Lie algebras #14901

tscrim · 2013-07-17T04:27:03Z

Initial implementation of Lie algebras in sage.

This will contain the following:

Free Lie algebras
- Hall basis
- Lyndon basis
Abelian Lie algebras
Lie algebras from an associative algebra
Lie algebras from structure coefficients
Finite type Lie algebras
- As matrices for types ABCD
- In the Chevalley basis
gl_n
Untwisted affine Lie algebras constructed from a finite type
Untwisted affine Kac-Moody Lie algebras (i.e. the above and the Lie derivative)
Universal enveloping algebras
- PBW bases
Quotient, sub, and direct sum Lie algebras and Lie algebra ideals
Other examples:
- Upper triangular matrices
- Strictly upper triangular matrices
- Heisenberg algebra
- Witt algebra
- Virasoro algebra
- Some nilpotent Lie algebras
Kac-Moody algebras based only on a (generalized) Cartan matrix
Lie algebra (co)homology
Chargeless and Verma modules for the Virasoro algebra
Fock space
su_n

There might also be the following:

recovery of Lie group from Lie algebra (Nilpotent Lie groups #26344 for nilpotent Lie groups)
connection between the quantum group and the Hall algebra (coming from representations of a quiver over F_q)

With this, one will be able to do basic computations, as well as compute things such as the lower central series (depending on the type).

Current ticket breakdown (more might follow):

Implement ABCs and categories for Lie algebras and finite dimensional Lie algebras given by structure coefficients #16820 - the categories and the basic framework and finite dimensional given by structure coefficients
Implement subalgebras of a Lie algebra #17416, Ideals and subalgebras of finite dimensional Lie algebras #26078 - subalgebras
Implement classical Lie algebras #16821 - classical Lie algebras
Implement PBW bases for the universal enveloping algebra #16822 - PBW basis for the universal enveloping algebra
Implement the free Lie algebra #16823 - the free Lie algebra (follow-up Hall basis of a free Lie algebra #25166)
Implement ideals, quotients, and finitely presented Lie algebras #16824 - quotients (Quotients of finite dimensional Lie algebras #26079 for quotients of finite-dimensional Lie algebras) and finitely presented Lie algebras
Implement untwisted affine Lie algebras #16825 - untwisted affine Lie algebras
Implement Kac-Moody algebras #16826 - Kac-Moody algebras
Implement Lie algebra (co)homology #22883 - Lie algebra (co)homology
Implement representations for Virasoro algebra #23037 - Chargeless and Verma modules for the Virasoro algebra
Add a method to contruct a finite-dimensional Lie algebra as a FiniteDimensionalAlgebra #22949 - Add method to construct a FiniteDimensionalAlgebra isomorphic to the Lie algebra
Implement Verma modules #23517 - Verma modules
Implement coercions between classical Lie algebras #25013 - Add coercions between classical Lie algebras
Implement interface between Lie algebras and gap #16860 - Interface with Gap (not part of the current branch)
Implement interface between Lie algebras and LiE #16861 - Interface with LiE (not part of the current branch)
Lie algebra morphism defined by values on a Lie generating subset #26036 - Morphisms defined only on generators
Implement compact real forms for matrix Lie algebras #28149 - Compact real forms (including su_n)

Bug tickets: #25007, #25008, #33836, #34006

Quantum group tickets:

Implement quantum group q-numbers #25067 - q-analogs for quantum groups
Implement modules for quantum (affine) minuscule and adjoint representations #25068 - minuscule and adjoint representations for finite/affine quantum groups
Implement tensor products of quantum group representations #25069 - tensor products of quantum group representations
Implement Fock space #15508 - Fock space and LLT algorithm for type A_n⁽¹⁾
Implement quantum groups by wrapping GAP's QuaGroup package #22623 - Quantum groups using GAP's QuaGroup package
Implement global bases from a crystal #16056 - Global crystal bases

Other related tickets:

Implement Jordan algebras #16055 - Jordan algebras
Clifford algebras and differential Weyl algebras #15300 - Weyl and Clifford algebras
Implement direct sum and tensor products for chain complexes and Koszul complexes #16851 - Koszul complexes
Implement Yangian for gl(n) #15484 - Yangians for gl_n
Implement fusion algebras #15485 - WZW fusion algebras
Implement derivation module basis for algebras #22900 - Methods to construct the derivation algebra
Symplectic derivation Lie algebra #30744 - Symplectic derivation Lie algebra
Implement universal commutative algebra of a finite-dimensional Lie algebra #30236 - Universal commutative algebra

Nilpotent Lie algebras tickets:

