Opened 4 years ago

Last modified 5 weeks ago

#14901 new task

Lie algebras

Reported by: tscrim Owned by: sage-combinat
Priority: major Milestone: sage-8.0
Component: algebra Keywords: Lie algebras, Kac Moody, days54
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) Commit: b8bcd7d206444e25d892d1821a1501851357be1b
Dependencies: #10963 #14898 #15151 #15289 #15384 #15827 #16820 #16821 #16822 #16823 #16824 #16825 #16826 #22883 #23037 Stopgaps:

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
  • Lie algebra (co)homology
  • Chargeless and Verma modules for the Virasoro algebra
  • Fock space

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).

Current ticket breakdown (more might follow):

  • #16820 - the categories and the basic framework and finite dimensional given by structure coefficients
  • #17416 - subalgebras
  • #16821 - classical Lie algebras
  • #16822 - PBW basis for the universal enveloping algebra
  • #16823 - the free Lie algebra
  • #16824 - ideals, quotients, and finitely presented Lie algebras
  • #16825 - untwisted affine Lie algebras
  • #16826 - Kac-Moody algebras
  • #22883 - Lie algebra (co)homology
  • #23037 - Chargeless and Verma modules for the Virasoro algebra
  • #22949 - Add method to construct a FiniteDimensionalAlgebra isomorphic to the Lie algebra
  • #16860 - Interface with Gap (not part of the current branch)
  • #16861 - Interface with LiE (not part of the current branch)

Related tickets:

  • #15508 - Fock space and LLT algorithm for type An(1)
  • #16055 - Jordan algebras
  • #15300 - Weyl and Clifford algebras
  • #16056 - Global crystal bases
  • #16851 - Koszul complexes
  • #15484 - Yangians for gln
  • #15485 - WZW fusion algebras
  • #22623 - Quantum groups using GAP's QuaGroup package
  • #22900 - Methods to construct the derivation algebra

Attachments (1)

trac_14901-lie_algebras-ts.patch (331.2 KB) - added by tscrim 4 years ago.

Download all attachments as: .zip

Change History (48)

comment:1 Changed 4 years ago by tscrim

  • Description modified (diff)

comment:2 Changed 4 years ago by darij

  • Cc darij added

comment:3 Changed 4 years ago by tscrim

  • Cc bsalisbury1 added

comment:4 Changed 4 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 4 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 4 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 4 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 4 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 4 years ago by tscrim

comment:9 Changed 4 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 4 years ago by tscrim (previous) (diff)

comment:10 Changed 4 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 4 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 4 years ago by tscrim

  • Keywords days54 added

comment:13 follow-up: Changed 4 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 4 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 4 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 4 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 4 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 4 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 4 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 4 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 4 years ago by tscrim

  • Description modified (diff)

comment:22 Changed 4 years ago by tscrim

Also for reference (and in case anyone can translate mathematica code into pseudocode or pseudo Sage) http://www.equaonline.com/math/SuperLie/.

Last edited 4 years ago by tscrim (previous) (diff)

comment:23 Changed 3 years ago by vbraun_spam

  • Milestone changed from sage-6.1 to sage-6.2

comment:24 Changed 3 years ago by git

  • Commit changed from d79c483ad3eed88edc5919e14284c9466a0b4998 to 3eb678411d781b02ea8a4aa80b4d96b7680c1850

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

6863447Merge branch 'master' into public/algebras/lie_algebras-14901
a970464Work on finite dimensional Lie algebras with bases.
b153849Merge branch 'master' into public/algebras/lie_algebras-14901
494ad67Merge branch 'develop' into public/algebras/lie_algebras-14901
4038bc8Starting work.
b09a669Fixed implementation, now to documentation.
2fdb307Started new file for PBW algebras and ideals.
6fb3c0dMerge branch 'develop' into public/algebras/fp_algebras
d2d1019Refactoring PBW ideals and FP algebras.
6213e71Merge branch 'develop' into public/algebras/fp_algebras

comment:25 Changed 3 years ago by bump

  • Cc bump added

comment:26 Changed 3 years ago by git

  • Commit changed from 3eb678411d781b02ea8a4aa80b4d96b7680c1850 to 4b66ac3a08008550ef1fac8e0fec92f18a54191a

