A website dedicated to the classification and properties of four-dimensional quadratic Artin–Schelter regular algebras (that are general in their family), organised around the known families and the data of arXiv:2511.08390.
So far 56 families are recorded here. See explained for the definitions and families for the full presentations. Click a column heading to sort.
Per-invariant overview tables: point schemes, line schemes, \(\mathrm{HH}^i_0(A)\), \(\mathrm{HH}^i(\operatorname{qgr} A)\), the centre, normal elements, the Kodaira–Spencer map, and the Nakayama automorphism.
| family | defined | parameters | dimension point scheme | dimension line scheme | \(\mathrm{HH}^i_0\) | \(\mathrm{HH}^i(\mathrm{qgr}\,A)\) | Kodaira–Spencer | \(\det \nu\) | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 1 | 2 | 3 | rank | inj. | surj. | ||||||
| commutative | ? | 0 | 3 | 4 | 16 | 60 | 80 | 15 | 45 | 35 | 0 | yes | no | \(1\) |
| Sklyanin | 1982 | 2 | 1 | 2 | 1 | 2 | 9 | 0 | 2 | 7 | 2 | yes | yes | \(1\) |
| skew | 1990 | 6 | 1 | 2 | 4 | 6 | 4 | 3 | 3 | 5 | 6 | yes | yes | \(1\) |
| Vancliff | 1994 | 3 | 2 | 3 | 3 | 4 | 3 | 2 | 4 | 7 | 3 | yes | no | \(1\) |
| Vancliff twist | 1994 | 3 | 1 | 2 | 3 | 4 | 3 | 2 | 4 | 7 | 3 | yes | no | \(1\) |
| Clifford | 1995 | 24 | 0 | 2 | 1 | 9 | 19 | 0 | 15 | 20 | 9 | no | yes | \(1\) |
| central extension of Sklyanin | 1996 | 12 | 1 | 2 | 1 | 7 | 19 | 0 | 7 | 12 | 7 | no | yes | \(1\) |
| Shelton–Tingey | 2001 | 0 | 0 | 1 | 1 | 1 | 7 | 0 | 16 | 21 | 0 | yes | no | \(1\) |
| Caines | 2005 | 4 | 0 | 1 | 1 | 8 | 21 | 0 | 14 | 19 | 1 | no | no | \(1\) |
| Cassidy–Goetz–Shelton | 2006 | 5 | 1 | 2 | 2 | 4 | 8 | 1 | 3 | 7 | 4 | no | yes | \(1\) |
| double Ore A | 2009 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 8 | 12 | 1 | yes | no | \(1\) |
| double Ore B | 2009 | 1 | 1 | 1 | 2 | 1 | 0 | 1 | 17 | 21 | 1 | yes | yes | \(1\) |
| double Ore C | 2009 | 1 | 1 | 2 | 2 | 1 | 0 | 1 | 0 | 4 | 1 | yes | yes | \(1\) |
| double Ore D | 2009 | 2 | 1 | 2 | 2 | 2 | 2 | 1 | 8 | 12 | 2 | yes | yes | \(1\) |
| double Ore E | 2009 | 1 | 1 | 2 | 2 | 1 | 0 | 2 | 3 | 6 | 1 | yes | yes | \(-1\) |
| double Ore F | 2009 | 1 | 1 | 1 | 2 | 1 | 0 | 1 | 16 | 20 | 1 | yes | yes | \(1\) |
| double Ore G | 2009 | 3 | 1 | 2 | 2 | 2 | 2 | 1 | 8 | 12 | 2 | no | yes | \(1\) |
| double Ore H | 2009 | 2 | 1 | 2 | 3 | 3 | 1 | 3 | 6 | 8 | 2 | yes | no | \(1\) |
| double Ore I | 2009 | 1 | 1 | 1 | 2 | 1 | 0 | 1 | 16 | 20 | 1 | yes | yes | \(1\) |
| double Ore J | 2009 | 1 | 1 | 2 | 2 | 1 | 0 | 2 | 3 | 6 | 1 | yes | yes | \(-1\) |
| double Ore K | 2009 | 3 | 1 | 2 | 3 | 3 | 1 | 2 | 1 | 4 | 3 | yes | yes | \(-1\) |
| double Ore L | 2009 | 3 | 1 | 2 | 3 | 3 | 1 | 3 | 6 | 8 | 3 | yes | yes | \(1\) |
| double Ore M | 2009 | 2 | 1 | 2 | 2 | 2 | 2 | 3 | 9 | 11 | 2 | yes | yes | \(1\) |
| double Ore N | 2009 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 15 | 19 | 2 | no | yes | \(1\) |
| double Ore O | 2009 | 2 | 1 | 2 | 2 | 2 | 2 | 3 | 9 | 11 | 2 | yes | yes | \(1\) |
