#17096: Implement categories for filtered algebras
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Reporter: tscrim | Owner: tscrim
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.4
Component: categories | Resolution:
Keywords: filtered algebras | Merged in:
Authors: Travis Scrimshaw | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
public/categories/filtered_algebras-17096|
bb234ec71d1276c6cc14320074b3de6fdc606192
Dependencies: | Stopgaps:
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Comment (by tscrim):
Replying to [comment:6 darij]:
> That `_element_constructor_` definitely shouldn't be a coercion, but
even not being a coercion it should be surrounded with big fat warning
signs for not being what one would expect.
You wouldn't expect it to be the canonical linear isomorphism (as
modules)? So given some `x + y` in the filtered algebra `A`, you wouldn't
expect it to return `x + y` in `gr A`? Instead you'd want it to return
only `x` (assuming it has larger degree)? This would be extremely
surprising to me (we have a method to remove the lower order terms too).
Or am I not understanding what you're saying?
> I don't understand your answer to 2). Even your own doc says that the
basis is graded:
> {{{
> + def basis(self, d=None):
> + """
> + Return the basis for (an homogeneous component of) ``self``.
> }}}
> This means precisely that the basis elements have nonnegative integers
ascribed to them, which stand for something like degree.
Well, any (additive) abelian group. Yet I'm not requiring that the `i`-th
element of the basis be the only element of degree `i`. Again, an element
is homogeneous of degree d if it belongs to F,,d,, but not F,,d-1,,.
However I am requiring that F,,d,, is a subspace.
> Generally, it seems to me that your filtered modules are precisely the
same as graded modules, and only the richer "sub"categories (filtered
algebras, filtered coalgebras etc.) differ from their graded counterparts.
If so, this is a perfectly fine design decision, but it would help to
document it explicitly.
In my (naive) world, filtrations are not really different than grading for
modules. That's not to say they aren't useful though because of things
like `I`-adic topology (TBH, this is wikipedia talking). However I don't
think we need doc on this. Nicolas, do you have any thoughts?
I could add something about the terminology for homogeneous in terms of
the filtration if that's non-standard or unclear.
Question, should we make Weyl and Clifford algebras filtered on this
ticket or on a followup since that's been closed? Same for group algebras
by the length function.
--
Ticket URL: <http://trac.sagemath.org/ticket/17096#comment:7>
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