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

Synopsis

• Usage:
fourDimSklyanin(R,params,varList)
• Inputs:
• 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 .