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[1] algebras of global dimension 3 in the 1980s.
[2] 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.
be a field with a primitive cube root of unity.
be the following subset of the projective plane
gives rise to a (quadratic 3-dimensional) Sklyanin algebra,
a degenerate Sklyanin algebra and whenever
[3] The non-degenerate case shares many properties with the commutative polynomial ring
, 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.
be a degenerate Sklyanin algebra.
be a non-degenerate Sklyanin algebra.
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.
be a degenerate Sklyanin algebra.
These two cases are Zhang twists of each other[3] and therefore have many properties in common.
[7] The commutative polynomial ring
is isomorphic to the non-degenerate Sklyanin algebra
{\displaystyle 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.
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.
in the definition of a non-degenerate Sklyanin algebra
are parametrised by an elliptic curve.
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.
is a Sklyanin algebra whose point modules are parametrised by an elliptic curve, then there exists an element
The point modules of degenerate Sklyanin algebras are not parametrised by a projective variety.