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

a278afaAdded cardinality methods.
f681606Added cardinality to free abelian monoid for consistancy.
3a0e50bMerge branch 'develop' into public/monoids/15289-indexed
d4606ccAdded more robustness to element creation.
991953aMerge branch 'develop' into public/monoids/15289-indexed
480188eMerge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901
cc0c2a5Merge branch 'develop' into public/algebras/lie_algebras-14901
c3e7244Added are_linearly_dependent.
b8ef03bMerge branch 'public/linear_algebra/linear_dep_check-15827' into public/algebras/lie_algebras-14901
4b66ac3Refactoring of Lie algebras with structure coefficients.

comment:27 Changed 3 years ago by tscrim

  • Dependencies changed from #10963 #14898 #15151 #15289 #15384 to #10963 #14898 #15151 #15289 #15384 #15827

comment:28 Changed 3 years ago by git

  • Commit changed from 4b66ac3a08008550ef1fac8e0fec92f18a54191a to f3562392e5f28d3db1a7f82c7b778bb97134b967

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

f356239Fixes from refactoring and (hopefully) a working version of finitely presented algebras.

comment:29 Changed 3 years ago by SimonKing

  • Cc SimonKing added

comment:30 Changed 3 years ago by git

  • Commit changed from f3562392e5f28d3db1a7f82c7b778bb97134b967 to 11c9bf3a512df00c991c051e69a89704dc76cb4f

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

7db4ef0Trac 10963: Fixed repr for the symmetric group algebra by improving a bit the generic repr methods in the categories
367d72eTrac 10963: fixed trivial doctest failure
8919180Trac 10963: reverted change that was meant to be put in #15801
d1b5374Trac 10963: reverted change that required #15801 for no good reason
4580db1Trac 10963: fixed trivial doctest failures
f44661fTrac 15801: added repr in HeckeModule_free_module (the generic one in modules with basis can't handle it anymore)
feab04amanual merge with 6.2.beta8
ce2193eMerge branch 'public/ticket/10963-doc-distributive' of trac.sagemath.org:sage into public/ticket/10963-doc-distributive
e5567c1Merge branch 'public/ticket/10963-doc-distributive' into public/algebras/lie_algebras-14901
11c9bf3Cleaning up bad merging.

comment:31 Changed 3 years ago by git

  • Commit changed from 11c9bf3a512df00c991c051e69a89704dc76cb4f to e30bf4eb0918544f3441707bd26452f79310adcc

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

6de0121Tweaks to elements and struct coeffs.
1c1b4ebChanged monomial_cmp to generator_cmp and added free (static)method to monoids and groups category.
47b5be7Removed __contains__ and fix monomial_cmp in indexed_monoid.py
4097a8cMerge branch 'develop' into public/monoids/15289-indexed
374ae3dMerge branch 'public/monoids/15289-indexed' into public/algebras/base_lie_algebras
211e1c7Merge branch 'public/algebras/base_lie_algebras' into public/algebras/lie_algebras-14901
23301a8Misc fixes and tweaks.
e30bf4eMerge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901

comment:32 Changed 3 years ago by vbraun_spam

  • Milestone changed from sage-6.2 to sage-6.3

comment:33 Changed 3 years ago by git

  • Commit changed from e30bf4eb0918544f3441707bd26452f79310adcc to c738b02cdbc10b5e1f241426873687dfe126627a

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

4f719e1Reverted FreeLieAlgebra error.
6993e66Fixes for removing of ngens() for morphisms.
c738b02Merge branch 'develop' into public/algebras/lie_algebras-14901

comment:34 Changed 3 years ago by git

  • Commit changed from c738b02cdbc10b5e1f241426873687dfe126627a to 850ab5c582dbaa6f47aebc6ecdbb900bb57c186b

