Implement categories for super algebras/modules #18044

tscrim · 2015-03-24T01:34:51Z

After some design discussion, a proposal for an implementation is to be a subcategory of Graded algebras/modules whose default even/odd considers the given grading as being ZZ / 2 ZZ compatible. Thus it will have the following methods:

is_even()
is_odd()
is_even_odd()

The API will be the user defines a degree method which is then called by is_even_odd() and taken mod 2. Therefore if the user wants to have a special grading group, then the user can override is_even_odd() to translate that into 0 or 1. The hook of is_even_odd also allows the user to change the super behavior away from the grading group (which will get used in #17096).

Depends on #18174

CC: @nthiery @darijgr

Component: categories

Keywords: super, sd67

Author: Travis Scrimshaw

Branch/Commit: 16a3791

Reviewer: Darij Grinberg

Issue created by migration from https://trac.sagemath.org/ticket/18044

The text was updated successfully, but these errors were encountered:

tscrim · 2015-03-24T02:55:30Z

Commit: cd6d21a

tscrim · 2015-03-24T02:55:30Z

New commits:

`cd6d21a`	`Created super categories for algebras and modules.`

tscrim · 2015-03-24T02:55:30Z

Branch: public/categories/super_categories-18044

darijgr · 2015-03-28T16:57:51Z

comment:2

+ def Super(self, base_ring=None):
+ r"""
+ Return the subcategory of the super objects of ``self``.

Is this really a subcategory? Super Lie algebras are not Lie algebras and cannot be reasonably made into such short of taking the even part...

tscrim · 2015-03-28T17:19:23Z

comment:3

Good point (this is different than Graded, which is where I copied things from). Think we should just replace the word "sub" with "associated"? If so, I can go make this change (in about 2 hours).

darijgr · 2015-03-28T17:25:14Z

comment:4

I'd just say "Return the super-analogue of self" (the word "analogue" is a bit weasel, but I think it's better to err on this side). In general, there should be a canonical embedding from C into C.Super(), not the other way round.

What do you mean by "Same as :meth:Graded"?

tscrim · 2015-03-28T23:42:02Z

comment:5

Replying to @darijgr:

I'd just say "Return the super-analogue of self" (the word "analogue" is a bit weasel, but I think it's better to err on this side). In general, there should be a canonical embedding from C into C.Super(), not the other way round.

It's more of there is not a forgetful functor from C.super() to C, but I'll make this change.

What do you mean by "Same as :meth:Graded"?

Meaning I copied it. <.< >.>

sagetrac-git · 2015-03-28T23:47:18Z

Branch pushed to git repo; I updated commit sha1. New commits:

`3d33730`	`Merge branch 'develop' into public/categories/super_categories-18044`
`0ce1cad`	`Changing the wording of the Super method.`

sagetrac-git · 2015-03-28T23:47:18Z

Changed commit from cd6d21a to 0ce1cad

tscrim · 2015-03-28T23:50:53Z

comment:7

I've made the change. Although actually, forgetting the "super" doesn't change the multiplication, just removes the notion of supercommuting elements and the degree. So there is a forgetful functor at least, right (or am I overthinking things)?

darijgr · 2015-03-29T14:41:29Z

comment:8

Forgetting the grading on a superalgebra gives you an algebra, yes. But forgetting the grading on a Lie superalgebra does not give you a Lie algebra; it gives something weird and unnamed. So forgetful functors should only exist in specific categories.

tscrim · 2015-03-29T17:57:22Z

comment:9

True, but I think for now we can live with the slight abuse since there are not plans currently (at least by me) to implement Lie superalgebras. Then again, IDK how much of the code I've written wouldn't work in the category for Lie superalgebras. ehh...I think we can cross that bridge if/when that's an issue.

