Hi Nurdin,
On 2014-06-26, Nurdin Takenov <
grea...@gmail.com> wrote:
> I've tried
> to do it like that:
>
> sage: F.<x,y,a> = FreeAlgebra(QQ,3)
> R.<x,y,a> = F.g_algebra({y*x: a*x*y, a*x: x*a, y*a: a*y})
>
> But it doesn't work. Is it because y*x=a*x*y is non-homogeneous? If so,
> what should I do?
The definition of a g-algebra contains some technical non-degeneracy
conditions, and it seems that they are violated when you define the
algebra in that way. However, since "a" is supposed to be a parameter,
it would make sense to consider it as an element of a function field,
and then define an algebra over the function field. Like this:
sage: P.<a> = QQ[]
sage: K = P.fraction_field()
sage: F.<x,y> = FreeAlgebra(K)
sage: R.<x,y> = F.g_algebra({y*x: a*x*y})
(Relations x*a=a*x are automatically satisfied, since the elements of K
commute with elements of F)
However, unfortunately this doesn't work either (gives an error about
coercion), even though libsingular (which is used in the background)
should be able to work with function fields as base ring. This comes as
a surprise to me, and perhaps it is a bug. To be investigated.
Best regards,
Simon