threeDimSklyanin -- Defines a three-dimensional Sklyanin with given parameters

Synopsis

Usage:

threeDimSklyanin(R,params,varList)
Inputs:
- R, a ring,
- params, a list,
- varList, a list,
Optional inputs:
- DegreeLimit => an integer, default value 6
Outputs:
- a ring,

Description

This method constructs a three dimensional Sklyanin algebra with parameters from the params list, and variables from varList (see here). If either list is not length three, then an error is thrown. The generic such algebra does not have a finite Groebner basis, so the optional parameter DegreeLimit 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 is chosen.

The following example is a PI algebra, and has a finite Groebner basis.

i1 : B = threeDimSklyanin(ZZ/101,{1,1,-1},{x,y,z})

o1 = B

o1 : FreeAlgebraQuotient

i2 : NCGB(ideal B,5)

o2 = | x2-yz-zy xz-y2+zx xy+yx-z2 yz2-z2y y2z-zy2 |

              ZZ            1       ZZ            5
o2 : Matrix (---<|x, y, z|>)  <-- (---<|x, y, z|>)
             101                   101

This is not generically true, however:

i3 : C = threeDimSklyanin(ZZ/101,{2,3,5},{a,b,c})

o3 = C

o3 : FreeAlgebraQuotient

i4 : NCGB(ideal C,5)

o4 = | a2-40bc+41cb ac-32b2-33ca ab-49ba-48c2 b2a-25bc2+16cbc-17c2b
     ------------------------------------------------------------------------
     b2c-41bcb+33cb2-32c2a bcb2+48bc2a-15cb3+39cbca-3c2ba-18c4
     ------------------------------------------------------------------------
     b4-24bcba+15bc3+24cbc2-14c2bc-22c3b
     ------------------------------------------------------------------------
     bcbc2-41bc2bc+27bc3b-38cbcbc+43cbc2b-8c2bcb+36c3b2-21c4a
     ------------------------------------------------------------------------
     bcbcb-20bc2b2+42c2b3+11c2bca+15c3ba-30c5
     ------------------------------------------------------------------------
     bcbca-35bc2ba+10bc4+32cbcba+46cbc3+37c2bc2-19c3bc+28c4b
     ------------------------------------------------------------------------
     bc2b3-12bc3ba-2cbc2ba-37cbc4+26c2bcba+50c2bc3+40c3bc2+3c4bc-11c5b
     ------------------------------------------------------------------------
     bc2bca-29bc3ba+48bc5+2cbc2ba+37cbc4+21c2bcba-14c2bc3-39c3bc2-12c4bc-
     ------------------------------------------------------------------------
     41c5b bc2bc2+5bc3bc+33bc4b-12cbc3b+33c2bcbc-40c2bc2b+33c3bcb+6c4b2+42c5a
     ------------------------------------------------------------------------
     |

              ZZ            1       ZZ            13
o4 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
             101                   101

In all cases, there is a degree three central regular element (a formula for which is given in the paper referenced above).

i5 : centralElements(B,3)

o5 = | -y3+yzx-zyx+z3 |

             1      1
o5 : Matrix B  <-- B

i6 : centralElements(C,3)

o6 = | 39b3+40bca+17cba+c3 |

             1      1
o6 : Matrix C  <-- C

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

i7 : apply(8, i -> numgens source ncBasis(i,C))

o7 = {1, 3, 6, 10, 15, 21, 28, 36}

o7 : List

i8 : apply(8, i -> binomial(i+2,2))

o8 = {1, 3, 6, 10, 15, 21, 28, 36}

o8 : List

Ways to use threeDimSklyanin :

threeDimSklyanin(Ring,List)
threeDimSklyanin(Ring,List,List)

For the programmer

The object threeDimSklyanin is a method function with options.