darijgr · 2015-03-29T22:49:47Z

comment:10

Sorry but the assumption that "super X are graded X" (or just "super X are X") is fundamentally wrong. The longer we keep it in, the harder it will be to cross that bridge. It already causes wrong results:

sage: C = HopfAlgebras(QQ).Super()
sage: C.super_categories()
[Category of hopf algebras over Rational Field,
 Category of super algebras over Rational Field]

Super Hopf algebras are not Hopf algebras. In a Hopf algebra,
(xy)_{(1)} \otimes (xy)_{(2)} = x_{(1)} y_{(1)} \otimes x_{(2)} y_{(2)}
(using sumfree Sweedler notation), while in a super Hopf algebra,
(xy)_{(1)} \otimes (xy)_{(2)} = (-1)^{\deg y_{(1)} \deg x_{(2)}} x_{(1)} y_{(1)} \otimes x_{(2)} y_{(2)}.

In other news: In the lift_module_morphism method for Clifford algebras, did you really mean Graded in

        return Cl.module_morphism(on_basis=f, codomain=self,
                                  category=AlgebrasWithBasis(self.base_ring()).Graded())

(and not Super)?

tscrim · 2015-03-29T23:42:07Z

comment:11

Then I guess we need to change the default super categories thingy and just be explicit about things.
So then the category hierarchy should be

SuperHopf    GradedHopf
    |           |
SuperAlg      Hopf
       \     /
         Alg

right? So we could do an explicit override of the SuperHopf's super category (at least, this is probably one of the very few instances where a super object is different notation from the graded).

darijgr · 2015-04-02T00:32:43Z

comment:12

Replying to @tscrim:

So we could do an explicit override of the SuperHopf's super category (at least, this is probably one of the very few instances where a super object is different notation from the graded).

Sorry, I just don't like this whole paradigm where a category liberally makes assumptions that the subcategory has to override if they are not satisfied. The category framework was created to mirror the mathematical structure of algebraic categories (or so I have been told). Mathematically, there is no reason why a superX should canonically be an X; the fact that this holds for many of the standard Xes (algebras, modules, etc.) is not inherent to the notion of "super"ness but to these Xes, so it shouldn't be reflected in any automatic inheritance. Does the way you implemented Super require SuperX to derive from X?

tscrim · 2015-04-02T04:39:13Z

comment:13

Currently yes, but I'm going to change it so that you have to explicitly set X to be supercategory of superX.

darijgr · 2015-04-02T04:41:42Z

comment:14

Thank you!

sagetrac-git · 2015-04-02T06:24:35Z

Branch pushed to git repo; I updated commit sha1. New commits:

`b29650b`	`Made it so you must explicitly designate graded as a super category`

sagetrac-git · 2015-04-02T06:24:35Z

Changed commit from 0ce1cad to b29650b

tscrim · 2015-04-02T06:48:43Z

comment:16

I'm not happy with this because of how certain things in the category hierarchy are respected, even in this new version. In particular:

sage: C = HopfAlgebras(ZZ).WithBasis().Super(); C
Category of super hopf algebras with basis over Integer Ring
sage: sorted(C.super_categories(), key=str)
[Category of hopf algebras with basis over Integer Ring,
 Category of super algebras over Integer Ring,
 Category of super algebras with basis over Integer Ring]

I don't know a way around this currently...

However I'm starting to wonder if this is what we want. If we want two notions of super, the first is via a functorial construction for things like modules and algebras (and rings once the graded versions get implemented), and for other things if we want the Super call to go to a Category_over_base_ring subclass. Mainly the first examples satisfy the conditions of the regressive covariant functorial construction, but the others, like Hopf and Lie algebras, don't. Doing things this way also seem like they will simplify the implementation as a fringe benefit. Thoughts (inc. from Nicolas)?

nthiery · 2015-04-07T12:31:21Z

comment:17

Replying to @tscrim:

(inc. from Nicolas)?

