Implement Yangian for gl(n) #15484

tscrim · 2013-12-04T21:57:09Z

Yangians can be seen as a deformation of U(gl[t]) and have many interesting connections to representation theory. See http://en.wikipedia.org/wiki/Yangian.

Depends on #23266
Depends on #23267
Depends on #23268
Depends on #23270

CC: @sagetrac-sage-combinat @anneschilling @darijgr @bsalisbury1

Component: algebra

Keywords: yangian, quantum groups

Author: Travis Scrimshaw

Branch/Commit: 8051da2

Reviewer: Ben Salisbury

Issue created by migration from https://trac.sagemath.org/ticket/15484

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

tscrim · 2013-12-04T21:58:42Z

Changed branch from public/algebras/yangian to public/algebras/yangians

tscrim · 2013-12-04T21:58:42Z

Last 10 new commits:

`[63f89cb](https://github.com/sagemath/sagetrac-mirror/commit/63f89cb)`	`Added doctests to full coverage.`
`[cd3e53e](https://github.com/sagemath/sagetrac-mirror/commit/cd3e53e)`	`Merge branch 'public/monoids/15289-indexed' into u/tscrim/yangians`
`[9dca526](https://github.com/sagemath/sagetrac-mirror/commit/9dca526)`	`Added comparison operations.`
`[a493bee](https://github.com/sagemath/sagetrac-mirror/commit/a493bee)`	`Merge branch 'master' into public/monoids/15289-indexed`
`[7e39c1f](https://github.com/sagemath/sagetrac-mirror/commit/7e39c1f)`	`Fixes for Yangian.`
`[bc1b55c](https://github.com/sagemath/sagetrac-mirror/commit/bc1b55c)`	`Merge branch 'public/monoids/15289-indexed' into u/tscrim/yangians`
`[435a303](https://github.com/sagemath/sagetrac-mirror/commit/435a303)`	`Merge branch 'public/ticket/10963' of ssh://trac.sagemath.org:2222/sage into u/tscrim/yangians`
`[80d55fe](https://github.com/sagemath/sagetrac-mirror/commit/80d55fe)`	`merging sage/categories/category.py`
`[b2064e6](https://github.com/sagemath/sagetrac-mirror/commit/b2064e6)`	`Merge branch 'u/tscrim/yangians' of ssh://trac.sagemath.org:2222/sage into u/tscrim/yangians`
`[b113acc](https://github.com/sagemath/sagetrac-mirror/commit/b113acc)`	`Merge branch 'master' into u/tscrim/yangians`

tscrim · 2013-12-04T21:58:42Z

Commit: 63f89cb

sagetrac-git · 2014-03-30T18:33:49Z

Changed commit from 63f89cb to 4fe436f

sagetrac-git · 2014-03-30T18:33:49Z

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

`bd82ed8`	`Merge branch 'develop' into public/monoids/15289-indexed`
`56703eb`	`Made Indexed* have entry points through Free*.`
`163df6e`	`Changed more _basis_keys to _indices, deprecated the former.`
`8db8e0a`	Changed `_an_element_` to indexed_monoid.py.
`760c939`	`Merge branch 'public/monoids/15289-indexed' of trac.sagemath.org:sage into public/monoids/15289-indexed`
`03057a4`	`Merge branch 'develop' into public/monoids/15289-indexed`
`a2996e0`	`Merge branch 'develop' into public/monoids/15289-indexed`
`c1cc341`	`Merge branch 'develop' into public/monoids/15289-indexed`
`c7baad9`	`Merge branch 'public/monoids/15289-indexed' into public/algebras/yangians`
`4fe436f`	`Merge branch 'public/algebras/yangians' of trac.sagemath.org:sage into public/algebras/yangians`

sagetrac-git · 2014-04-02T01:46:01Z

Changed commit from 4fe436f to c467001

sagetrac-git · 2014-04-02T01:46:01Z

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

`49068b2`	`Merge branch 'develop' into public/monoids/15289-indexed`
`c467001`	`Merge branch 'public/algebras/yangians' of trac.sagemath.org:sage into public/algebras/yangian`

sagetrac-git · 2014-07-25T07:15:22Z

Changed commit from c467001 to 0962d33

sagetrac-git · 2014-07-25T07:15:22Z

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

`0962d33`	`Merge branch 'public/algebras/yangians' of trac.sagemath.org:sage into public/algebras/yangians`

