Graded-commutative ideals

[trevorkarn]

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.

CC: @tscrim @trevorkarn @jhpalmieri

Component: algebra

Keywords: gsoc2021 gradedcommutative ideal graded

Author: Trevor K. Karn

Branch/Commit: u/tkarn/sc-ideals-32249 @ 1a79ed0

Issue created by migration from https://trac.sagemath.org/ticket/32249

[trevorkarn]

Description changed:

--- 
+++ 
@@ -1 +1 @@
-A supercommutative 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 supercommutative rings inside of `sage.rings.ideal` via the class `Ideal_supercommutative`. 
+A supercommutative 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 supercommutative rings inside of `sage.rings.noncommutative_ideals` via the class `Ideal_supercommutative`. 

[trevorkarn]

New commits:

1a79ed0

Initial commit of supercommutative ideal

[trevorkarn]

Commit: 1a79ed0

[trevorkarn]

Branch: u/tkarn/sc-ideals-32249

[nbruin]

comment:3

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.

[trevorkarn]

comment:4

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.

[trevorkarn]

Description changed:

--- 
+++ 
@@ -1 +1 @@
-A supercommutative 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 supercommutative rings inside of `sage.rings.noncommutative_ideals` via the class `Ideal_supercommutative`. 
+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`. 

[trevorkarn]

Changed keywords from gsoc2021 supercommutative ideal to gsoc2021 gradedcommutative ideal graded

[trevorkarn]

comment:6

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

[mjungmath]

comment:7

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

[mjungmath]

comment:8

This algebra also supports ideals.

[tscrim]

comment:9

Replying to @mjungmath:

This algebra also supports ideals.

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

[mjungmath]

comment:10

Replying to @tscrim:

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

Indeed.

[mathzeta]

comment:11

Maybe I confuse the terms, or this is entirely anecdotal, but I have encountered the name superalgebra few times (e.g. in [https://icerm.brown.edu/materials/Slides/tw-18-1/Lie_Superalgebras_andSage_Dan_Bump,_Stanford_University.pdf|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.

[trevorkarn]

comment:12

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.

[tscrim]

comment:14

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.