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

## Synopsis

• Usage:
threeDimSklyanin(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 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 .