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 formula_35 and formula_36 in the definition of a non-degenerate Sklyanin algebra formula_37, then the point modules of formula_18 are parametrised by an elliptic curve. If the parameters formula_39 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 formula_18 is a Sklyanin algebra whose point modules are parametrised by an elliptic curve, then there exists an element formula_41 which annihilates all point modules i.e. formula_42 for all point modules formula_43 of formula_18.
Degenerate Sklyanin algebras.
The point modules of degenerate Sklyanin algebras are not parametrised by a projective variety. 