| double Ore P | 2009 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 15 | 19 | 2 | yes | yes | \(1\) |
| double Ore Q | 2009 | 1 | 1 | 2 | 2 | 1 | 0 | 1 | 1 | 5 | 1 | yes | yes | \(-1\) |
| double Ore R | 2009 | 1 | 2 | 3 | 2 | 1 | 0 | 2 | 6 | 9 | 1 | yes | yes | \(1\) |
| double Ore S | 2009 | 1 | 1 | 1 | 2 | 1 | 0 | 1 | 17 | 21 | 1 | yes | yes | \(1\) |
| double Ore T | 2009 | 1 | 1 | 2 | 2 | 1 | 0 | 2 | 8 | 11 | 1 | yes | yes | \(-1\) |
| double Ore U | 2009 | 1 | 1 | 2 | 2 | 1 | 0 | 2 | 8 | 11 | 1 | yes | yes | \(-1\) |
| double Ore V | 2009 | 1 | 1 | 2 | 2 | 1 | 0 | 1 | 1 | 5 | 1 | yes | yes | \(-1\) |
| double Ore W | 2009 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 15 | 19 | 2 | yes | yes | \(1\) |
| double Ore X | 2009 | 1 | 1 | 2 | 3 | 3 | 1 | 3 | 6 | 8 | 1 | yes | no | \(1\) |
| double Ore Y | 2009 | 2 | 1 | 2 | 2 | 3 | 4 | 1 | 16 | 20 | 1 | no | no | \(1\) |
| double Ore Z | 2009 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 15 | 19 | 2 | yes | yes | \(1\) |
| Generalized Clifford 1 | 2010 | 1 | 0 | 1 | 1 | 1 | 7 | 0 | 16 | 21 | 1 | yes | yes | \(1\) |
| Generalized Clifford 2 | 2010 | 4 | 0 | 1 | 1 | 9 | 19 | 0 | 16 | 21 | 3 | no | no | \(1\) |
| Generalized Clifford 3 | 2010 | 4 | 0 | 2 | 1 | 9 | 7 | — | — | — | 4 | yes | no | \(-1\) |
| Ore extension of commutative | 2015 | 6 | 1 | 2 | 4 | 9 | 9 | 3 | 6 | 8 | 9 | no | yes | \(1\) |
| Jordan | 2016 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 0 | 4 | 1 | yes | yes | \(1\) |
| \(\mathrm{S}_{d,i}\) | 2016 | 3 | 1 | 1 | 1 | 3 | 9 | 0 | 4 | 9 | 3 | yes | yes | \(1\) |
| \(\mathrm{S}_{d,i}\) twist | 2016 | 3 | 0 | 1 | 1 | 8 | 17 | 0 | 14 | 19 | 3 | yes | no | \(1\) |
| \(\mathrm{S}_\infty\) | 2016 | 2 | 1 | 2 | 1 | 2 | 5 | 0 | 2 | 7 | 2 | yes | yes | \(1\) |
| \(\mathrm{S}_\infty\) twist | 2016 | 2 | 0 | 1 | 1 | 8 | 17 | 0 | 14 | 19 | 2 | yes | no | \(1\) |
| Sklyanin twist | 2016 | 2 | 0 | 1 | 1 | 8 | 21 | 0 | 14 | 19 | 2 | yes | no | \(1\) |
| Goetz–Kirkman–Moore–Vashaw R | 2024 | 0 | 1 | 2 | 2 | 12 | 22 | 3 | 21 | 23 | 0 | yes | no | \(1\) |
| Goetz–Kirkman–Moore–Vashaw S | 2024 | 0 | 0 | 1 | 1 | 4 | 9 | 0 | 15 | 20 | 0 | yes | no | \(1\) |
| Goetz–Kirkman–Moore–Vashaw T | 2024 | 0 | 0 | 1 | 1 | 4 | 9 | 0 | 15 | 20 | 0 | yes | no | \(1\) |
| \(\mathrm{A}_5\) | 2025 | 4 | 1 | 2 | 2 | 1 | 0 | 1 | 1 | 5 | 1 | no | yes | \(i\) |
| central extension of Sklyanin twist | 2025 | 5 | 1 | 2 | 1 | 4 | 9 | 0 | 7 | 12 | 4 | no | yes | \(1\) |
| deformed skew \((x_3x_2;\ x_4^2)\) | ? | 5 | 1 | 2 | 3 | 4 | 3 | 2 | 4 | 7 | 4 | no | yes | \(1\) |
| deformed skew \((x_2x_1;\ x_3x_4)\) | ? | 5 | 1 | 2 | 3 | 4 | 3 | 2 | 4 | 7 | 4 | no | yes | \(1\) |
| deformed skew \((x_3x_2;\ x_4^2, x_1^2)\) | ? | 5 | 1 | 2 | 2 | 4 | 8 | 1 | 3 | 7 | 3 | no | no | \(1\) |
| deformed skew \((x_3x_1, x_3x_2;\ x_4^2)\) | ? | 4 | 1 | 2 | 2 | 2 | 2 | 1 | 8 | 12 | 2 | no | yes | \(1\) |
| deformed skew \((x_2x_1, x_3x_2;\ x_3^2, x_4^2)\) | ? | 4 | 1 | 2 | 2 | 2 | 2 | 1 | 8 | 12 | 2 | no | yes | \(1\) |