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[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
Let [math]\displaystyle{ k }[/math] be a field with a primitive cube root of unity. Let [math]\displaystyle{ \mathfrak{D} }[/math] be the following subset of the projective plane [math]\displaystyle{ \textbf{P}_k^2 }[/math]:
[math]\displaystyle{ \mathfrak{D} = \{ [1:0:0], [0:1:0], [0:0:1] \} \sqcup \{ [a:b:c ] \big| a^3=b^3=c^3\}. }[/math]
Each point [math]\displaystyle{ [a:b:c] \in \textbf{P}_k^2 }[/math] gives rise to a (quadratic 3-dimensional) Sklyanin algebra,
[math]\displaystyle{ S_{a,b,c} = k \langle x,y,z \rangle / (f_1, f_2, f_3), }[/math]
where,
[math]\displaystyle{ f_1 = ayz + bzy + cx^2, \quad f_2 = azx + bxz + cy^2, \quad f_3 = axy + b yx + cz^2. }[/math]
Whenever [math]\displaystyle{ [a:b:c ] \in \mathfrak{D} }[/math] we call [math]\displaystyle{ S_{a,b,c} }[/math] a degenerate Sklyanin algebra and whenever [math]\displaystyle{ [a:b:c] \in \textbf{P}^2 \setminus \mathfrak{D} }[/math] we say the algebra is non-degenerate.[3]
Properties
The non-degenerate case shares many properties with the commutative polynomial ring [math]\displaystyle{ k[x,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 [math]\displaystyle{ S_{\text{deg}} }[/math] be a degenerate Sklyanin algebra.
- [math]\displaystyle{ S_{\text{deg}} }[/math] contains non-zero zero divisors.[4]
- The Hilbert series of [math]\displaystyle{ S_{\text{deg}} }[/math] is [math]\displaystyle{ H_{S_{\text{deg}}} = \frac{1+t}{1-2t} }[/math].[4]
- Degenerate Sklyanin algebras have infinite Gelfand–Kirillov dimension.[4]
- [math]\displaystyle{ S_{\text{deg}} }[/math] is neither left nor right Noetherian.[4]
- [math]\displaystyle{ S_{\text{deg}} }[/math] is a Koszul algebra.[4]
- Degenerate Sklyanin algebras have infinite global dimension.[4]
Properties of non-degenerate Sklyanin algebras
Let [math]\displaystyle{ S }[/math] be a non-degenerate Sklyanin algebra.
- [math]\displaystyle{ S }[/math] contains no non-zero zero divisors.[5]
- The hilbert series of [math]\displaystyle{ S }[/math] is [math]\displaystyle{ H_{S} = \frac{1}{(1-t)^3} }[/math].[5]
- Non-degenerate Sklyanin algebras are Noetherian.[5]
- [math]\displaystyle{ S }[/math] is Koszul.[5]
- Non-degenerate Sklyanin algebras are Artin-Schelter [1] regular.[5] Therefore, they have global dimension 3 and Gelfand–Kirillov dimension 3.[1]
- There exists a normal central element in every non-degenerate Sklyanin algebra.[6]
Examples
Degenerate Sklyanin algebras
The subset [math]\displaystyle{ \mathfrak{D} }[/math] 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 [math]\displaystyle{ S_{\text{deg}} = S_{a,b,c} }[/math] be a degenerate Sklyanin algebra.
- If [math]\displaystyle{ a=b }[/math] then [math]\displaystyle{ S_{\text{deg}} }[/math] is isomorphic to [math]\displaystyle{ k \langle x,y,z \rangle /(x^2,y^2,z^2) }[/math], which is the Sklyanin algebra corresponding to the point [math]\displaystyle{ [0:0:1] \in \mathfrak{D} }[/math].
