On May 19, 3:55 pm, "Nicolas M. Thiery" <nicolas.thi...@u-psud.fr>
wrote:
> Hi John!
>
> On Wed, May 19, 2010 at 03:40:44PM -0700, John H Palmieri wrote:
> > On May 19, 3:02 pm, "Nicolas M. Thiery" <nicolas.thi...@u-psud.fr>
> > wrote:
> > > On Wed, May 19, 2010 at 02:25:22PM -0700, John Palmieri wrote:
> > > > - Should I do anything with the new categories framework? I've
> > > > defined a "category" method for the algebra, but should I do anything
> > > > else?
>
> > > > - I'm working on a differential graded algebra, in fact. It is a
> > > > "graded algebra with basis" (and it's actually Z x Z graded, so a
> > > > "bigraded algebra with basis"); do I need to do anything special about
> > > > this? I've defined a "basis" method already which returns the basis
> > > > in each bidegree as a Python list.
>
> > > As a short first answer, please try:
>
> > > sage: A = AlgebrasWithBasis(QQ).example()
>
> > Yes, I saw that. It looks like I should try to define my algebra
> > (call it L) as a CombinatorialFreeModule and implement at least
> > one_basis, product_on_basis, and algebra_generators. Unfortunately,
> > there is no corresponding example for GradedAlgebrasWithBasis, and the
> > framework for graded objects doesn't seem very complete. So should I
> > specify the basis for the CombinatorialFreeModule using some family,
> > and then attach some sort of grading to the basis elements to induce a
> > grading on L? That seems awkward -- it is important, at least for
> > this example, to be able to easily extract the basis in any given
> > (bi)degree, so it seems better to define a function specifying the
> > basis in each degree. What do you think?
>
> > If that's right, then I don't see how to conveniently construct L as a
> > CombinatorialFreeModule; rather, it is an infinite direct sum of
> > (finite-dimensional) such objects, and I need to be able to consider
> > each piece as well as the whole.
>
> Yes, real support for graded enumerated sets / algebras is still to
> come. Now I really need to jump to bed; in the mean time, you may want
> to have a look at DisjointUnionEnumeratedSets!
I'm sorry, but I can't see how to use this. I'm stuck with the whole
thing, in fact. (I'm not stuck with a Sage implementation, but I am
stuck with an implementation using the new categories framework.)
Let me provide some details: my algebra L is a bigraded algebra over
GF(p) (there is one algebra L for each prime p). I don't know a
simple way to list the basis elements; instead, if I want a basis in a
particular bidegree (s,t), I build it recursively using the algebra
generators and the bases in bidegrees (s', t') for various s' < s and
t' < t. Thus, for example, I don't know offhand the dimension in each
bidegree. As a result, I don't see how to implement this with
families or DisjointUnionEnumeratedSets.
(In case it helps: at the prime 2, the basis in bidegree (s,t) is in
bijection with length s partitions (i_1, i_2, ..., i_s) of t-s that
don't increase too fast: for each j, 2i_j >= i_{j+1}. It's similar
but a bit more complicated at odd primes.)
Anyway, if you have more suggestions, I would appreciate it.
--
John
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