# Noncommuting variables

 2 I am extremely new to Sage, and even newer to this site, so I apologize if anything is not up to standards. I am dealing with a multivariable polynomial ring over $\mathbb{Z}$ with noncommuting variables. Is there a way to implement this with Sage? The closest thing I have found is FreeAlgebra, where the variables are noncommutative, but I have not found any way to impose relations that I want. As stated before, I am extremely new to all of this so don't assume that I know anything, and don't hesitate to give any and all suggestions. asked Dec 06 '10 Eric A Bunch 53 ● 2 ● 8 Kelvin Li 443 ● 10 ● 17

 2 Perhaps FreeAlgebraQuotient will be useful? Here's the first part of the docstring: sage: FreeAlgebraQuotient? Type: type Base Class: String Form: Namespace: Interactive File: /Applications/sage/local/lib/python2.6/site-packages/sage/algebras/free_algebra_quotient.py Definition: FreeAlgebraQuotient(self, x) Docstring: Returns a quotient algebra defined via the action of a free algebra A on a (finitely generated) free module. The input for the quotient algebra is a list of monomials (in the underlying monoid for A) which form a free basis for the module of A, and a list of matrices, which give the action of the free generators of A on this monomial basis. EXAMPLES: Quaternion algebra defined in terms of three generators: sage: n = 3 sage: A = FreeAlgebra(QQ,n,'i') sage: F = A.monoid() sage: i, j, k = F.gens() sage: mons = [ F(1), i, j, k ] sage: M = MatrixSpace(QQ,4) sage: mats = [M([0,1,0,0, -1,0,0,0, 0,0,0,-1, 0,0,1,0]), M([0,0,1,0, 0,0,0,1, -1,0,0,0, 0,-1,0,0]), M([0,0,0,1, 0,0,-1,0, 0,1,0,0, -1,0,0,0]) ] sage: H3. = FreeAlgebraQuotient(A,mons,mats) sage: x = 1 + i + j + k sage: x 1 + i + j + k sage: x**128 -170141183460469231731687303715884105728 + 170141183460469231731687303715884105728*i + 170141183460469231731687303715884105728*j + 170141183460469231731687303715884105728*k  posted Dec 06 '10 niles 3605 ● 7 ● 45 ● 101 http://nilesjohnson.net/ Thanks! This is just the thing Eric A Bunch (Dec 07 '10)

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