sagetrac-git · 2014-07-25T07:18:59Z

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

`d5954ea`	`Fix from changes in #15289.`

sagetrac-git · 2014-07-25T07:18:59Z

Changed commit from 0962d33 to d5954ea

sagetrac-git · 2014-08-01T17:46:40Z

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

`6f98761`	`Merge branch 'develop' into public/algebras/yangians`
`f8fe06c`	`Some output tweaks.`
`629604f`	`Many fixes for Yangians and added to the documentation.`

sagetrac-git · 2014-08-01T17:46:40Z

Changed commit from d5954ea to 629604f

sagetrac-git · 2014-09-21T17:11:12Z

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

`377d266`	`Merge branch 'public/algebras/yangians' of trac.sagemath.org:sage into public/algebras/yangians`
`36d0701`	`Added associated graded algebras to the Yangians.`

sagetrac-git · 2014-09-21T17:11:12Z

Changed commit from 629604f to 36d0701

sagetrac-git · 2014-11-30T02:57:51Z

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

`25c0fbc`	`looked through clifford_algebra and filtered_algebras*; many TODOs left`
`2a62c3b`	`lift_* methods in algebras/clifford_algebra.py should remember the graded-filtered choice`
`15cf0dc`	`fix and a first doctest for induced_graded_map`
`5029956`	`another pitfall documented`
`292ef5f`	`another doctest`
`c2e84e1`	`remaining doctests for induced_graded_map`
`03bd4cf`	`possibly controversial: graded_algebra and the three methods forming its interface are now cached_methods`
`8a747c0`	`further edits`
`237daea`	`Merge branch 'public/categories/filtered_algebras-17096' of trac.sagemath.org:sage into public/algebras/yangians`
`99a92e3`	`Added to catalog and refactored over #17096.`

sagetrac-git · 2014-11-30T02:57:51Z

Changed commit from 36d0701 to 99a92e3

tscrim · 2014-11-30T03:10:32Z

comment:12

Refactored over #17096.

sagetrac-git · 2017-04-02T21:24:12Z

Changed commit from 8227771 to 5336bec

sagetrac-git · 2017-05-21T14:57:27Z

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

`a107deb`	`Merge branch 'develop' into t/15484/public/algebras/yangians-15484`
`8aa9ef9`	`Added to documentation and fixed one doctest`

sagetrac-git · 2017-05-21T14:57:27Z

Changed commit from 5336bec to 8aa9ef9

bsalisbury1 · 2017-05-21T14:58:49Z

Reviewer: Ben Salisbury

bsalisbury1 · 2017-05-21T14:58:49Z

comment:28

Hi Travis,

I added some more to the documentation. Please make sure you are okay with it. I will continue the review soon.

Ben

tscrim · 2017-05-21T15:01:22Z

comment:30

Looks good. :)

sagetrac-git · 2017-05-21T20:36:38Z

Changed commit from 8aa9ef9 to 9b97798

sagetrac-git · 2017-05-21T20:36:38Z

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

`9b97798`	`More documentation additions`

bsalisbury1 · 2017-05-21T20:45:55Z

comment:32

Hi Travis,

I'm still going through and adding more to the documentation, but some things (including some corrections) may be better completed by you. (I have the arXiv versions of Molev's references that you've listed, but I do not have a copy of the book you referenced.) For example:

You've implemented the method defining_polynomial, but I cannot find that in my references. Moreover, in the references I do have, the notation you've used for the defining polynomial seems to be reserved for matrices whose entries are power series.
I think there is an antipode method missing from the main Yangian class (and the counit_on_basis method, according to the documentation, returns the antipode).
Add definitions for the coproduct, counit, and antipode. I added the coproduct definition already for the Yangian class.
Can/should the Yang R-matrix also be added to this module?

tscrim · 2017-05-21T23:46:56Z

comment:33

Replying to @bsalisbury1:

You've implemented the method defining_polynomial, but I cannot find that in my references. Moreover, in the references I do have, the notation you've used for the defining polynomial seems to be reserved for matrices whose entries are power series.

This is the level \ell version, so they become polynomials instead of power series.

I think there is an antipode method missing from the main Yangian class (and the counit_on_basis method, according to the documentation, returns the antipode).

