Opened 9 years ago

Last modified 8 weeks ago

#14901 new task

Lie algebras — at Version 21

Reported by: tscrim Owned by: sage-combinat
Priority: major Milestone: sage-feature
Component: algebra Keywords: Lie algebras, Kac Moody, days54, fpsac2019
Cc: sage-combinat, aschilling, nthiery, darij, bsalisbury1, bump, SimonKing Merged in:
Authors: Travis Scrimshaw Reviewers:
Report Upstream: N/A Work issues:
Branch: public/algebras/lie_algebras-14901 (Commits, GitHub, GitLab) Commit: d79c483ad3eed88edc5919e14284c9466a0b4998
Dependencies: #10963 #14898 #15151 #15289 #15384 Stopgaps:

Status badges

Description (last modified by tscrim)

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
  • gln
  • 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
  • Fock space #15508

There might also be the following:

  • sun
  • recovery of Lie group from Lie algebra
  • connection between the quantum group and the Hall algebra (coming from representations of a quiver over Fq)

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).

Change History (22)

comment:1 Changed 9 years ago by tscrim

  • Description modified (diff)

comment:2 Changed 9 years ago by darij

  • Cc darij added

comment:3 Changed 9 years ago by tscrim

  • Cc bsalisbury1 added

comment:4 Changed 9 years ago by tscrim

  • Description modified (diff)

Other (poosible) additions:

  • LLT basis and Fock spaces for Uq(sln) (arbitrary g?)
  • implementation of the quantum group
  • connection between the quantum group and the Hall algebra (coming from representations of a quiver over Fq)
  • 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

comment:5 Changed 9 years ago by darij

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.

comment:6 Changed 9 years ago by tscrim

  • Dependencies changed from #10963 to #10963 #14898

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

comment:7 Changed 9 years ago by tscrim

  • Dependencies changed from #10963 #14898 to #10963 #14898 #15151

For defining the action on Fock space needs #15151.

comment:8 Changed 9 years ago by tscrim

  • Dependencies changed from #10963 #14898 #15151 to #10963 #14898 #15151 #15289

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

Changed 9 years ago by tscrim

comment:9 Changed 9 years ago by tscrim

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
  • gln
  • 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 Uq(sln) [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 sln [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 Fq) [I would need to understand Hall algebras better to do this]
  • sun [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]
Last edited 9 years ago by tscrim (previous) (diff)

comment:10 Changed 9 years ago by tscrim

  • Branch set to public/algebras/lie_algebras-14901
  • Dependencies changed from #10963 #14898 #15151 #15289 to #10963 #14898 #15151 #15289 #15384

comment:11 Changed 9 years ago by git

  • Commit set to 9d3326862166e28f7424271a7c8a5d18dedb7e13

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

9d33268#14901: Initial implementation of Lie algebras.
db19b7cMerge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901
643bfcbMerge branch 'public/combinat/root_systems/improvements' into public/algebras/lie_algebras-14901
362fd5e# Tue Oct 29 20:14:19 2013 +0100
b2914f3# Sun Oct 27 13:58:49 2013 +0100
9d9cae3# Sat Oct 19 11:50:04 2013 +0200
0251a33Trac #13394: Implement faster and safer WeakValueDictionary?
c1b5afe#15289: Implemented indexed monoids and groups.
6fd33b2#15169: Fix FreeAlgebra? element constructor from a base field.
0fa598aimported patch root_system_data-ts.patch

comment:12 Changed 9 years ago by tscrim

  • Keywords days54 added

comment:13 follow-up: Changed 9 years ago by kcrisman

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

comment:14 in reply to: ↑ 13 Changed 9 years ago by tscrim

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/

comment:15 Changed 9 years ago by git

  • Commit changed from 9d3326862166e28f7424271a7c8a5d18dedb7e13 to 658e41d5b948fba42eb55a12edcca8880502716e

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

658e41dMerge branch 'master' into public/algebras/lie_algebras-14901

comment:16 Changed 9 years ago by git

  • Commit changed from 658e41d5b948fba42eb55a12edcca8880502716e to efecd11e40e7528972576cbcf9edabdf6ac449af

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

efecd11Fixes to printing of PBW basis elements.
eec85adMerge branch 'public/combinat/root_systems/improvements-15384' into public/algebras/lie_algebras-14901
b4c7865Merge branch 'master' into public/combinat/root_systems/improvements-15384
e40f892Changed horizontal to basic_untwisted.
930782bFixed indentation error.
f87789dMerge branch 'master' into public/combinat/root_systems/improvements-15384

comment:17 Changed 9 years ago by git

  • Commit changed from efecd11e40e7528972576cbcf9edabdf6ac449af to 4466b12a784901161b4700ebb06b4120022cf7d5

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

4466b12Merge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901
9dca526Added comparison operations.
a493beeMerge branch 'master' into public/monoids/15289-indexed

comment:18 Changed 9 years ago by git

  • Commit changed from 4466b12a784901161b4700ebb06b4120022cf7d5 to 84b241fc071155f21f29c7169b45a6505a02bb23

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

84b241fMerge branch 'public/functorial_constructions-10963' into public/algebras/lie_algebras-14901
80d55femerging sage/categories/category.py

comment:19 Changed 9 years ago by tscrim

  • Description modified (diff)
  • Type changed from enhancement to task

I've separated out Fock space into #15508. More splittings to come.

I'd also like to get Lie cohomology and connections with the exterior algebra (#15300) and De Rahm cohomology done as well.

comment:20 Changed 9 years ago by git

  • Commit changed from 84b241fc071155f21f29c7169b45a6505a02bb23 to d79c483ad3eed88edc5919e14284c9466a0b4998

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

d79c483Merge branch 'public/combinat/root_systems/improvements-15384' into public/algebras/lie_algebras-14901
585729aMerge branch 'public/modules/fock_space' into public/algebras/lie_algebras-14901
702f96aMerge branch 'u/tscrim/lie_algebras-14901' of trac.sagemath.org:sage into public/algebras/lie_algebras-14901
4548a5bMerge branch 'public/modules/fock_space' of trac.sagemath.org:sage into public/modules/fock_space
31b1cf2Finished GW algorithm and finalized files.
0374b6fMerge branch 'master' into public/algebras/lie_algebras-14901
20af672Merge branch 'master' into public/modules/fock_space
1d8057aMerge branch 'u/tscrim/lie_algebras-14901' of ssh://trac.sagemath.org:2222/sage into public/algebras/lie_algebras-14901
63ac373Tweaks to Virasoro algebra documentation.
90424b3More work on GW algorithm.

comment:21 Changed 9 years ago by tscrim

  • Description modified (diff)
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