Subcategories and base class for nilpotent Lie algebras #26074 - Subcategories and base class for nilpotent Lie algebras
Free nilpotent Lie algebras #26076 - Free nilpotent Lie algebras
Ideals and subalgebras of finite dimensional Lie algebras #26078 - Ideals of finite dimensional Lie algebras
Quotients of finite dimensional Lie algebras #26079 - Quotients of finite dimensional Lie algebras
The Baker-Campbell-Hausdorff formula for nilpotent Lie algebras #26080 - The Baker-Campbell-Hausdorff formula for nilpotent Lie algebras
Zeroth Tanaka prolongation of free nilpotent Lie algebras #26081 - Zeroth Tanaka prolongation of free nilpotent Lie algebras
Nilpotent Lie groups #26344 - Nilpotent Lie groups

Depends on #16824
Depends on #16826

CC: @sagetrac-sage-combinat @anneschilling @nthiery @darijgr @bsalisbury1 @dwbump @simon-king-jena

Component: algebra

Keywords: Lie algebras, Kac Moody, days54, fpsac2019

Author: Travis Scrimshaw

Branch/Commit: public/algebras/lie_algebras-14901 @ b8bcd7d

Issue created by migration from https://trac.sagemath.org/ticket/14901

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

tscrim · 2013-07-20T14:52:32Z

comment:4

Other (poosible) additions:

LLT basis and Fock spaces for U_q(sl_n) (arbitrary g?)
implementation of the quantum group
connection between the quantum group and the Hall algebra (coming from representations of a quiver over F_q)
recovery of Lie group from Lie algebra (where it makes sense)
PBW(-type) bases of universal enveloping algebra (quantum group)
canonical bases of quantum groups

darijgr · 2013-07-20T15:04:28Z

comment:5

That's a hell of a project!

At some point, when linear algebra works in combinatorial free modules, I'll want (and probably make) the Lie algebra of primitives of a Hopf algebra.

tscrim · 2013-08-03T11:39:28Z

Changed dependencies from #10963 to #10963 #14898

tscrim · 2013-08-03T11:39:28Z

comment:6

I'm currently setting #14898 as a dependency since that will likely be done before this patch and because they should play together.

tscrim · 2013-09-03T19:25:13Z

Changed dependencies from #10963 #14898 to #10963 #14898 #15151

tscrim · 2013-09-03T19:25:13Z

comment:7

For defining the action on Fock space needs #15151.

tscrim · 2013-10-16T23:06:46Z

Changed dependencies from #10963 #14898 #15151 to #10963 #14898 #15151 #15289

tscrim · 2013-10-16T23:06:46Z

comment:8

To do PBW(-type) bases, I'll need #15289.

tscrim · 2013-11-05T08:43:54Z

Attachment: trac_14901-lie_algebras-ts.patch.gz

tscrim · 2013-11-05T09:02:09Z

comment:9

Okay, here's the current version of the patch for reference for those don't want to get/use the combinat queue. It does about 80% or so of the functionality I want, but it probably could use some restructuring. It's also missing a lot of documentation (most importantly doctests). Here's what's done and (mostly) working:

Free Lie algebras in the Hall basis
The Lyndon basis for the free Lie algebra
Abelian Lie algebras
Lie algebras from an associative algebra
Lie algebras from structure coefficients
Finite type Lie algebras
- As matrices for types ABCD
- In the Chevalley basis
gl_n
The Lie algebra of strictly upper triangular matrices
The Lie algebra of upper triangular matrices
Untwisted affine Lie algebras constructed from a finite type
Untwisted affine Kac-Moody Lie algebras (i.e. the above and the Lie derivative) [done]
Universal enveloping algebras
Kac-Moody algebras based only on a (generalized) Cartan matrix
LLT basis and Fock spaces for U_q(sl_n) [done]
PBW(-type) bases of universal enveloping algebra (quantum group) [done for Lie algebras up to abstracting internal structure]
canonical bases of quantum groups [90% done]
the Heisenberg Lie algebras and some other misc examples
quotient, sub, and direct sum Lie algebras and Lie algebra ideals [doesn't always seem to reduce properly]

Everything else:

twisted affine Lie and Kac-Moody algebras via loop groups/diagram automorphims [I want to do this, but haven't started and might push to a later ticket]
the Goodman and Wenzl modified LLT algorithm [haven't started yet]
polynomial representations for sl_n [doesn't give the correct results]
implementation of quantum groups [65% done]
connection between the quantum group and the Hall algebra (coming from representations of a quiver over F_q) [I would need to understand Hall algebras better to do this]
su_n [possibly to be dropped as the Sage infrastructure does not seem to be there]
recovery of Lie group from Lie algebra (where it makes sense) [0% done, likely to be dropped]

tscrim · 2013-11-08T20:49:42Z

Changed dependencies from #10963 #14898 #15151 #15289 to #10963 #14898 #15151 #15289 #15384

