Opened 13 months ago
Last modified 5 months ago
#32249 new enhancement
Graded-commutative ideals
Reported by: | tkarn | Owned by: | |
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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: |
Description (last modified by )
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
- Description modified (diff)
comment:2 Changed 13 months ago by
- Branch set to u/tkarn/sc-ideals-32249
- Commit set to 1a79ed0f7861180ec7f1c0ae1f0239eb6b7a3e17
comment:3 Changed 13 months ago by
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
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
- 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
Here is a reference: https://d-nb.info/1012919684/34
comment:7 Changed 13 months ago by
Please notice that graded superalgebras already exist, though under a different name (see here).
comment:8 follow-up: ↓ 9 Changed 13 months ago by
This algebra also supports ideals.
comment:9 in reply to: ↑ 8 ; follow-up: ↓ 10 Changed 13 months ago by
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
comment:11 Changed 13 months ago by
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
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
- Cc jhpalmieri added
comment:14 Changed 13 months ago by
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
- Milestone changed from sage-9.5 to sage-9.6
comment:16 Changed 5 months ago by
- Milestone changed from sage-9.6 to sage-9.7
New commits:
Initial commit of supercommutative ideal