Basis of a Finite Dimensional Algebra with a Finitely Generated Relation Set By Computer - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T00:01:59Z http://mathoverflow.net/feeds/question/87338 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87338/basis-of-a-finite-dimensional-algebra-with-a-finitely-generated-relation-set-by-c Basis of a Finite Dimensional Algebra with a Finitely Generated Relation Set By Computer Mihail Matrix 2012-02-02T16:05:58Z 2012-02-04T11:57:11Z <p>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. </p> <p>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?</p> <p>Basic examples would be particularily helpful.</p> http://mathoverflow.net/questions/87338/basis-of-a-finite-dimensional-algebra-with-a-finitely-generated-relation-set-by-c/87520#87520 Answer by Leandro Vendramin for Basis of a Finite Dimensional Algebra with a Finitely Generated Relation Set By Computer Leandro Vendramin 2012-02-04T11:57:11Z 2012-02-04T11:57:11Z <p>As it was mentioned in my comment, you can use <a href="http://www.gap-system.org/" rel="nofollow">GAP</a> and the noncommutative Gröbner bases package <a href="http://mathdox.org/products/gbnp/" rel="nofollow">gbnp</a>, written by Arjeh M. Cohen and Jan Willem Knopper. </p> <p>Here you have an <strong>example</strong>:</p> <p>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\$.</p> <p>(This algebra is related to Schubert calculus and it was first discovered by Fomin and Kirillov, see <a href="http://www.ams.org/mathscinet-getitem?mr=1667680" rel="nofollow">MR1667680 (2001a:05152)</a>.)</p> <pre><code>gap&gt; 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&gt; A := FreeAssociativeAlgebraWithOne(Rationals, "a", "b", "c");; gap&gt; a := A.a;; gap&gt; b := A.b;; gap&gt; c := A.c;; gap&gt; rels := [a^2, b^2, c^2, a*b+c*a+b*c, b*a+c*b+a*c];; gap&gt; K := GP2NPList(rels);; gap&gt; G := SGrobner(K);; gap&gt; Display(DimQA(G,3)); 12 gap&gt; PrintNPList(BaseQA(G, 3, 0)); 1 a b c ab ac ba bc aba abc bac abac </code></pre> <p>Here is the complete reference related to this algebra:</p> <p>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. <a href="http://www.ams.org/mathscinet-getitem?mr=1667680" rel="nofollow">MR1667680 (2001a:05152)</a>.)</p>