tscrim · 2013-11-08T20:49:42Z

Branch: public/algebras/lie_algebras-14901

sagetrac-git · 2013-11-08T20:50:41Z

Commit: 9d33268

sagetrac-git · 2013-11-08T20:50:41Z

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

`[9d33268](https://github.com/sagemath/sagetrac-mirror/commit/9d33268)`	`#14901: Initial implementation of Lie algebras.`
`[db19b7c](https://github.com/sagemath/sagetrac-mirror/commit/db19b7c)`	`Merge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901`
`[643bfcb](https://github.com/sagemath/sagetrac-mirror/commit/643bfcb)`	`Merge branch 'public/combinat/root_systems/improvements' into public/algebras/lie_algebras-14901`
`[362fd5e](https://github.com/sagemath/sagetrac-mirror/commit/362fd5e)`	`# Tue Oct 29 20:14:19 2013 +0100`
`[b2914f3](https://github.com/sagemath/sagetrac-mirror/commit/b2914f3)`	`# Sun Oct 27 13:58:49 2013 +0100`
`[9d9cae3](https://github.com/sagemath/sagetrac-mirror/commit/9d9cae3)`	`# Sat Oct 19 11:50:04 2013 +0200`
`[0251a33](https://github.com/sagemath/sagetrac-mirror/commit/0251a33)`	`Trac #13394: Implement faster and safer WeakValueDictionary`
`[c1b5afe](https://github.com/sagemath/sagetrac-mirror/commit/c1b5afe)`	`#15289: Implemented indexed monoids and groups.`
`[6fd33b2](https://github.com/sagemath/sagetrac-mirror/commit/6fd33b2)`	`#15169: Fix FreeAlgebra element constructor from a base field.`
`[0fa598a](https://github.com/sagemath/sagetrac-mirror/commit/0fa598a)`	`imported patch root_system_data-ts.patch`

tscrim · 2013-11-09T00:12:09Z

Changed keywords from Lie algebras Kac Moody to Lie algebras Kac Moody, days54

kcrisman · 2013-11-13T02:13:59Z

comment:13

You might find this link interesting and relevant. Apparently this person implemented quite a bit...

tscrim · 2013-11-13T17:39:32Z

comment:14

Replying to @kcrisman:

You might find this link interesting and relevant. Apparently this person implemented quite a bit...

Thanks. That has the branching rules and operations on the algebras themselves, which is useful, than doing computations within them.

Also as a reference for myself: http://coropa.sourceforge.net/

sagetrac-git · 2013-11-18T07:26:51Z

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

`[658e41d](https://github.com/sagemath/sagetrac-mirror/commit/658e41d)`	`Merge branch 'master' into public/algebras/lie_algebras-14901`

sagetrac-git · 2013-11-18T07:26:51Z

Changed commit from 9d33268 to 658e41d

sagetrac-git · 2013-11-19T02:07:39Z

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

`[efecd11](https://github.com/sagemath/sagetrac-mirror/commit/efecd11)`	`Fixes to printing of PBW basis elements.`
`[eec85ad](https://github.com/sagemath/sagetrac-mirror/commit/eec85ad)`	`Merge branch 'public/combinat/root_systems/improvements-15384' into public/algebras/lie_algebras-14901`
`[b4c7865](https://github.com/sagemath/sagetrac-mirror/commit/b4c7865)`	`Merge branch 'master' into public/combinat/root_systems/improvements-15384`
`[e40f892](https://github.com/sagemath/sagetrac-mirror/commit/e40f892)`	`Changed horizontal to basic_untwisted.`
`[930782b](https://github.com/sagemath/sagetrac-mirror/commit/930782b)`	`Fixed indentation error.`
`[f87789d](https://github.com/sagemath/sagetrac-mirror/commit/f87789d)`	`Merge branch 'master' into public/combinat/root_systems/improvements-15384`

sagetrac-git · 2013-11-19T02:07:39Z

Changed commit from 658e41d to efecd11

sagetrac-git · 2013-12-03T19:59:35Z

Changed commit from efecd11 to 4466b12

sagetrac-git · 2013-12-03T19:59:35Z

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

`[4466b12](https://github.com/sagemath/sagetrac-mirror/commit/4466b12)`	`Merge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901`
`[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`

videlec · 2018-08-03T19:20:18Z

comment:53

update milestone 8.3 -> 8.4

ehaka · 2018-08-17T07:41:03Z

comment:55

Added a subsection of tickets related to various constructions related to nilpotent Lie algebras. I have some existing implementations for these in a codebase that needs to be cleaned up and refactored in order to be imported to Sage.

tscrim · 2019-07-10T07:14:20Z

Changed keywords from Lie algebras, Kac Moody, days54 to Lie algebras, Kac Moody, days54, fpsac2019

tscrim · 2022-06-17T02:27:05Z

Changed dependencies from #10963 #14898 #15151 #15289 #15384 #15827 #16820 #16821 #16822 #16823 #16824 #16825 #16826 #22883 #23037 #23517 to #16824 #16826

Lie algebras #14901

Lie algebras #14901

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