Clifford algebras and differential Weyl algebras

[tscrim]

Initial implementations of Weyl and Clifford algebras.

Depends on #16037

CC: @darijgr @sagetrac-sage-combinat

Component: algebra

Keywords: days54

Author: Travis Scrimshaw

Branch/Commit: ff27bdc

Reviewer: Darij Grinberg, John Palmieri

Issue created by migration from https://trac.sagemath.org/ticket/15300

[tscrim]

Attachment: trac_15300-weyl_clifford_algebras-ts.patch.gz

[mguaypaq]

Branch: u/mguaypaq/weyl-clifford

[6bbcde06-8197-41f1-b9a3-c998bb839000]

Changed branch from u/mguaypaq/weyl-clifford to u/bump/ticket/15300

[darijgr]

Description changed:

--- 
+++ 
@@ -1 +1,3 @@
 Initial implementations of Weyl and Clifford algebras.
+
+Test!

[anneschilling]

Changed branch from u/bump/ticket/15300 to u/aschilling/ticket/15300

[anneschilling]

New commits:

[changeset:e3ff430]	Merge branch 'u/bump/ticket/15300' of git://trac.sagemath.org/sage into ticket/15300
[changeset:427f2fe]	changed formatting of list
[changeset:0734921]	changed spaces
[changeset:1c7458a]	#15300: Implement Weyl and Clifford algebras.

[anneschilling]

Commit: e3ff430

[darijgr]

Changed branch from u/aschilling/ticket/15300 to u/darij/ticket/15300

[darijgr]

Description changed:

--- 
+++ 
@@ -1,3 +1 @@
 Initial implementations of Weyl and Clifford algebras.
-
-Test!

[a9a57cdf-9183-46c4-a1f6-41c1b5271dce]

Changed branch from u/darij/ticket/15300 to u/gmoose05/ticket/15300

[darijgr]

Changed commit from e3ff430 to 8138949

[darijgr]

comment:10

Attachment: trac_15300-todos-dg.patch.gz

The patch I've posted contains a couple corrections and comments. Sorry for being this slow; the cold isn't very beneficial for my concentration.

EDIT: The commit list below has nothing to do with my post; it seems that trac automatically appends it to whatever post is made first after the commits. Anyway, for everyone who is not at Davis: The git commits on this ticket are a sandbox for people merging git; the true work is being done in the hg patches. I'm very positive this one is going to be reviewed way before we move over to git.

---

New commits:

[changeset:8138949]	Merge branch 'u/darij/ticket/15300' of trac.sagemath.org:sage into gregg-change
[changeset:9f8c4a1]	merged with Anne
[changeset:529d73c]	Merge branch 'u/bump/ticket/15300' of git://trac.sagemath.org/sage into ticket/15300
[changeset:9086aca]	added spaces
[changeset:911906c]	#15300: Implement Weyl and Clifford algebras.
[changeset:2033f23]	trying out git (pyflakes corrections)

[tscrim]

Changed branch from u/gmoose05/ticket/15300 to public/algebras/weyl_clifford-15300

[tscrim]

New commits:

[c2f2fef](https://github.com/sagemath/sagetrac-mirror/commit/c2f2fef)

Merge branch 'master' into public/algebras/weyl_clifford-15300

[tscrim]

Changed commit from 8138949 to c2f2fef

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

[802459a](https://github.com/sagemath/sagetrac-mirror/commit/802459a)

trac #15300: some remarks

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from c2f2fef to 802459a

[tscrim]

Changed keywords from none to days54

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 802459a to dce70bb

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

[dce70bb](https://github.com/sagemath/sagetrac-mirror/commit/dce70bb)	Fixes to Clifford algebras and expanded to have a functor method lift_morphism()
[728019a](https://github.com/sagemath/sagetrac-mirror/commit/728019a)	Merge branch 'master' into public/algebras/weyl_clifford-15300

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from dce70bb to 6b69822

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

[6b69822](https://github.com/sagemath/sagetrac-mirror/commit/6b69822)	More docstring tweaks to Clifford algebra.
[f759608](https://github.com/sagemath/sagetrac-mirror/commit/f759608)	Added lift_isometry method.

[darijgr]

comment:16

Is there an established class for bilinear forms (not necessarily symmetric!) on free modules? I'd need one to get anywhere towards the Clifford-exterior iso.

The scalar function as it stands is rather useless, since on the exterior algebra it just returns the product of the constant coefficients of self and other. What should be made (I can do that) is the lift of a bilinear (not quadratic) form on V to the exterior algebra of V (using the Gram matrix). But that requires bilinear forms, too, so I'd love to hear whether they exist before doing any changes.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 6b69822 to 760de0b

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

[760de0b](https://github.com/sagemath/sagetrac-mirror/commit/760de0b)

some doc fixes

[tscrim]

comment:18

Replying to @darijgr:

Is there an established class for bilinear forms (not necessarily symmetric!) on free modules? I'd need one to get anywhere towards the Clifford-exterior iso.

