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Sklyanin algebra

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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.[2]

Formal definition

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Let be a field with a primitive cube root of unity. Let be the following subset of the projective plane :

Each point gives rise to a (quadratic 3-dimensional) Sklyanin algebra,

where,

Whenever we call a degenerate Sklyanin algebra and whenever we say the algebra is non-degenerate.[3]

Properties

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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.

Properties of degenerate Sklyanin algebras

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Let be a degenerate Sklyanin algebra.

Properties of non-degenerate Sklyanin algebras

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Let be a non-degenerate Sklyanin algebra.

Examples

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Degenerate Sklyanin algebras

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The subset 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 be a degenerate Sklyanin algebra.

  • If then is isomorphic to , which is the Sklyanin algebra corresponding to the point .
  • If then is isomorphic to , which is the Sklyanin algebra corresponding to the point .[3]

These two cases are Zhang twists of each other[3] and therefore have many properties in common.[7]

Non-degenerate Sklyanin algebras

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The commutative polynomial ring is isomorphic to the non-degenerate Sklyanin algebra and is therefore an example of a non-degenerate Sklyanin algebra.

Point modules

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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.[2]

Non-degenerate Sklyanin algebras

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Whenever and in the definition of a non-degenerate Sklyanin algebra , the point modules of are parametrised by an elliptic curve.[2] If the parameters 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.[8] If is a Sklyanin algebra whose point modules are parametrised by an elliptic curve, then there exists an element which annihilates all point modules i.e. for all point modules of .

Degenerate Sklyanin algebras

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The point modules of degenerate Sklyanin algebras are not parametrised by a projective variety.[4]

References

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  1. ^ a b c Artin, Michael; Schelter, William F. (1987-11-01). "Graded algebras of global dimension 3". Advances in Mathematics. 66 (2): 171–216. doi:10.1016/0001-8708(87)90034-X. ISSN 0001-8708.
  2. ^ a b c d Rogalski, D. (2014-03-12). "An introduction to Noncommutative Projective Geometry". arXiv:1403.3065 [math.RA].
  3. ^ a b c Smith, S. Paul (15 May 2012). ""Degenerate" 3-dimensional Sklyanin algebras are monomial algebras". Journal of Algebra. 358: 74–86. arXiv:1112.5809. doi:10.1016/j.jalgebra.2012.01.039.
  4. ^ a b c d e f g Walton, Chelsea (2011-12-23). "Degenerate Sklyanin algebras and Generalized Twisted Homogeneous Coordinate rings". Journal of Algebra. 322 (7): 2508–2527. arXiv:0812.0609. doi:10.1016/j.jalgebra.2009.02.024.
  5. ^ a b c d e Tate, John; van den Bergh, Michel (1996-01-01). "Homological properties of Sklyanin algebras". Inventiones Mathematicae. 124 (1): 619–648. Bibcode:1996InMat.124..619T. doi:10.1007/s002220050065. ISSN 1432-1297. S2CID 121438487.
  6. ^ De Laet, Kevin (October 2017). "On the center of 3-dimensional and 4-dimensional Sklyanin algebras". Journal of Algebra. 487: 244–268. arXiv:1612.06158. doi:10.1016/j.jalgebra.2017.05.032.
  7. ^ Zhang, J. J. (1996). "Twisted Graded Algebras and Equivalences of Graded Categories". Proceedings of the London Mathematical Society. s3-72 (2): 281–311. doi:10.1112/plms/s3-72.2.281. hdl:2027.42/135651. ISSN 1460-244X.
  8. ^ Artin, Michael; Tate, John; Van den Bergh, M. (2007), Cartier, Pierre; Illusie, Luc; Katz, Nicholas M.; Laumon, Gérard (eds.), "Some Algebras Associated to Automorphisms of Elliptic Curves", The Grothendieck Festschrift: A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck, Progress in Mathematics, Boston, MA: Birkhäuser, pp. 33–85, doi:10.1007/978-0-8176-4574-8_3, ISBN 978-0-8176-4574-8, retrieved 2021-04-28