On Tue, Sep 20, 2011 at 12:40:30PM +0200, Nicolas M. Thiery wrote:
> Here is a suggestion of implementation plan:
>
> - Low level kernel: implement efficient "mutable" subspaces. That is
> we want the usual subspaces:
> [..]
> - Implement generically (in ModulesWithBasis) a method:
> [..]
> - Use it to implement (in AlgebrasWithBasis) a method:
> [..]
> - Make sure that all sage vector spaces and algebras are in the
> appropriate categories!
Hi there.
this looks familiar (in a way).
some time ago i've started implementing path (or 'quiver') algebras in a
similar way. a path algebra is the algebra generated by the paths in a
directed graph -- think of it as a generalization of a free algebra
(where the graph has only one vertex).
i haven't come beyond elements, multiplication and (graded) quotients,
which are currently brute force and quite slow.
nevertheless these basic things work:
(from memory, see doctests for more examples/details):
sage: G = DiGraph({'e':['s'], 's':['e']}, name="some graph");
sage: A = PathAlgebra(GF(3),G);
sage: X=A.gens();X
(e,s,es,se)
sage: e,s,es,se=X
sage: es*e
0
sage: es*s
es
sage: es*se
es_se
sage: A(1)
e + s
sage: I = A.ideal([es*se])
sage: Q = A.quotient(I)
sage: Y=Q.gens();Y
[[e], [s], [es], [se]]
sage: e,s,es,se=Y
sage: Q.gens(2)
[[se_es]]
sage: Q.graded_comp(2).dimension()
1
sage: es+se
[es+se]
sage: e*es
[es]
sage: a=es*se;a
[es_se]
sage: a==0
True
sage: (es*se).simplify()
0
you can find the code here:
http://tool.em.cs.uni-frankfurt.de/~felix/misc/path_algebra-0.0.0.tbz2
regards
felix
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