Question

Let $A$ be a noncommutative finitely generated algebra with a finitely generated set of relations. Moreover, assume that $A$ is finite dimensional as a vector space.

What I want to know is, can Mathematica (or any other package) be used to find the dimension of $A$ given only the generators of $A$ and the generators of the set of relations? Or, even better, can Mathematica (or any other package) be used to find a basis of $A$ given only the generators of $A$ and the generators of the set of relations?

Basic examples would be particularily helpful.

Answer 1

As it was mentioned in my comment, you can use GAP and the noncommutative Gröbner bases package gbnp, written by Arjeh M. Cohen and Jan Willem Knopper.

Here you have an example:

Assume that you want to compute the dimension and a basis for the algebra $A$ with generators $a,b,c$ and relations $a^2 =b^2=c^2=0$, $ab + ca + bc = 0$ and $ba + cb + ac = 0$.

(This algebra is related to Schubert calculus and it was first discovered by Fomin and Kirillov, see MR1667680 (2001a:05152).)

gap> LoadPackage("gbnp");
-----------------------------------------------------------------------------
Loading  GBNP 0.9.5 (Non-commutative Gröbner bases)
by A.M. Cohen (http://www.win.tue.nl/~amc) and
   D.A.H. Gijsbers (D.A.H.Gijsbers@tue.nl).
-----------------------------------------------------------------------------
true
gap> A := FreeAssociativeAlgebraWithOne(Rationals, "a", "b", "c");;
gap> a := A.a;;
gap> b := A.b;;
gap> c := A.c;;
gap> rels := [a^2, b^2, c^2, a*b+c*a+b*c, b*a+c*b+a*c];;
gap> K := GP2NPList(rels);;                             
gap> G := SGrobner(K);;
gap> Display(DimQA(G,3));
12
gap> PrintNPList(BaseQA(G, 3, 0));
 1 
 a 
 b 
 c 
 ab 
 ac 
 ba 
 bc 
 aba 
 abc 
 bac 
 abac 

Here is the complete reference related to this algebra:

Fomin, Sergey; Kirillov, Anatol N. Quadratic algebras, Dunkl elements, and Schubert calculus. Advances in geometry, 147--182, Progr. Math., 172, Birkhäuser Boston, Boston, MA, 1999. MR1667680 (2001a:05152).)