On Tue, 22 Mar 2011 11:22:28 -0700 (PDT) John H Palmieri <jhpalmier...@gmail.com> wrote:
> On Tuesday, March 22, 2011 1:34:44 AM UTC-7, Nicolas M. Thiéry wrote: > > > > Is it already possible or would it be easy to implement a quotient > > > of the free algebra by specifying relations between the > > > generators? > > > > Unless Singular can provide something (but I guess that would be > > more for skew-commutative algebras; Simon: can you confirm?), the > > proper way to do this would be to use gap's KBMAG package. It > > should be fairly straightforward, but not instantaneous either. > > > > I don't know anything about KBMAG; I should look into that. > Meanwhile, Singular does let you define certain quotients of free > algebras. See > <http://www.singular.uni-kl.de/Manual/3-1-0/sing_404.htm>. In Sage, > I can define a GF(2)-algebra S to be the free algebra on x and y > subject to the relation [x,y] = y^2: > > sage: singular.LIB('ncall.lib') > sage: R=singular.ring(2,'(x,y)') > sage: C = singular.matrix(2, 2, '(1,1,1,1)') > sage: D = singular.matrix(2, 2, '(0, y*y, 0, 0)') > sage: S = C.nc_algebra(D) > sage: S.set_ring() > sage: x = singular('x') > sage: y = singular('y') > sage: x*y > x*y > sage: y*x > x*y+y^2 > > There are limitations on the sorts of algebras which can be defined > this way -- I think they need to have a PBW basis, basically -- see > <http://www.singular.uni-kl.de/Manual/3-1-0/sing_420.htm#SEC461> -- > but Singular does give you some quotients of free algebras. 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 Cheers, Burcin -- 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.
