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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.

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  • $\begingroup$ I use GAP (gap-system.org) and the non-commutative Gröbner basis package gbnp (mathdox.org/products/gbnp). You will find examples in the manual. $\endgroup$
    – Leandro Vendramin
    Commented Feb 2, 2012 at 16:16
  • $\begingroup$ I used to use Steve Linton's Vector Enumeration but this is now deprecated. $\endgroup$
    – Bruce Westbury
    Commented Feb 2, 2012 at 16:18
  • $\begingroup$ Not really related, but still: mathoverflow.net/questions/85400/… $\endgroup$
    – Alexander Chervov
    Commented Feb 2, 2012 at 17:48
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    $\begingroup$ If you know A to be a basic algebra then this can be done in Magma. magma.maths.usyd.edu.au/magma/handbook/text/898 Magma may also be able to do more general types of algebra, have a look at the Algebras section of the handbook. There is an online calculator if you don't have institutional access to Magma but it is limited to computations taking at most a minute. $\endgroup$
    – M T
    Commented Feb 2, 2012 at 18:46

1 Answer 1

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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 ([email protected]).
-----------------------------------------------------------------------------
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).)

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  • $\begingroup$ Great. That looks like just what I want! But I'm having installing the package. On the package page it says: %%%%%%%%%%%%%% unpack GBNP-1.0.1.tar.gz in the pkg subdirectory of your GAP installation (or in the pkg subdirectory of any other GAP root directory, for example one added with the -l argument) with the following command: tar -xvzf GBNP-1.0.1.tar.gz. %%%%%%%%%%%% I typed this into GAP and got an error. What/Where is the pkg subdirectory? $\endgroup$
    – Mihail Matrix
    Commented Feb 4, 2012 at 19:02
  • $\begingroup$ Where have you installed GAP? Usually the GAP package directory is gap4r4/pkg/. $\endgroup$
    – Leandro Vendramin
    Commented Feb 4, 2012 at 19:27
  • $\begingroup$ @Leandro, do you know if there is a way to do this sort of thing in GAP when the algebra is defined over a field of rational functions? $\endgroup$
    – MTS
    Commented Feb 25, 2014 at 0:24
  • $\begingroup$ @MTS, Maybe not, but I am not sure. What about Magma? $\endgroup$
    – Leandro Vendramin
    Commented Feb 25, 2014 at 6:53

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