I don't think so. I think this is usually worked (hacked) around using dedicated methods.

The scalar function as it stands is rather useless, since on the exterior algebra it just returns the product of the constant coefficients of self and other. What should be made (I can do that) is the lift of a bilinear (not quadratic) form on V to the exterior algebra of V (using the Gram matrix). But that requires bilinear forms, too, so I'd love to hear whether they exist before doing any changes.

I'd just have a function Element.scalar(other) (or some other name) which projects down self and other to the exterior algebra and does the computation there via some (predefined) scalar() method.

[darijgr]

comment:19

Do you think it makes sense to add a BilinearForm class in analogy to QuadraticForm, or should I just use matrices to represent bilinear forms?

[nthiery]

comment:20

It depends what feature about bilinear forms you need for now. If it's just about computing it on some elements, I would stick for now with implementing appropriate "scalar" methods. Granted, it's a bit rudimentary. In the long run we want to have proper multivariate morphisms, with support from the coercion model. See #8900 and http://trac.sagemath.org/wiki/CategoriesRoadMap.

Cheers, Nicolas

[darijgr]

comment:21

What I want is to change the WeylAlgebra class to depend upon a space with a bilinear form, not on a polynomial ring. So the bilinear form will be a parameter. Still OK if it is a matrix or should I rather have an extra class?

[tscrim]

comment:22

Replying to @darijgr:

What I want is to change the WeylAlgebra class to depend upon a space with a bilinear form, not on a polynomial ring. So the bilinear form will be a parameter. Still OK if it is a matrix or should I rather have an extra class?

I think a matrix would be best for now (although I would like to keep the polynomial ring version).

Thanks,

Travis

[darijgr]

comment:23

Will be a matrix then.

Yeah, I'm not going to deprecate your Weyl algebra; I might end up renaming it, though. Thing is, the Weyl algebra of an antisymmetric bilinear form is the most correct analogue of the Clifford algebra known to me (there might be better ones, though) -- far closer than the Weyl algebra of a polynomial ring (which is just the Weyl algebra of the usual antisymmetric form which is

0 1
-1 0

as a block matrix.

[tscrim]

comment:24

I'm okay with renaming the class, but I'd like to keep the global entry point the same. In other words, I'd like to create it via WeylAlgebra(QQ, 4), where the absence of the bilinear form says to use the polynomial representation, or perhaps with a keyword such as polynomial=True. Thanks.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 760de0b to e9902b9

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

e9902b9

Merge branch 'public/algebras/weyl_clifford-15300' of trac.sagemath.org:sage into public/algebras/weyl_clifford-15300

[darijgr]

comment:27

There is a stupid __repr__ bug I've no idea how to fix.

sage: A = CliffordAlgebra(QuadraticForm(QQ, 3, [1,0,-1,3,-4,5]))
sage: A.basis()
Finite family {(0, 1): e0*e1, (1, 2): ee2, (0,): e0, (1,): e1, (0, 1, 2): e0*ee2, (2,): e2, (): 1, (0, 2): e0*e2}

See the ee2? It's an e1*e2, which the __repr__ routine seems to simplify to ee2 because 1* can be omitted, right? One way to deal with it would be to not use the multiplication signs; what do you think about that?

(Also, the doctests of lift_isometry are wrong -- could you compute them by hand and re-insert them? Thank you.)

EDIT: Oh, I see, the bug is in repr_from_monomials of weyl_algebra.py, not in some big free-module module.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from e9902b9 to ba596fa

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

c288a0c	Merge branch 'public/algebras/weyl_clifford-15300' of trac.sagemath.org:sage into public/algebras/weyl_clifford-15300
015ce74	Fix the repr.
ba596fa	Fixed doctests in clifford_algebra.py.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

adc2fb1	rename currently existing Weyl algebra classes, so as to later introduce something more general
dad0bce	review of repr_from_monomials

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from ba596fa to dad0bce

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

8749f21	Added homology to exterior algebras.
6909a23	Merge branch 'public/algebras/weyl_clifford-15300' of trac.sagemath.org:sage into public/algebras/weyl_clifford-15300
5109135	Fixed doctests.

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from dad0bce to 5109135

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 5109135 to 9bdf752

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

9bdf752

various improvements to clifford_algebra.py

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 9bdf752 to 9f5bb97

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

9f5bb97

fix merge

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

ab37f83

tuple(S) for S a set doesn't always return the increasing list

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from 9f5bb97 to ab37f83

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

Changed commit from ab37f83 to a2354c6

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

a2354c6

minor edits

[7ed8c4ca-6d56-4ae9-953a-41e42b4ed313]

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

8914ccc

further clifford edits