#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 darij):
Replying to [comment:7 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?
Well, I wouldn't expect to have any map from `A` to `gr(A)` when `A` is a
filtered module/algebra, and the closest thing that comes to such a map
would be a sequence of maps `p_0, p_1, p_2, ...` where `p_n` sends
degree-\leq n elements of `A` to the `n`-th graded component of `gr(A)`.
However, when `A` is a filtered module/algebra *with basis*, then your
`_element_constructor` can be viewed as a canonical map from `A` to
`gr(A)` indeed (it depends on the basis, but this is no problem because
the basis is part of `A`'s data). You may find this a squeamish
distinction, but the way most algebraists think about filtrations is not
the way you do. For most algebraists, a filtered algebra has an associated
graded algebra even if it does not have a basis or has several natural
bases; and when things depend on a choice of basis, one regards these
things as properties of the basis rather than properties of the algebra.
This is why I want you to document this all so carefully.
> 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`.
Oh! I think we misunderstood each other here.
> In my (naive) world, filtrations are not really different than grading
for modules.
Once again, this is good (I think this is the best we can do explicitly in
a CAS, whereas the algebraists' notion of a filtered algebra would be some
indiscrete lazy object) -- but this absolutely needs to be doced. This is
plainly not the way algebraists think.
> 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.
Followup, definitely. The ticket has been closed already and I don't think
this one will be done too quickly.
--
Ticket URL: <http://trac.sagemath.org/ticket/17096#comment:8>
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