I don't have the big picture at this point. Let's discuss this in Montreal!

tscrim · 2015-04-13T14:27:03Z

Dependencies: #18174

tscrim · 2015-04-13T14:27:03Z

comment:18

#18174 will simplify this a bit.

sagetrac-git · 2015-04-14T21:19:32Z

Branch pushed to git repo; I updated commit sha1. New commits:

`7554dbd`	`1 step forward, 2 steps back.`

sagetrac-git · 2015-04-14T22:00:14Z

Changed commit from 7554dbd to caada13

tscrim · 2015-04-14T22:26:08Z

Changed keywords from super to super, sd67

sagetrac-git · 2015-04-16T05:30:15Z

Changed commit from caada13 to 26859cb

sagetrac-git · 2015-04-16T05:30:15Z

Branch pushed to git repo; I updated commit sha1. New commits:

`26859cb`	`manual merge with sage 6.7.beta1`

sagetrac-git · 2015-04-16T05:39:11Z

Branch pushed to git repo; I updated commit sha1. New commits:

`6051865`	`add a doctest which hopefully works (hard to check while compiling)`

sagetrac-git · 2015-04-16T05:39:11Z

Changed commit from 26859cb to 6051865

sagetrac-git · 2015-08-09T12:20:01Z

Changed commit from 6051865 to b91cd82

sagetrac-git · 2015-08-09T12:20:01Z

Branch pushed to git repo; I updated commit sha1. New commits:

`570bc49`	`Merge branch 'public/categories/super_categories-18044' into 6.9.b1`
`b91cd82`	`trac #18044 correct one typo in the doc`

sagetrac-git · 2015-10-03T23:08:41Z

Changed commit from b91cd82 to 0579337

sagetrac-git · 2015-10-03T23:08:41Z

Branch pushed to git repo; I updated commit sha1. New commits:

`7fd1df2`	`Merge branch 'public/categories/super_categories-18044' of trac.sagemath.org:sage into public/categories/super_categories-18044`
`0579337`	`Some reviewer tweaks and doc additions.`

sagetrac-git · 2015-10-03T23:14:47Z

Changed commit from 0579337 to aec22cc

sagetrac-git · 2015-10-03T23:14:47Z

Branch pushed to git repo; I updated commit sha1. New commits:

`aec22cc`	`Added one more test.`

darijgr · 2015-10-03T23:15:59Z

comment:27

LGTM now; thanks for the commit!

For posterity: There are design issues that I don't fully like, but that are standard for Sage.

tscrim · 2015-10-03T23:16:27Z

Reviewer: Darij Grinberg

sagetrac-git · 2015-10-06T02:13:31Z

Branch pushed to git repo; I updated commit sha1 and set ticket back to needs_review. New commits:

`16a3791`	`Marked a doctest as indirect doctest.`

sagetrac-git · 2015-10-06T02:13:31Z

Changed commit from aec22cc to 16a3791

tscrim · 2015-10-06T02:14:23Z

comment:30

One test should have been marked as # indirect doctest.

vbraun · 2015-10-12T07:16:38Z

Changed branch from public/categories/super_categories-18044 to 16a3791

tscrim added this to the sage-6.6 milestone Mar 24, 2015

tscrim added c: categories labels Mar 24, 2015

tscrim self-assigned this Mar 24, 2015

tscrim added the s: needs review label Mar 24, 2015

tscrim added s: positive review and removed s: needs review labels Oct 3, 2015

tscrim modified the milestones: sage-6.6, sage-6.9 Oct 3, 2015

sagetrac-git mannequin added s: needs review and removed s: positive review labels Oct 6, 2015

tscrim added s: positive review and removed s: needs review labels Oct 6, 2015

vbraun removed the s: positive review label Oct 12, 2015

vbraun closed this as completed in a677b85 Oct 12, 2015

This was referenced Dec 29, 2022

Implement categories for differential algebras #16513

Open

Implement categories for filtered algebras #17096

Closed

Implement categories for super algebras/modules #18044

Implement categories for super algebras/modules #18044

Comments

tscrim commented Mar 24, 2015

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