Opened 13 months ago

Last modified 5 months ago

#32249 new enhancement

Graded-commutative ideals

Reported by: tkarn Owned by:
Priority: major Milestone: sage-9.7
Component: algebra Keywords: gsoc2021 gradedcommutative ideal graded
Cc: tscrim, tkarn, jhpalmieri Merged in:
Authors: Trevor K. Karn Reviewers:
Report Upstream: N/A Work issues:
Branch: u/tkarn/sc-ideals-32249 (Commits, GitHub, GitLab) Commit: 1a79ed0f7861180ec7f1c0ae1f0239eb6b7a3e17
Dependencies: Stopgaps:

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Description (last modified by tkarn)

A graded-commutative ring is a mildly noncommutative ring in which x*y = (-1)**(x.deg()*y.deg())*y*x for the right notion of degree. This ticket aims to implement ideals of graded-commutative rings inside of sage.rings.noncommutative_ideals via the class Ideal_gc.

Change History (16)

comment:1 Changed 13 months ago by tkarn

  • Description modified (diff)

comment:2 Changed 13 months ago by tkarn

  • Branch set to u/tkarn/sc-ideals-32249
  • Commit set to 1a79ed0f7861180ec7f1c0ae1f0239eb6b7a3e17

New commits:

1a79ed0Initial commit of supercommutative ideal

comment:3 Changed 13 months ago by nbruin

Searching on google for the synonymous term "graded-commutative" returns "About 5,310,000 results" whereas for "supercommutative" it returns "About 13,400 results". This suggests to me that the former is the more entrenched term and therefore is probably the one to use in a general-purpose CAS such as sage.

comment:4 Changed 13 months ago by tkarn

It is also a more general idea, I had the definition of supercommutative slightly off. It is an instance of a graded-commutative algebra with only degree 0 and degree 1 things. I agree the name should change.

comment:5 Changed 13 months ago by tkarn

  • Description modified (diff)
  • Keywords gradedcommutative graded added; supercommutative removed
  • Summary changed from Supercommutative ideals to Graded-commutative ideals

comment:6 Changed 13 months ago by tkarn

Here is a reference: https://d-nb.info/1012919684/34

comment:7 Changed 13 months ago by gh-mjungmath

Please notice that graded superalgebras already exist, though under a different name (see here).

comment:8 follow-up: Changed 13 months ago by gh-mjungmath

This algebra also supports ideals.

comment:9 in reply to: ↑ 8 ; follow-up: Changed 13 months ago by tscrim

Replying to gh-mjungmath:

This algebra also supports ideals.

Not really very well. In particular, it does not compute Gröbner bases.

comment:10 in reply to: ↑ 9 Changed 13 months ago by gh-mjungmath

Replying to tscrim:

Not really very well. In particular, it does not compute Gröbner bases.

Indeed.

comment:11 Changed 13 months ago by mathzeta2

Maybe I confuse the terms, or this is entirely anecdotal, but I have encountered the name superalgebra few times (e.g. in slides, publication list and here). If this is indeed the case, and these are the same, can the documentation include the string "also known as a superalgebra" in the right place? This should help a user searching for "sage superalgebra" finding more than SuperAlgebras from the category framework.

For Gröbner bases, it might be worth looking at the unfinished #31446 where they are computed using the GAP package GBNP.

comment:12 Changed 13 months ago by tkarn

From my understanding, the concepts are very closely related, but not quite the same. Superalgebras are one generalization of supercommutative algebras and graded-commutative algebras are different generalizations of supercommutative algebras where we have finer information about what products look like.

It would be correct to say that a supercommutative algebra (also known as a "commutative superalgebra" although the algebra need not satisfy ab=ba so I would suggest we use supercommutative) is a graded commutative algebra where the grading is given by Z/2Z. It certainly would be good to include this in the documentation, thanks for the suggestion.

comment:13 Changed 13 months ago by gh-mjungmath

  • Cc jhpalmieri added

comment:14 Changed 13 months ago by tscrim

A graded commutative algebra (as defined here) with G=Z grading is a G-graded supercommutative superalgebra with the Z/2Z grading (for the superalgebra part) induced from the natural quotient map of the gradings. For a multigraded case, this also holds but we take the sum of the degrees (as explained in the documentation).

Thus a graded commutative algebra is a special case of a superalgebra, but superalgebras do not necessarily have to be graded nor supercommutative.

comment:15 Changed 8 months ago by mkoeppe

  • Milestone changed from sage-9.5 to sage-9.6

comment:16 Changed 5 months ago by mkoeppe

  • Milestone changed from sage-9.6 to sage-9.7
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