Opened 9 years ago
Last modified 3 years ago
#14901 new task
Lie algebras
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: | b8bcd7d206444e25d892d1821a1501851357be1b |
Dependencies: | #10963 #14898 #15151 #15289 #15384 #15827 #16820 #16821 #16822 #16823 #16824 #16825 #16826 #22883 #23037 #23517 | Stopgaps: |
Description (last modified by )
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
- sun
There might also be the following:
- recovery of Lie group from Lie algebra (#26344 for nilpotent Lie groups)
- 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, #26078 - subalgebras
- #16821 - classical Lie algebras
- #16822 - PBW basis for the universal enveloping algebra
- #16823 - the free Lie algebra (follow-up #25166)
- #16824 - quotients (#26079 for quotients of finite-dimensional Lie algebras) 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 - #23517 - Verma modules
- #25013 - Add coercions between classical Lie algebras
- #16860 - Interface with Gap (not part of the current branch)
- #16861 - Interface with LiE (not part of the current branch)
- #26036 - Morphisms defined only on generators
- #28149 - Compact real forms (including sun)
Quantum group tickets:
- #25067 -
q
-analogs for quantum groups - #25068 - minuscule and adjoint representations for finite/affine quantum groups
- #25069 - tensor products of quantum group representations
- #15508 - Fock space and LLT algorithm for type An(1)
- #22623 - Quantum groups using GAP's QuaGroup package
- #16056 - Global crystal bases
Other related tickets:
- #16055 - Jordan algebras
- #15300 - Weyl and Clifford algebras
- #16851 - Koszul complexes
- #15484 - Yangians for gln
- #15485 - WZW fusion algebras
- #22900 - Methods to construct the derivation algebra
Nilpotent Lie algebras tickets:
- #26074 - Subcategories and base class for nilpotent Lie algebras
- #26076 - Free nilpotent Lie algebras
- #26078 - Ideals of finite dimensional Lie algebras
- #26079 - Quotients of finite dimensional Lie algebras
- #26080 - The Baker-Campbell-Hausdorff formula for nilpotent Lie algebras
- #26081 - Zeroth Tanaka prolongation of free nilpotent Lie algebras
- #26344 - Nilpotent Lie groups
Attachments (1)
Change History (59)
comment:1 Changed 9 years ago by
- Description modified (diff)
comment:2 Changed 9 years ago by
- Cc darij added
comment:3 Changed 9 years ago by
- Cc bsalisbury1 added
comment:4 Changed 9 years ago by
- Description modified (diff)
comment:5 Changed 9 years ago by
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
- 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
- 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
- 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
comment:9 Changed 9 years ago by
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]
comment:10 Changed 9 years ago by
- 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
- Commit set to 9d3326862166e28f7424271a7c8a5d18dedb7e13
Branch pushed to git repo; I updated commit sha1. Last 10 new commits:
9d33268 | #14901: Initial implementation of Lie algebras. |
db19b7c | Merge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901 |
643bfcb | Merge 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 |
0251a33 | Trac #13394: Implement faster and safer WeakValueDictionary? |
c1b5afe | #15289: Implemented indexed monoids and groups. |
6fd33b2 | #15169: Fix FreeAlgebra? element constructor from a base field. |
0fa598a | imported patch root_system_data-ts.patch |
comment:12 Changed 9 years ago by
- Keywords days54 added
comment:13 follow-up: ↓ 14 Changed 9 years ago by
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
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
- Commit changed from 9d3326862166e28f7424271a7c8a5d18dedb7e13 to 658e41d5b948fba42eb55a12edcca8880502716e
Branch pushed to git repo; I updated commit sha1. New commits:
658e41d | Merge branch 'master' into public/algebras/lie_algebras-14901 |
comment:16 Changed 9 years ago by
- Commit changed from 658e41d5b948fba42eb55a12edcca8880502716e to efecd11e40e7528972576cbcf9edabdf6ac449af
Branch pushed to git repo; I updated commit sha1. New commits:
efecd11 | Fixes to printing of PBW basis elements. |
eec85ad | Merge branch 'public/combinat/root_systems/improvements-15384' into public/algebras/lie_algebras-14901 |
b4c7865 | Merge branch 'master' into public/combinat/root_systems/improvements-15384 |
