Implement Yangian for gl(n)

[tscrim]

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.

[tscrim]

Last 10 new commits:

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

[git]

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

[git]

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

[git]

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

[git]

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

​d5954ea

Fix from changes in #15289.

[git]

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.

[git]

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.

[git]

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.

[tscrim]

Refactored over #17096.

[git]

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

​6051865	add a doctest which hopefully works (hard to check while compiling)
​570bc49	Merge branch 'public/categories/super_categories-18044' into 6.9.b1
​b91cd82	trac #18044 correct one typo in the doc
​7fd1df2	Merge branch 'public/categories/super_categories-18044' of trac.sagemath.org:sage into public/categories/super_categories-18044
​0579337	Some reviewer tweaks and doc additions.
​aec22cc	Added one more test.
​4b2046f	Merge branch 'public/categories/super_categories-18044' into public/categories/filtered_algebras-17096
​3f67b6b	Fixing trivial doctest due to changes in category heirarchy.
​fa476dd	Fixing double-colon in INPUT block.
​1770d3f	Merge branch 'public/algebras/yangians' of trac.sagemath.org:sage into public/algebras/yangians

[tscrim]

New commits:

​976c3a5	imported patch yangians-ts.patch
​446e08b	Fixes for Yangian.
​6d21520	Added doctests to full coverage.
​b0c1bd1	Fix from changes in #15289.
​62acea7	Some output tweaks.
​03281d0	Many fixes for Yangians and added to the documentation.
​f5e13bc	Added associated graded algebras to the Yangians.
​8ce5567	Added to catalog and refactored over #17096.
​ea127f1	Fixing the Yangian due to recent changes in Sage.

[git]

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

​6b8bcd3

Merge branch 'public/algebras/yangians-15484' of trac.sagemath.org:sage into public/algebras/yangians-15484

[jdemeyer]

See patchbot (note that I don't personally care about this ticket, sorry)

[git]

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

​236b896	Merge branch 'public/algebras/yangians-15484' of trac.sagemath.org:sage into public/algebras/yangians-15484
​f297f33	Use sorting_key instead of generator_cmp.

[tscrim]

I understand; thanks for the heads-up.

[tscrim]

Just in case you're interested Andrew.

[andrew.mathas]

Replying to tscrim:

Just in case you're interested Andrew.

Sorry Travis, but I don't play at all with Yangians and don't know much about them...

[darij]

REFERENCES:: please!

I happen to be (slowly) reading my way through a paper on Capelli identities right now, and I'd be really interested in what seems to be the theory behind them, but I probably won't learn much from the code in itself :)

[git]

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

​ec4e344

Added references.

[tscrim]

Andrew, no problem.

Darij, I've added some references. A good reference is the MNO94 arXiv version.

[git]

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

​8227771

Merge branch 'public/algebras/yangians-15484' in 7.5.b6

[git]

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

​9fbaa14	Merge branch 'public/algebras/yangians-15484' of git://trac.sagemath.org/sage into public/algebras/yangians-15484
​5336bec	Moving references.

[git]

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

[bsalisbury1]

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]

Looks good. :)

[git]

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

​9b97798

More documentation additions

[bsalisbury1]

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]

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.

[git]

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

[tscrim]

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).

[git]

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

​5967289

Fixing typo antipode -> counit.

[git]

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

​adb8d97

Fixed documentation errors

[git]

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]

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.