Clifford algebras and differential Weyl algebras #15300

tscrim · 2013-10-16T23:16:37Z

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

The text was updated successfully, but these errors were encountered:

tscrim · 2013-11-02T21:17:22Z

Attachment: trac_15300-weyl_clifford_algebras-ts.patch.gz

mguaypaq · 2013-11-05T21:57:04Z

Branch: u/mguaypaq/weyl-clifford

dwbump · 2013-11-05T22:56:47Z

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

anneschilling · 2013-11-05T23:12:51Z

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

anneschilling · 2013-11-05T23:12:51Z

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 · 2013-11-05T23:12:51Z

Commit: e3ff430

darijgr · 2013-11-05T23:32:50Z

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

sagetrac-gmoose05 · 2013-11-05T23:52:41Z

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

darijgr · 2013-11-07T08:43:21Z

Changed commit from e3ff430 to 8138949

darijgr · 2013-11-07T08:43:21Z

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 · 2013-11-08T20:27:15Z

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

tscrim · 2013-11-08T20:27:15Z

New commits:

`[c2f2fef](https://github.com/sagemath/sagetrac-mirror/commit/c2f2fef)`	`Merge branch 'master' into public/algebras/weyl_clifford-15300`

tscrim · 2013-11-08T20:27:15Z

Changed commit from 8138949 to c2f2fef

sagetrac-git · 2013-11-08T20:32:11Z

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

`[802459a](https://github.com/sagemath/sagetrac-mirror/commit/802459a)`	`trac #15300: some remarks`

sagetrac-git · 2013-11-08T20:32:11Z

Changed commit from c2f2fef to 802459a

tscrim · 2013-11-08T22:52:46Z

Changed keywords from none to days54

sagetrac-git · 2013-11-30T05:39:57Z

Changed commit from 802459a to dce70bb

sagetrac-git · 2013-11-30T05:39:57Z

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`

sagetrac-git · 2013-11-30T07:56:23Z

Changed commit from dce70bb to 6b69822

sagetrac-git · 2013-11-30T07:56:23Z

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 · 2013-12-03T01:47:43Z

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.

sagetrac-git · 2013-12-03T03:52:30Z

Changed commit from 6b69822 to 760de0b

tscrim · 2014-10-23T03:23:28Z

comment:156

Thank you both!

sagetrac-git · 2014-10-23T14:46:55Z

Changed commit from 339b71d to 8698275

sagetrac-git · 2014-10-23T14:46:55Z

Branch pushed to git repo; I updated commit sha1 and set ticket back to needs_review. New commits:

`3ce3f6a`	`Merge branch 'public/algebras/weyl_clifford-15300' of git://trac.sagemath.org/sage into clifford`
`8698275`	`lifting bilinear forms onto exterior algebras`

darijgr · 2014-10-23T14:56:23Z

comment:158

Thanks from me, too, John and Travis! Care to review one final piece of code? I have added the lifted bilinear form on the exterior algebra.

TODO list for further patches:

WEYL:

Weyl algebras in general. This will be mostly copypasting the structure of the Clifford algebra class.
Differential Weyl algebra as a particular case of a Weyl algebra (I think the matrix is the block matrix
(0 -I
I 0)
here). Is this best achieved by having DifferentialWeylAlgebra inherit from WeylAlgebra, or better by coercions from one to the other? (I hope it's the former.)

CLIFFORD:

The canonical isomorphism between a Clifford algebra and an exterior algebra induced by a bilinear (! not quadratic !) form. This isomorphism, and its inverse, are particular cases of a common construction (Bourbaki's "Algèbre IX", §9, no. 2-3; Ricardo Baeza's "Quadratic Forms over Semilocal Rings", Lecture Notes in Mathematics 655, Springer 1978): If f and g are two bilinear forms on a module V, and if F and G are their corresponding quadratic forms (so F(x) = f(x, x) and G(x) = g(x, x)), then there is a canonical isomorphism from Cl(V, F) to Cl(V, F + G), which however depends on g and not just on G.
Use this Clifford-exterior isomorphism to define constant coefficients and scalar products on Clifford algebras. This will depend on a bilinear form.
Check if the scalar product on the Clifford algebra allows a faster way to compute the lifted bilinear form on the exterior algebra.