No, that is because I could not implement the antipode as it is defined in terms of an inverse matrix of power series. There is a ticket (I don't remember the number offhand) that gives a default implementation of the antipode in connected graded Hopf algebras.

In the counit_on_basis, that is a typo from copy/pasting.

Add definitions for the coproduct, counit, and antipode. I added the coproduct definition already for the Yangian class.

Let me know when you're done with your changed, and I can add them.

Can/should the Yang R-matrix also be added to this module?

It would make for a good follow-up.

sagetrac-git · 2017-06-18T14:48:07Z

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

`e0f9c88`	`Use monomial_coefficients instead of iteration for consistancy.`
`f61aae2`	`Better mutliplication of PBW elements.`
`9cea97d`	`Merge branch 'public/lie_algebras/pbw_mult_monomial_coeffs-23268' into public/algebras/yangians-15484`
`35aa493`	`Added monomial_coefficients for gl.`
`eb19d0e`	`Merge branch 'public/lie_algebras/monomial_gl-23267' into public/algebras/yangians-15484`
`a89bfcc`	`Adding an additional doctest checking U(gl_n) homomorphism.`
`5d87394`	`Trivial doctest fixes.`
`707c5f1`	`Merge branch 'public/linear_algebra/matrix_space_with_basis-22955' into public/algebras/yangians-15484`
`522d374`	`Adding Filtered category to PBW basis.`
`f26834b`	`Merge branch 'public/lie_algebras/pbw_filtered-23270' into public/algebras/yangians-15484`

sagetrac-git · 2017-06-18T14:48:07Z

Changed commit from 9b97798 to f26834b

tscrim · 2017-06-18T14:49:03Z

Dependencies: #23266, #23267, #23268, #23270

tscrim · 2017-06-18T14:49:03Z

comment:35

I've updated the doc to be a more complete overview. I've also added a doctest that tests the homomorphism as filtered algebras with U(gl_n) (although this uncovered #23266, #23267, #23268, #23270).

sagetrac-git · 2017-06-18T14:49:59Z

Changed commit from f26834b to 5967289

sagetrac-git · 2017-06-18T14:49:59Z

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

`5967289`	`Fixing typo antipode -> counit.`

sagetrac-git · 2017-06-20T00:50:43Z

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

`adb8d97`	`Fixed documentation errors`

sagetrac-git · 2017-06-20T00:50:43Z

Changed commit from 5967289 to adb8d97

sagetrac-git · 2017-06-20T00:53:11Z

Changed commit from adb8d97 to 8051da2

sagetrac-git · 2017-06-20T00:53:11Z

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

`6393c5f`	`Fixed doctest errors`
`8051da2`	`Merge branch 't/23267/public/lie_algebras/monomial_gl-23267' into t/15484/public/algebras/yangians-15484`

bsalisbury1 · 2017-06-20T00:56:16Z

comment:39

Thanks for making those additions! I added some additional clarification and fixed some typos. In particular, now the PDF documentation will build. HTML documentation also builds and all tests pass.

vbraun · 2017-06-25T15:45:31Z

Changed branch from public/algebras/yangians-15484 to 8051da2

tscrim added this to the sage-6.1 milestone Dec 4, 2013

tscrim added c: algebra labels Dec 4, 2013

tscrim assigned sagetrac-sage-combinat Dec 4, 2013

tscrim added the s: needs review label Dec 4, 2013

sagetrac-vbraun-spam mannequin modified the milestones: sage-6.1, sage-6.2 Jan 30, 2014

sagetrac-vbraun-spam mannequin modified the milestones: sage-6.2, sage-6.3 May 6, 2014

sagetrac-vbraun-spam mannequin modified the milestones: sage-6.3, sage-6.4 Aug 10, 2014

bsalisbury1 modified the milestones: sage-7.4, sage-8.0 May 21, 2017

bsalisbury1 added s: positive review and removed s: needs review labels Jun 20, 2017

vbraun removed the s: positive review label Jun 25, 2017

vbraun closed this as completed in ec4b5bb Jun 25, 2017

This was referenced Jun 17, 2022

Lie algebras #14901

Open

Implement categories for filtered algebras #17096

Closed

Implement Yangian for gl(n) #15484

Implement Yangian for gl(n) #15484

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