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 ${\displaystyle k}$ be a field with a primitive cube root of unity. Let ${\displaystyle {\mathfrak {D}}}$ be the following subset of the projective plane ${\displaystyle {\textbf {P}}_{k}^{2}}$:

${\displaystyle {\mathfrak {D}}=\{[1:0:0],[0:1:0],[0:0:1]\}\sqcup \{[a:b:c]{\big |}a^{3}=b^{3}=c^{3}\}.}$

Each point ${\displaystyle [a:b:c]\in {\textbf {P}}_{k}^{2}}$ gives rise to a (quadratic 3-dimensional) Sklyanin algebra,

${\displaystyle S_{a,b,c}=k\langle x,y,z\rangle /(f_{1},f_{2},f_{3}),}$

where,

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

Whenever ${\displaystyle [a:b:c]\in {\mathfrak {D}}}$ we call ${\displaystyle S_{a,b,c}}$ a degenerate Sklyanin algebra and whenever ${\displaystyle [a:b:c]\in {\textbf {P}}^{2}\setminus {\mathfrak {D}}}$ we say the algebra is non-degenerate. ^[3]

Properties

The non-degenerate case shares many properties with the commutative polynomial ring ${\displaystyle k[x,y,z]}$, 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 ${\displaystyle S_{\text{deg}}}$ be a degenerate Sklyanin algebra.

• ${\displaystyle S_{\text{deg}}}$ contains non-zero zero divisors. ^[4]
• The Hilbert series of ${\displaystyle S_{\text{deg}}}$ is ${\displaystyle H_{S_{\text{deg}}}={\frac {1+t}{1-2t}}}$. ^[4]
• Degenerate Sklyanin algebras have infinite Gelfand–Kirillov dimension. ^[4]
• ${\displaystyle S_{\text{deg}}}$ is neither left nor right Noetherian. ^[4]
• ${\displaystyle S_{\text{deg}}}$ is a Koszul algebra. ^[4]
• Degenerate Sklyanin algebras have infinite global dimension. ^[4]

Properties of non-degenerate Sklyanin algebras

Let ${\displaystyle S}$ be a non-degenerate Sklyanin algebra.

• ${\displaystyle S}$ contains no non-zero zero divisors. ^[5]
• The hilbert series of ${\displaystyle S}$ is ${\displaystyle H_{S}={\frac {1}{(1-t)^{3}}}}$. ^[5]
• Non-degenerate Sklyanin algebras are Noetherian. ^[5]
• ${\displaystyle S}$ 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 ${\displaystyle {\mathfrak {D}}}$ 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 ${\displaystyle S_{\text{deg}}=S_{a,b,c}}$ be a degenerate Sklyanin algebra.

• If ${\displaystyle a=b}$ then ${\displaystyle S_{\text{deg}}}$ is isomorphic to ${\displaystyle k\langle x,y,z\rangle /(x^{2},y^{2},z^{2})}$, which is the Sklyanin algebra corresponding to the point ${\displaystyle [0:0:1]\in {\mathfrak {D}}}$.
• If ${\displaystyle a\neq b}$ then ${\displaystyle S_{\text{deg}}}$ is isomorphic to ${\displaystyle k\langle x,y,z\rangle /(xy,yx,zx)}$, which is the Sklyanin algebra corresponding to the point ${\displaystyle [1:0:0]\in {\mathfrak {D}}}$. ^[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 ${\displaystyle k[x,y,z]}$ 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.

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 ${\displaystyle abc\neq 0}$ and ${\displaystyle \left({\frac {a^{3}+b^{3}+c^{3}}{3abc}}\right)^{3}\neq 1}$ in the definition of a non-degenerate Sklyanin algebra ${\displaystyle S=S_{a,b,c}}$, the point modules of ${\displaystyle S}$ are parametrised by an elliptic curve. ^[2] If the parameters ${\displaystyle a,b,c}$ 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. ^[8] If ${\displaystyle S}$ is a Sklyanin algebra whose point modules are parametrised by an elliptic curve, then there exists an element ${\displaystyle g\in S}$ which annihilates all point modules i.e. ${\displaystyle Mg=0}$ for all point modules ${\displaystyle M}$ of ${\displaystyle S}$.

Degenerate Sklyanin algebras

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

References

1. 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. Rogalski, D. (2014-03-12). "An introduction to Noncommutative Projective Geometry". arXiv: 1403.3065 [math.RA].
3. 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. 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. 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