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.

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

[tscrim]

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.

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

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

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

Changed commit from 9b97798 to f26834b

[tscrim]

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

[tscrim]

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).

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

Changed commit from f26834b to 5967289

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

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

5967289

Fixing typo antipode -> counit.

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

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

adb8d97

Fixed documentation errors

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

Changed commit from 5967289 to adb8d97

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

Changed commit from adb8d97 to 8051da2

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

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]

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]

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