#16508: Add Commutative graded differential algebras.
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Reporter: mmarco | Owner:
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.3
Component: algebra | Resolution:
Keywords: sd58, sd59, | Merged in:
algebras, nonconmutative, graded | Reviewers:
Authors: mmarco | Work issues:
Report Upstream: N/A | Commit:
Branch: | 61487e692af1d715e7cc4281714b7fb0705c542f
u/mmarco/ticket/16508 | Stopgaps:
Dependencies: |
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Comment (by jhpalmieri):
Here is a new branch: u/jhpalmieri/DGA_new. (Is there some sort of markup
I can give to make that an active link? Should we make it the official
branch attached to this ticket?)
This is a complete rewrite based on Miguel's version, and so built on top
of `QuotientRing_nc`. Changes:
- I removed all of the category stuff. That can go on a separate ticket
(#16513).
- I added the option to multi-grade the objects, using commands like
{{{
A.<x,y,z> = CDGAlgebra(QQ, degrees=((1,0), (0,1), (2,0)))
}}}
- Tinkered with documentation. Added the main file to the reference
manual.
- Various code modifications and clean up. Among other things:
- Sped up `GCAlgebra.homogeneous_part`. The old version was
`GCAlgebra.homogeneous_part`, the new version `GCAlgebra.basis`. In my
current code, there is some caching going on, but when doing these
timings, I disabled the caching.
{{{
sage: A.<t,x,y,z> = CDGAlgebra(QQ, degrees=(1,1,2,4))
sage: A.homogeneous_part(10)
[y*z^2, t*x*z^2, y^3*z, t*x*y^2*z, y^5, t*x*y^4]
sage: A.basis(10)
[t*x*z^2, t*x*y^2*z, t*x*y^4, y*z^2, y^3*z, y^5]
sage: sorted(A.homogeneous_part(10)) == sorted(A.basis(10))
True
sage: sorted(A.homogeneous_part(40)) == sorted(A.basis(40))
True
sage: timeit('A.homogeneous_part(40)') # original version
5 loops, best of 3: 440 ms per loop
sage: timeit('A.basis(40)') # new version
25 loops, best of 3: 12.9 ms per loop
}}}
- Sped up `_call_` method for `CDGA_Differential`:
{{{
sage: A.<t,x,y,z> = CDGAlgebra(QQ, degrees=(1,1,2,4))
sage: B = CDGAlgebra(A, differential={t:y, x:y})
sage: a = t*x*y**20*z**20
sage: timeit('B.differential()._call1_(a)') # original version
25 loops, best of 3: 19.6 ms per loop
sage: timeit('B.differential()._call3_(a)') # new version
125 loops, best of 3: 2.38 ms per loop
}}}
- Rewrote `is_homogeneous` based on other such methods elsewhere in Sage.
It's slightly faster this way.
- Should it be "commutative graded algebra" or "graded commutative
algebra"? I think the second is more standard – for example, the phrase
"graded commutativity" is commonly used – and after some thought, that's
what I used in the rewrite. ("Commutative graded algebra" sounds like a
graded algebra that happens to be commutative, implying strict
commutativity. In "graded commutative algebra", "graded" modifies both
"commutative" and the algebra, so the algebra is graded and the
commutativity has a sign coming from the grading. According to Google
search, there are about four times as many hits for "graded commutative
algebra" compared to "commutative graded algebra". Similarly for a search
on http://mathoverflow.net.)
- On the other hand, "differential graded algebra" is completely standard,
and so is "commutative differential graded algebra". So the adjectives are
not ordered consistently in the different types of algebras.
- So the main classes and functions are `GCAlgebra` and `CDGAlgebra`. The
second of these is the only thing imported at the top-level.
Comments and questions:
1. `GCAlgebra._element_constructor_` works unexpectedly: if B and C are
quotients of A, then you can convert elements of B to elements of C. For
example:
{{{
sage: A.<x,y,z> = CDGAlgebra(QQ, degrees=(1, 2, 1))
sage: B = A.quotient(A.ideal(z))
sage: C = A.quotient(A.ideal(x*y))
sage: C.inject_variables()
Defining x, y, z
sage: x*y
0
sage: B(x*y)
0
sage: B(x)*B(y)
x*y
}}}
However, it looks like this is just a slightly modified version of the
same method from `rings/quotient_ring.py`, so maybe we should leave it as
is. Maybe that method could use some documentation, though.
2. When initializing a graded commutative algebra, what should we do if
neither ``names`` or ``degrees`` is specified (as in `GCAlgebra(QQ)`)?
Right now I'm raising an error.
3. `TestSuite` fails on these algebras:
{{{
The following tests failed: _test_elements, _test_pickling
}}}
Is that important?
--
Ticket URL: <http://trac.sagemath.org/ticket/16508#comment:25>
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