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Quotient of non-commutative ring

asked 2019-02-16 14:54:16 -0500

franslag gravatar image

I'm trying to make a non-commutative ring, where X^2 = -1 ; XY = -YX and Y^2 = 0, something like this ;

                               R[X,Y] / (X^2 + 1 , XY + YX , Y^2)

To get the non-commutative part, I did :

A.<x,y> = FreeAlgebra(QQ,2)

P.<x,y> = A.g_algebra(relations={yx:-xy})

What is the next command to have the quotient by (X^2+1 , Y^2) ? Or maybe an other way to build the entire ring, if there exists a simpler method :) thanks !

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answered 2019-02-18 03:03:52 -0500

rburing gravatar image

You want to quotient by the two-sided ideal generated by x^2 + 1 and y^2:

R.<x,y> = P.quo(P*[x^2 + 1, y^2]*P)

Note this re-defines x and y once again, as you also did in your code.

Then you get:

sage: x^2
-1
sage: y^2
0
sage: y*x
-x*y
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This was helpful, thanks !!

franslag gravatar imagefranslag ( 2019-02-27 03:50:26 -0500 )edit

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Asked: 2019-02-16 14:54:16 -0500

Seen: 49 times

Last updated: Feb 18