e40f892 | Changed horizontal to basic_untwisted. |
930782b | Fixed indentation error. |
f87789d | Merge branch 'master' into public/combinat/root_systems/improvements-15384 |
comment:17 Changed 8 years ago by
- Commit changed from efecd11e40e7528972576cbcf9edabdf6ac449af to 4466b12a784901161b4700ebb06b4120022cf7d5
comment:18 Changed 8 years ago by
- Commit changed from 4466b12a784901161b4700ebb06b4120022cf7d5 to 84b241fc071155f21f29c7169b45a6505a02bb23
comment:19 Changed 8 years ago by
- Description modified (diff)
- Type changed from enhancement to task
comment:20 Changed 8 years ago by
- Commit changed from 84b241fc071155f21f29c7169b45a6505a02bb23 to d79c483ad3eed88edc5919e14284c9466a0b4998
Branch pushed to git repo; I updated commit sha1. Last 10 new commits:
d79c483 | Merge branch 'public/combinat/root_systems/improvements-15384' into public/algebras/lie_algebras-14901 |
585729a | Merge branch 'public/modules/fock_space' into public/algebras/lie_algebras-14901 |
702f96a | Merge branch 'u/tscrim/lie_algebras-14901' of trac.sagemath.org:sage into public/algebras/lie_algebras-14901 |
4548a5b | Merge branch 'public/modules/fock_space' of trac.sagemath.org:sage into public/modules/fock_space |
31b1cf2 | Finished GW algorithm and finalized files. |
0374b6f | Merge branch 'master' into public/algebras/lie_algebras-14901 |
20af672 | Merge branch 'master' into public/modules/fock_space |
1d8057a | Merge branch 'u/tscrim/lie_algebras-14901' of ssh://trac.sagemath.org:2222/sage into public/algebras/lie_algebras-14901 |
63ac373 | Tweaks to Virasoro algebra documentation. |
90424b3 | More work on GW algorithm. |
comment:21 Changed 8 years ago by
- Description modified (diff)
comment:22 Changed 8 years ago by
Also for reference (and in case anyone can translate mathematica code into pseudocode or psuedo Sage) http://www.equaonline.com/math/SuperLie/.
comment:23 Changed 8 years ago by
- Milestone changed from sage-6.1 to sage-6.2
comment:24 Changed 8 years ago by
- Commit changed from d79c483ad3eed88edc5919e14284c9466a0b4998 to 3eb678411d781b02ea8a4aa80b4d96b7680c1850
Branch pushed to git repo; I updated commit sha1. Last 10 new commits:
6863447 | Merge branch 'master' into public/algebras/lie_algebras-14901
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a970464 | Work on finite dimensional Lie algebras with bases.
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b153849 | Merge branch 'master' into public/algebras/lie_algebras-14901
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494ad67 | Merge branch 'develop' into public/algebras/lie_algebras-14901
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4038bc8 | Starting work.
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b09a669 | Fixed implementation, now to documentation.
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2fdb307 | Started new file for PBW algebras and ideals.
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6fb3c0d | Merge branch 'develop' into public/algebras/fp_algebras
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d2d1019 | Refactoring PBW ideals and FP algebras.
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6213e71 | Merge branch 'develop' into public/algebras/fp_algebras
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comment:25 Changed 8 years ago by
- Cc bump added
comment:26 Changed 8 years ago by
- Commit changed from 3eb678411d781b02ea8a4aa80b4d96b7680c1850 to 4b66ac3a08008550ef1fac8e0fec92f18a54191a
Branch pushed to git repo; I updated commit sha1. Last 10 new commits:
a278afa | Added cardinality methods.
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f681606 | Added cardinality to free abelian monoid for consistancy.
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3a0e50b | Merge branch 'develop' into public/monoids/15289-indexed
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d4606cc | Added more robustness to element creation.
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991953a | Merge branch 'develop' into public/monoids/15289-indexed
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480188e | Merge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901
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cc0c2a5 | Merge branch 'develop' into public/algebras/lie_algebras-14901
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c3e7244 | Added are_linearly_dependent.
