On Wed, 23 Mar 2011 09:17:35 -0700 Anne Schilling <a...@math.ucdavis.edu> wrote:
> > This is also accessible through the patch at #4539: > > > > http://trac.sagemath.org/sage_trac/ticket/4539 > > > > Here is how you can do the example above with that patch: > > > > sage: A.<x,y> = FreeAlgebra(QQ, 2) > > sage: H = A.g_algebra({y*x: x*y + y^2}) > > sage: H.inject_variables() > > sage: x*y > > x*y > > sage: y*x > > x*y + y^2 > > The example you sent indeed works fine: > > sage: A.<x,y> = FreeAlgebra(QQ, 2) > sage: sage: H = A.g_algebra({y*x: x*y + y**2}) > > However, what are the restrictions on the relations? The following > example does not work: > > sage: A.<x,y> = FreeAlgebra(QQ, 2) > sage: H = A.g_algebra({y*x*x: x*y*x}) <boom> Indeed, this is not a G-algebra as defined here: http://www.singular.uni-kl.de/Manual/latest/sing_436.htm#SEC476 Letterplace should be able to handle quotients of free algebras: http://www.singular.uni-kl.de/Manual/latest/sing_441.htm#SEC493 Simon King is working on a patch to make this available in Sage. I can't find his last message to sage-algebra in the google groups archive, so I included it below. Cheers, Burcin On Wed, 23 Mar 2011 06:33:34 -0700 (PDT) Simon King <simon.k...@uni-jena.de> wrote: > Hi all! > > On 22 Mrz., 13:18, Simon King <simon.k...@uni-jena.de> wrote: > > Features: > > - Arithmetic in free algebras that is a lot faster than the current > > implementation (but: restricted to the homogeneous case). > > - One- and twosided ideals of non-commutative rings. > > - Twosided ideals of free algebras, with degree-wise Gröbner basis > > computation. > > - Normal forms/ideal containment for two-sided ideals of free > > algebras. > > > > I do not plan to implement quotients of free algebras, yet. > > However, I think it would be straight forward, on top of the above > > features. > > It *is* fairly straight forward. I did implement a notion of one- and > twosided ideals in non-commutative rings, and then it is easy to > change sage.rings.quotient_ring.QuotientRing to not require that the > base ring is commutative but to require that the ideal is two-sided. > > The following will work with my to-be-submitted patch: > sage: F.<x,y,z> = FreeAlgebra(QQ, implementation='letterplace') > sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F > sage: F.quo(I) > Quotient of Free Associative Unital Algebra on 3 generators ('x', 'y', > 'z') over Rational Field by the ideal (x*y + y*z, x*x + x*y - y*x - > y*y) > sage: Q = F.quo(I) > sage: Q.0 > xbar > sage: Q.1 > ybar > sage: Q.2 > zbar > sage: Q.0*Q.1 > -ybar*zbar > sage: Q.0^2+Q.0*Q.1 > ybar*xbar + ybar*ybar > > It is restricted to homogeneous ideals, but I guess that's better than > nothing. > > Cheers, > Simon -- You received this message because you are subscribed to the Google Groups "sage-combinat-devel" group. To post to this group, send email to sage-combinat-devel@googlegroups.com. To unsubscribe from this group, send email to sage-combinat-devel+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/sage-combinat-devel?hl=en.
