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 (missing documentation) => an integer, default value 5,
Outputs:
- an instance of the type NCRing,

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 5. 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(QQ,{1,1,-1},{x,y,z})
--Calling Bergman for NCGB calculation.
Complete!

o1 = B

o1 : NCQuotientRing

i2 : ncGroebnerBasis ideal B

      2    2                2
o2 = y x-xy ; Lead Term = (y x, 1)
       2  2                  2
     yx -x y; Lead Term = (yx , 1)
         2
     zx-y +xz; Lead Term = (zx, 1)
            2
     zy+yz-x ; Lead Term = (zy, 1)
      2                      2
     z -yx-xy; Lead Term = (z , 1)

o2 : NCGroebnerBasis

This is not generically true, however:

i3 : C = threeDimSklyanin(QQ,{a,b,c})
--Calling Bergman for NCGB calculation.
Complete!

o3 = C

o3 : NCQuotientRing

i4 : ncGroebnerBasis ideal C

           4049035857  2    479504274903  4 513457984      324   3   10609 2  2  6956383927  3    63565111 4
o4 = babac+----------ba bc+-------------ba +---------ababc+---aba - ------a ba +------------a ba+---------a b; Lead Term = (babac, 1)
           7406276800      1540505574400     80115975      103      173056      149977105200     161787600
             572085999   2 2   7469841911  3   6551866359 2 3  671915007  2       75087  3     75087 5
     babab- -----------ba b - -----------ba c+-----------a b +-----------a bac- --------a bc- ------a ; Lead Term = (babab, 1)
            59825459200       29912729600     59825459200     29912729600       35952800      691400
         2   617137  2     561286   3   9518912       617137   2   103 2    10874 3 2 2379728 4                    2
     baba - -------ba ba- --------ba b+--------ababa+-------aba b- ---a bab+-----a b +-------a c; Lead Term = (baba , 1)
            7000425       80115975     80115975      8901775        81      86425     7000425
      4  1664      103  3  9    2   10609 2    86425 3                 4
     b - ----babc- ---ba +---aba - ------a ba- -----a b; Lead Term = (b , 1)
          729      104    104      682344      75816
      2 2                      2
     c +-ba+2ab; Lead Term = (c , 1)
        9
            9 2
     cb+9bc+-a ; Lead Term = (cb, 1)
            2
        1 2 1
     ca+-b +-ac; Lead Term = (ca, 1)
        2   9
      2  832     1  2  208 2                 2
     b a+---bab- -ab - ---a c; Lead Term = (b a, 1)
          81     9      81
      2  103  2   4      103 2                 2
     b c+---ba - --aba- ----a b; Lead Term = (b c, 1)
         208     81     1872
        2  926721   2    927   3  1458     80887239 2   96408 4                  2
     bab +--------ba c- -----ab +-----abac+--------a bc+-----a ; Lead Term = (bab , 1)
          35952800      86425    86425     35952800     86425

o4 : NCGroebnerBasis

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 = | y^3-y*x*z+x*y*z-x^3 |

o5 : NCMatrix

i6 : centralElements(C,3)

o6 = | -1/2*b^3+81/103*b*a*c+a*b*c-729/206*a^3 |

o6 : NCMatrix

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 : hilbertBergman B
--Calling bergman for HS computation.
Complete!

                2      3      4      5      6      7      8      9      10
o7 = 1 + 3T + 6T  + 10T  + 15T  + 21T  + 28T  + 36T  + 45T  + 55T  + 66T

o7 : ZZ[T]

i8 : hilbertBergman(C,DegreeLimit=>5)
--Calling bergman for HS computation.
Complete!

                2      3      4      5
o8 = 1 + 3T + 6T  + 10T  + 15T  + 21T

o8 : ZZ[T]

Ways to use threeDimSklyanin :

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

For the programmer

The object threeDimSklyanin is a method function with options.