- If [math]\displaystyle{ a \neq b }[/math] then [math]\displaystyle{ S_{\text{deg}} }[/math] is isomorphic to [math]\displaystyle{ k \langle x,y,z \rangle /(xy,yx,zx) }[/math], which is the Sklyanin algebra corresponding to the point [math]\displaystyle{ [1:0:0] \in \mathfrak{D} }[/math].[3]
These two cases are Zhang twists of each other[3] and therefore have many properties in common.[7]
Non-degenerate Sklyanin algebras
The commutative polynomial ring [math]\displaystyle{ k[x,y,z] }[/math] is isomorphic to the non-degenerate Sklyanin algebra [math]\displaystyle{ S_{1,-1,0} = k \langle x,y,z \rangle /( xy-yx, yz-zy, zx- xz) }[/math] 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.[2]
Non-degenerate Sklyanin algebras
Whenever [math]\displaystyle{ abc \neq 0 }[/math] and [math]\displaystyle{ \left( \frac{a^3+b^3+c^3}{3abc} \right) ^3 \neq 1 }[/math] in the definition of a non-degenerate Sklyanin algebra [math]\displaystyle{ S=S_{a,b,c} }[/math], the point modules of [math]\displaystyle{ S }[/math] are parametrised by an elliptic curve.[2] If the parameters [math]\displaystyle{ a,b,c }[/math] 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 [math]\displaystyle{ S }[/math] is a Sklyanin algebra whose point modules are parametrised by an elliptic curve, then there exists an element [math]\displaystyle{ g \in S }[/math] which annihilates all point modules i.e. [math]\displaystyle{ Mg = 0 }[/math] for all point modules [math]\displaystyle{ M }[/math] of [math]\displaystyle{ S }[/math].
Degenerate Sklyanin algebras
The point modules of degenerate Sklyanin algebras are not parametrised by a projective variety.[4]
References
- ↑ 1.0 1.1 1.2 Artin, Michael; Schelter, William F. (1987-11-01). "Graded algebras of global dimension 3" (in en). Advances in Mathematics 66 (2): 171–216. doi:10.1016/0001-8708(87)90034-X. ISSN 0001-8708.
- ↑ 2.0 2.1 2.2 2.3 Rogalski, D. (2014-03-12). "An introduction to Noncommutative Projective Geometry". arXiv:1403.3065 [math.RA].
- ↑ 3.0 3.1 3.2 Smith, S. Paul (15 May 2012). ""Degenerate" 3-dimensional Sklyanin algebras are monomial algebras" (in en). Journal of Algebra 358: 74–86. doi:10.1016/j.jalgebra.2012.01.039.
- ↑ 4.0 4.1 4.2 4.3 4.4 4.5 4.6 Walton, Chelsea (2011-12-23). "Degenerate Sklyanin algebras and Generalized Twisted Homogeneous Coordinate rings". Journal of Algebra 322 (7): 2508–2527. doi:10.1016/j.jalgebra.2009.02.024.
- ↑ 5.0 5.1 5.2 5.3 5.4 Tate, John; van den Bergh, Michel (1996-01-01). "Homological properties of Sklyanin algebras" (in en). Inventiones Mathematicae 124 (1): 619–648. doi:10.1007/s002220050065. ISSN 1432-1297. Bibcode: 1996InMat.124..619T. https://doi.org/10.1007/s002220050065.
- ↑ De Laet, Kevin (October 2017). "On the center of 3-dimensional and 4-dimensional Sklyanin algebras". Journal of Algebra 487: 244–268. doi:10.1016/j.jalgebra.2017.05.032.
- ↑ Zhang, J. J. (1996). "Twisted Graded Algebras and Equivalences of Graded Categories" (in en). Proceedings of the London Mathematical Society s3-72 (2): 281–311. doi:10.1112/plms/s3-72.2.281. ISSN 1460-244X. https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-72.2.281.
- ↑ Artin, Michael; Tate, John; Van den Bergh, M. (2007), Cartier, Pierre; Illusie, Luc; Katz, Nicholas M. et al., eds., "Some Algebras Associated to Automorphisms of Elliptic Curves" (in en), 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, https://doi.org/10.1007/978-0-8176-4574-8_3, retrieved 2021-04-28
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