vbraun · 2014-10-23T15:23:04Z

comment:159

sage -t --long --warn-long 38.2 src/sage/combinat/descent_algebra.py
git log**********************************************************************
File "src/sage/combinat/descent_algebra.py", line 292, in sage.combinat.descent_algebra.DescentAlgebra.D.to_B_basis
Failed example:
    map(B, D.basis()) # indirect doctest
Expected:
    [B[4],
     B[1, 3] - B[4],
     B[2, 2] - B[4],
     B[1, 1, 2] - B[1, 3] - B[2, 2] + B[4],
     B[3, 1] - B[4],
     B[1, 2, 1] - B[1, 3] - B[3, 1] + B[4],
     B[2, 1, 1] - B[2, 2] - B[3, 1] + B[4],
     B[1, 1, 1, 1] - B[1, 1, 2] - B[1, 2, 1] + B[1, 3]
      - B[2, 1, 1] + B[2, 2] + B[3, 1] - B[4]]
Got:
    [B[4],
     B[1, 3] - B[4],
     B[2, 2] - B[4],
     B[3, 1] - B[4],
     B[1, 1, 2] - B[1, 3] - B[2, 2] + B[4],
     B[1, 2, 1] - B[1, 3] - B[3, 1] + B[4],
     B[2, 1, 1] - B[2, 2] - B[3, 1] + B[4],
     B[1, 1, 1, 1] - B[1, 1, 2] - B[1, 2, 1] + B[1, 3] - B[2, 1, 1] + B[2, 2] + B[3, 1] - B[4]]
 **********************************************************************
1 item had failures:
   1 of   5 in sage.combinat.descent_algebra.DescentAlgebra.D.to_B_basis

sagetrac-git · 2014-10-23T15:30:27Z

Changed commit from 8698275 to a002f4b

sagetrac-git · 2014-10-23T15:30:27Z

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

`a002f4b`	`Some tweaks to lifted_bilinear_form and fixed doctest.`

tscrim · 2014-10-23T15:33:17Z

comment:161

I made some tweaks to the lifted_bilinear_form. Most importantly I made it return a PoorManMap so the repr is nicer, and I changed the doc. I also fixed the doctest Volker noted.

tscrim · 2014-10-23T15:47:05Z

comment:162

Darij could you check my changes (and add a space to that last line in that descent_algebra.py doctest since all my cython files are recompiling currently)?

darijgr · 2014-10-23T15:56:52Z

comment:163

Done and done. I had to change the domain of the PoorManMap, though, and I don't like the result very much.

sagetrac-git · 2014-10-23T15:57:18Z

Changed commit from a002f4b to ff27bdc

sagetrac-git · 2014-10-23T15:57:18Z

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

`ff27bdc`	`further fixes`

tscrim · 2014-10-23T16:11:16Z

comment:165

Ah yea. Duh. Thanks.

vbraun · 2014-10-24T10:14:33Z

Changed branch from public/algebras/weyl_clifford-15300 to ff27bdc

tscrim added this to the sage-6.1 milestone Oct 16, 2013

tscrim added c: algebra labels Oct 16, 2013

tscrim self-assigned this Oct 16, 2013

tscrim added the s: needs review label Oct 16, 2013

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jhpalmieri added the s: positive review label Oct 23, 2014

sagetrac-git mannequin added s: needs review and removed s: positive review labels Oct 23, 2014

darijgr changed the title ~~Weyl and Clifford Algebras~~ Clifford algebras and differential Weyl algebras Oct 23, 2014

vbraun added s: needs work and removed s: needs review labels Oct 23, 2014

tscrim added s: needs review and removed s: needs work labels Oct 23, 2014

tscrim added s: positive review and removed s: needs review labels Oct 23, 2014

vbraun removed the s: positive review label Oct 24, 2014

vbraun closed this as completed in c7053a5 Oct 24, 2014

This was referenced Jun 17, 2022

Lie algebras #14901

Open

Implement categories for differential algebras #16513

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Closed

jdemeyer mentioned this issue Oct 24, 2014

Improve formula for Bell numbers #17157

Closed

Clifford algebras and differential Weyl algebras #15300

Clifford algebras and differential Weyl algebras #15300

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