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

f9f8b9fMerge branch 'public/combinat/root_systems/improvements-15384' into public/algebras/lie_algebras-14901
56e3891Putting out some fires.
329ea79Merge branch 'develop' into public/algebras/base_lie_algebras
6da634bMerge branch 'develop' into public/algebras/base_lie_algebras
dc6747bBetter repr/latex and actually use IndexedGenerators.
2346150Merge branch 'public/algebras/base_lie_algebras' into public/algebras/lie_algebras-14901
4785df5Some fixes from the merge.
21e7b0bRemoved s from affine Lie algebras module name.
1edf001Merge branch 'public/algebras/base_lie_algebras' into public/algebras/lie_algebras-14901
850ab5cPutting out more fires. (Untwisted) Affine Lie algebras working agian.

comment:35 Changed 3 years ago by git

  • Commit changed from 850ab5c582dbaa6f47aebc6ecdbb900bb57c186b to b8bcd7d206444e25d892d1821a1501851357be1b

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

ecdc00fFixed doctests because of different processing of input.
909196eMerge branch 'develop' into public/monoids/15289-indexed
1db7c1dMerge branch 'develop' into public/monoids/15289-indexed
73a4e48Implemented other changes/improvements Nicolas suggested.
b9f6f18Merge branch 'develop' into public/monoids/15289-indexed
45ee958Merge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901
399bb1eRebased over changes from indexed generators.
ab4fa91Removed duplicate code and fixed ideals.
469a5b2Putting out more fires.
b8bcd7dMerge branch 'public/algebras/lie_algebras-14901' of trac.sagemath.org:sage into public/algebras/lie_algebras-14901

comment:36 Changed 3 years ago by vbraun_spam

  • Milestone changed from sage-6.3 to sage-6.4

comment:37 Changed 3 years ago by tscrim

  • Dependencies changed from #10963 #14898 #15151 #15289 #15384 #15827 to #10963 #14898 #15151 #15289 #15384 #15827 #16819 #16820 #16821 #16822 #16823 #16824 #16825 #16826
  • Description modified (diff)
  • Keywords changed from Lie algebras Kac Moody, days54 to Lie algebras, Kac Moody, days54

comment:38 Changed 3 years ago by tscrim

  • Description modified (diff)

FYI - #16819 has been split off and is ready for review.

comment:39 Changed 3 years ago by tscrim

  • Description modified (diff)

comment:40 Changed 3 years ago by tscrim

  • Description modified (diff)

comment:41 Changed 2 years ago by tscrim

More references/information/future work:

Atlas of Lie groups and representations - http://www.liegroups.org/

comment:42 Changed 2 years ago by tscrim

  • Dependencies changed from #10963 #14898 #15151 #15289 #15384 #15827 #16819 #16820 #16821 #16822 #16823 #16824 #16825 #16826 to #10963 #14898 #15151 #15289 #15384 #15827 #16820 #16821 #16822 #16823 #16824 #16825 #16826
  • Description modified (diff)

comment:43 Changed 3 months ago by tscrim

  • Description modified (diff)

comment:44 Changed 8 weeks ago by tscrim

  • Dependencies changed from #10963 #14898 #15151 #15289 #15384 #15827 #16820 #16821 #16822 #16823 #16824 #16825 #16826 to #10963 #14898 #15151 #15289 #15384 #15827 #16820 #16821 #16822 #16823 #16824 #16825 #16826 #22883
  • Description modified (diff)

Lie algebra (co)homology is done in #22883. It is not the most efficient implementation and doesn't seem work very well for Lie algebras of dimension larger than 10 due to the amount of information needed to be computed (i.e., large dimensional vector spaces).

comment:45 Changed 8 weeks ago by tscrim

  • Description modified (diff)
  • Milestone changed from sage-6.4 to sage-8.0

comment:46 Changed 7 weeks ago by tscrim

  • Description modified (diff)

comment:47 Changed 5 weeks ago by tscrim

  • Dependencies changed from #10963 #14898 #15151 #15289 #15384 #15827 #16820 #16821 #16822 #16823 #16824 #16825 #16826 #22883 to #10963 #14898 #15151 #15289 #15384 #15827 #16820 #16821 #16822 #16823 #16824 #16825 #16826 #22883 #23037
  • Description modified (diff)

#23037 adds a little bit more to the representation theory in Sage. Good follow-ups will include trivial, adjoint, and tensor product representations for Lie algebras.

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