ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Fri, 09 Nov 2012 04:55:30 +0100Skew commuting variableshttps://ask.sagemath.org/question/9512/skew-commuting-variables/I want to work in the ring `QQ<x0, x1, x2> / (xi*xj = -xj*xi for i \neq j)`. (In particular, xi^2 \neq 0; this is not the exterior algebra.)
I seems like FreeAlgebraQuotient is the thing to use, but I am not sure how. In the documentation for FreeAlgebraQuotient, the algebras are 4-dimensional as modules over QQ. However in my application, the algebra is infinite-dimensional as a module, so I can't write down the matrices for the action of the generators.
Is there another way to obtain this ring?
Thanks.Wed, 07 Nov 2012 18:22:17 +0100https://ask.sagemath.org/question/9512/skew-commuting-variables/Answer by burcin for <p>I want to work in the ring <code>QQ<x0, x1, x2> / (xi*xj = -xj*xi for i \neq j)</code>. (In particular, xi^2 \neq 0; this is not the exterior algebra.)</p>
<p>I seems like FreeAlgebraQuotient is the thing to use, but I am not sure how. In the documentation for FreeAlgebraQuotient, the algebras are 4-dimensional as modules over QQ. However in my application, the algebra is infinite-dimensional as a module, so I can't write down the matrices for the action of the generators.</p>
<p>Is there another way to obtain this ring?
Thanks.</p>
https://ask.sagemath.org/question/9512/skew-commuting-variables/?answer=14239#post-id-14239You can use our wrapper for [Singular](http://www.singular.uni-kl.de/)'s non-commutative component [Plural](http://www.singular.uni-kl.de/Manual/latest/sing_445.htm#SEC485). In particular, you can create a [G-algebra](http://www.singular.uni-kl.de/Manual/latest/sing_510.htm#SEC550) for this ring as follows:
sage: A.<x0,x1,x2> = FreeAlgebra(QQ, 3)
sage: A
Free Algebra on 3 generators (x0, x1, x2) over Rational Field
sage: R.<x0,x1,x2> = A.g_algebra({x1*x0: -x0*x1, x2*x0: -x0*x2, x2*x1: -x1*x2})
sage: R
Noncommutative Multivariate Polynomial Ring in x0, x1, x2 over Rational Field, nc-relations: {x2*x1: -x1*x2, x2*x0: -x0*x2, x1*x0: -x0*x1}
sage: x2*x1
-x1*x2
sage: x1*x2
x1*x2
Thu, 08 Nov 2012 07:00:46 +0100https://ask.sagemath.org/question/9512/skew-commuting-variables/?answer=14239#post-id-14239Comment by OliverP for <p>You can use our wrapper for <a href="http://www.singular.uni-kl.de/">Singular</a>'s non-commutative component <a href="http://www.singular.uni-kl.de/Manual/latest/sing_445.htm#SEC485">Plural</a>. In particular, you can create a <a href="http://www.singular.uni-kl.de/Manual/latest/sing_510.htm#SEC550">G-algebra</a> for this ring as follows:</p>
<pre><code>sage: A.<x0,x1,x2> = FreeAlgebra(QQ, 3)
sage: A
Free Algebra on 3 generators (x0, x1, x2) over Rational Field
sage: R.<x0,x1,x2> = A.g_algebra({x1*x0: -x0*x1, x2*x0: -x0*x2, x2*x1: -x1*x2})
sage: R
Noncommutative Multivariate Polynomial Ring in x0, x1, x2 over Rational Field, nc-relations: {x2*x1: -x1*x2, x2*x0: -x0*x2, x1*x0: -x0*x1}
sage: x2*x1
-x1*x2
sage: x1*x2
x1*x2
</code></pre>
https://ask.sagemath.org/question/9512/skew-commuting-variables/?comment=18741#post-id-18741Thanks! This is exactly what I needed!Thu, 08 Nov 2012 12:39:33 +0100https://ask.sagemath.org/question/9512/skew-commuting-variables/?comment=18741#post-id-18741Comment by burcin for <p>You can use our wrapper for <a href="http://www.singular.uni-kl.de/">Singular</a>'s non-commutative component <a href="http://www.singular.uni-kl.de/Manual/latest/sing_445.htm#SEC485">Plural</a>. In particular, you can create a <a href="http://www.singular.uni-kl.de/Manual/latest/sing_510.htm#SEC550">G-algebra</a> for this ring as follows:</p>
<pre><code>sage: A.<x0,x1,x2> = FreeAlgebra(QQ, 3)
sage: A
Free Algebra on 3 generators (x0, x1, x2) over Rational Field
sage: R.<x0,x1,x2> = A.g_algebra({x1*x0: -x0*x1, x2*x0: -x0*x2, x2*x1: -x1*x2})
sage: R
Noncommutative Multivariate Polynomial Ring in x0, x1, x2 over Rational Field, nc-relations: {x2*x1: -x1*x2, x2*x0: -x0*x2, x1*x0: -x0*x1}
sage: x2*x1
-x1*x2
sage: x1*x2
x1*x2
</code></pre>
https://ask.sagemath.org/question/9512/skew-commuting-variables/?comment=18734#post-id-18734Let me know if there are any problems with the wrapper. It's not very polished at the moment, some things might not work as intended. I already have a few patches I need to push upstream, but I'd appreciate hearing bug reports and use cases.Fri, 09 Nov 2012 04:55:30 +0100https://ask.sagemath.org/question/9512/skew-commuting-variables/?comment=18734#post-id-18734