fourDimSklyanin -- Defines a four-dimensional Sklyanin with given parameters

Synopsis

Usage:

fourDimSklyanin(R,params,varList)
Inputs:
- R, a ring,
- params, a list,
- varList, a list,
Optional inputs:
- DegreeLimit => an integer, default value 6
Outputs:
- a quotient of a free algebra,

Description

This method constructs a four dimensional Sklyanin algebra with parameters from the params list, and variables from varList (see here). If either list is not the appropriate length, then an error is thrown. The generic such algebra has a fairly complicated Groebner basis, so the optional parameter DegreeLimit may be provided to limit the maximum of a generator of a Groebner basis found. This value has been defaulted to 6. If only one list is provided, it is used for the variable names, and a random choice for each parameter (satisfying the nondegeneracy condition given below) is chosen.

In order to not get a degenerate example, one should ensure that the parameters provided satisfy \alpha + \beta + \gamma + \alpha\beta\gamma = 0. This method does not check this condition, since the degenerate examples are of interest as well. If no parameters are provided, however a generic choice of \alpha,\beta and \gamma satisfying the equation above are selected.

i1 : C = fourDimSklyanin(ZZ/32003,{a,b,c,d})

o1 = C

o1 : FreeAlgebraQuotient

In all nondegenerate cases, there is are two central elements of degree two which form a regular sequence on the four dimensional Sklyanin (this was proven by Paul Smith and Toby Stafford in a paper in Compositio).

i2 : centralElements(C,2)

o2 = | -742a2+10313b2+c2 741a2-10312b2+d2 |

             1      2
o2 : Matrix C  <-- C

These algebras also all AS-regular and as such have the same Hilbert series as a commutative polynomial algebra in four variables, as we can see here:

i3 : apply(8, i -> numgens source ncBasis(i,C))

o3 = {1, 4, 10, 20, 35, 56, 84, 120}

o3 : List

i4 : apply(8, i -> binomial(i+3,3))

o4 = {1, 4, 10, 20, 35, 56, 84, 120}

o4 : List

Ways to use fourDimSklyanin :

fourDimSklyanin(Ring,List)
fourDimSklyanin(Ring,List,List)

For the programmer

The object fourDimSklyanin is a method function with options.