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2021-01-15 18:26:19 +0100 received badge  Student (source)
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2016-01-13 02:26:06 +0100 asked a question Confused about FreeAlgebra quotients

I think I'm misunderstanding how quotients of free algebras work. I tried to make a free algebra on two generators x, y and mod out by xy = yx, so you get a polynomial algebra -- but that's not what happened.

sage: R.<x,y> = FreeAlgebra(QQ)
sage: I = R*[x*y-y*x]*R
sage: Q.<a,b> = R.quo(I)
sage: a*b is b*a
False
sage: Q.is_commutative()
False

Relatedly, the documentation for free_module_quotient gives an example (constructing the quaternions as a free quotient):

sage: n = 2
sage: A = FreeAlgebra(QQ,n,'x')
sage: F = A.monoid()
sage: i, j = F.gens()
sage: mons = [ F(1), i, j, i*j ]
sage: r = len(mons)
sage: M = MatrixSpace(QQ,r)
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]) ]
sage: H2.<i,j> = A.quotient(mons,mats)

but I'm confused exactly how the matrices in the penultimate line describe the (multiplication) action? I.e. if I have some relation, say x^2 = 0, that I want to mod out by, how do I accomplish that using matrices?