Is what you want something like this?:
sage: P = FreeAlgebra(QQ,7,'x1,x2,x3,x4,y1,y2,y3')
sage: (x1, x2, x3, x4, y1, y2, y3) = P.gens()
sage: I =
P.ideal([y1*y2+y2*y1,y2*y3+y3*y2,y1*y3+y3*y1,x1*x2-x2*x1,x1*x3-x3*x1,x1*x4-x4*x1,x2*x3-x3*x2,x2*x4-x4*x2,x3*x4-x4*x3])
sage: P.quotient_ring(I)
Quotient of Free Algebra on 7 generators (x1, x2, x3, x4, y1, y2, y3) over
Rational Field by the ideal (y1*y2 + y2*y1, y2*y3 + y3*y2, y1*y3 + y3*y1,
x1*x2 - x2*x1, x1*x3 - x3*x1, x1*x4 - x4*x1, x2*x3 - x3*x2, x2*x4 - x4*x2,
x3*x4 - x4*x3)
sage: Q = P.quotient_ring(I)
sage: Q.gens()
(x1bar, x2bar, x3bar, x4bar, y1bar, y2bar, y3bar)
I think that the problem is that no non-commutative Grobner bases or
'normal form' are implemented here:
sage: (x1bar, x2bar, x3bar, x4bar, y1bar, y2bar, y3bar) = Q.gens()
sage: y2bar*y1bar+y1bar*y2bar # should be 0
y1bar*y2bar + y2bar*y1bar
On Friday, 10 February 2017 10:45:25 UTC-5, Biswajit Ransingh wrote:
>
> Dear friends,
>
>
> P.<x0, x1, x2, x3, x4, y1, y2,y3> = LaurentPolynomialRing(QQ)
>
> Can we make rich algebraic structure using the above P?
>
> The variables *x0, x1, x2, x3, x4 are commutative *
> and *y1, y2,y3 are non commutative.*
>
> if we ask for relation y1*y2 = -y2*y1 it gives
> SyntaxError: can't assign to operator.
>
> Can we get something like: sage: P multivariate Laurent polynomial Ring in
> x0, x1, x2, x3, x4, y1, y2,y3 over Rational Field, nc→relations: {y1*y2:
> -y2*y1,y2*y3:-y3*y2,y1*
> y3:-y3*y1,x1*x2:-x2*x1}
>
> For example: Can we extend the below code to above information
>
> sage: A. = FreeAlgebra(QQ, 3)
> sage: P. = A.g_algebra(relations={y*x:-x*y}, order = 'lex')
> sage: P Noncommutative Multivariate Polynomial Ring in x, y, z over
> Rational Field, nc- ˓→relations: {y*x: -x*y}
>
>
>
>
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