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 (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 the appropriate length, then an error is thrown. The generic such algebra has a fairly complicated 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.

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(QQ,{a,b,c,d})
--Calling Bergman for NCGB calculation.
Complete!

o1 = C

o1 : NCQuotientRing

i2 : ncGroebnerBasis ideal C

         2  1818276801    2 2143385  2   14253260827  3   150884721  4  10166949  4  2560916417  2 2  444796  4  372650850976079      166872692193833   2   172755830058223 2     26773655493 3 2 138932082197701 3 2  3633135120329 3 2                   2
o2 = babc - ----------baba +-------ba ba+-----------ba b- ---------ad - --------ac - ----------ab c - ------ab - ---------------ababa+---------------aba b- ---------------a bab- -----------a d +---------------a c - -------------a b ; Lead Term = (babc , 1)
             116742400       177408       525340800        51074800      1276870      849756985       638435      58417371619200       9789200467200         12981638137600        3283403200      11358933370400       93875482400
         1    4
     db- -bd- -ac; Lead Term = (db, 1)
         3    3
        2    1
     ca+-bd- -ac; Lead Term = (ca, 1)
        3    3
      2   154     468  2 792  2  1089  2                2
     b a- ---bab- ---ad +---ac - ----ab ; Lead Term = (b a, 1)
          919     919    919      919
        3 324950129    2  844561  2    324950129  3  124834203  4 16823214  4 8946325044  2 2 728827  4 34474710943933       16892056950761   2  15141244924951 2    24103521 3 2  125076396897 3 2 325473383 3 2                  3
     bab +---------baba - ------ba ba- ---------ba b+---------ad +--------ac +----------ab c +------ab +--------------ababa- --------------aba b+--------------a bab+--------a d - ------------a c +---------a b ; Lead Term = (bab , 1)
          160520800       271040       160520800     140455700     7022785    9347326835      638435    17849752439200        2991144587200       8924876219600       9825200       33990279400      25537400
        3   3890140   2   12943        43  2    911  3   1204  3    35878612        9267643   2    1314768  2 3    9717484  2 2  4958833111 2      71950438 3      7012096   4                   3
     bac - --------bab c- -----babad- ---ba bd- ---ba c- ----ab c- ----------ababd+--------aba c- ---------a d - ----------a b d+----------a bac- ---------a bc- -----------a d; Lead Term = (bac , 1)
           10280853       49720       120       720      2757      4294592685      37384920       477176965      1431530895      4580898864       139352565      15746839845
        11    7
     dc+--ba+--ab; Lead Term = (dc, 1)
        18   18
         40    7
     da- --bc+--ad; Lead Term = (da, 1)
         33   33
         7   26
     cb+--bc+--ad; Lead Term = (cb, 1)
        33   33
          7    11
     cd- --ba- --ab; Lead Term = (cd, 1)
         18    18
       2  20  2  11  2   7    919 2                  2
     bd - --bc - --ba - --aba+---a b; Lead Term = (bd , 1)
          11      4     18    396
        2  2260   2  799760   2 10109  3  4480    2  745360  3   713377   2  80758031 2    822640 3                   2
     bad - ----bac +-------bab +-----ba +------abc +-------ab - -------aba - --------a ba+-------a b; Lead Term = (bad , 1)
           1419     1066959      4644    118551     1066959     2133918      46946196     1066959
       3  8 3   116489     1153  2    11739  3   347053  2   370237      108715153 2     31304  3                  3
     bc +--b c- ------babd+----ba c- ------ad - -------ab d+-------abac- ---------a bc- -------a d; Lead Term = (bc , 1)
         99     330840      720      101090     1213080     3639240       80063280      1667985
           726822279  2 2  730237383  2 2  18634  2 2  776151559  4     2 324950129   3  1008 2  2 18634 2 3 74611264729 2  2  230516 3   20566 4
     babab+---------ba d - ---------ba c - -----ba b - ---------ba -abab +---------aba +-----a bc +-----a b +-----------a ba - ------a ba+-----a b; Lead Term = (babab, 1)
           561822800       309002540       91205       204299200          160520800     91205      91205     24720203200        91205     91205

o2 : NCGroebnerBasis

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.

i3 : centralElements(C,2)

o3 = | c^2-11/9*b^2+13/9*a^2 d^2+20/9*b^2-22/9*a^2 |

o3 : NCMatrix

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:

i4 : hilbertBergman(C, DegreeLimit => 6)
--Calling bergman for HS computation.
Complete!

                 2      3      4      5      6
o4 = 1 + 4T + 10T  + 20T  + 35T  + 56T  + 84T

o4 : ZZ[T]

Ways to use fourDimSklyanin :

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

For the programmer

The object fourDimSklyanin is a method function with options.