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b8ef03b | Merge branch 'public/linear_algebra/linear_dep_check-15827' into public/algebras/lie_algebras-14901
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4b66ac3 | Refactoring of Lie algebras with structure coefficients.
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comment:27 Changed 8 years ago by
- Dependencies changed from #10963 #14898 #15151 #15289 #15384 to #10963 #14898 #15151 #15289 #15384 #15827
comment:28 Changed 8 years ago by
- Commit changed from 4b66ac3a08008550ef1fac8e0fec92f18a54191a to f3562392e5f28d3db1a7f82c7b778bb97134b967
Branch pushed to git repo; I updated commit sha1. New commits:
f356239 | Fixes from refactoring and (hopefully) a working version of finitely presented algebras.
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comment:29 Changed 8 years ago by
- Cc SimonKing added
comment:30 Changed 8 years ago by
- Commit changed from f3562392e5f28d3db1a7f82c7b778bb97134b967 to 11c9bf3a512df00c991c051e69a89704dc76cb4f
Branch pushed to git repo; I updated commit sha1. Last 10 new commits:
7db4ef0 | Trac 10963: Fixed repr for the symmetric group algebra by improving a bit the generic repr methods in the categories
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367d72e | Trac 10963: fixed trivial doctest failure
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8919180 | Trac 10963: reverted change that was meant to be put in #15801
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d1b5374 | Trac 10963: reverted change that required #15801 for no good reason
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4580db1 | Trac 10963: fixed trivial doctest failures
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f44661f | Trac 15801: added repr in HeckeModule_free_module (the generic one in modules with basis can't handle it anymore)
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feab04a | manual merge with 6.2.beta8
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ce2193e | Merge branch 'public/ticket/10963-doc-distributive' of trac.sagemath.org:sage into public/ticket/10963-doc-distributive
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e5567c1 | Merge branch 'public/ticket/10963-doc-distributive' into public/algebras/lie_algebras-14901
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11c9bf3 | Cleaning up bad merging.
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comment:31 Changed 8 years ago by
- Commit changed from 11c9bf3a512df00c991c051e69a89704dc76cb4f to e30bf4eb0918544f3441707bd26452f79310adcc
Branch pushed to git repo; I updated commit sha1. New commits:
6de0121 | Tweaks to elements and struct coeffs.
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1c1b4eb | Changed monomial_cmp to generator_cmp and added free (static)method to monoids and groups category.
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47b5be7 | Removed __contains__ and fix monomial_cmp in indexed_monoid.py
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4097a8c | Merge branch 'develop' into public/monoids/15289-indexed
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374ae3d | Merge branch 'public/monoids/15289-indexed' into public/algebras/base_lie_algebras
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211e1c7 | Merge branch 'public/algebras/base_lie_algebras' into public/algebras/lie_algebras-14901
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23301a8 | Misc fixes and tweaks.
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e30bf4e | Merge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901
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comment:32 Changed 8 years ago by
- Milestone changed from sage-6.2 to sage-6.3
comment:33 Changed 8 years ago by
- Commit changed from e30bf4eb0918544f3441707bd26452f79310adcc to c738b02cdbc10b5e1f241426873687dfe126627a
comment:34 Changed 8 years ago by
- Commit changed from c738b02cdbc10b5e1f241426873687dfe126627a to 850ab5c582dbaa6f47aebc6ecdbb900bb57c186b
Branch pushed to git repo; I updated commit sha1. Last 10 new commits:
f9f8b9f | Merge branch 'public/combinat/root_systems/improvements-15384' into public/algebras/lie_algebras-14901
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56e3891 | Putting out some fires.
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329ea79 | Merge branch 'develop' into public/algebras/base_lie_algebras
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6da634b | Merge branch 'develop' into public/algebras/base_lie_algebras
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dc6747b | Better repr/latex and actually use IndexedGenerators.
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2346150 | Merge branch 'public/algebras/base_lie_algebras' into public/algebras/lie_algebras-14901
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4785df5 | Some fixes from the merge.
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21e7b0b | Removed s from affine Lie algebras module name.
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1edf001 | Merge branch 'public/algebras/base_lie_algebras' into public/algebras/lie_algebras-14901
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850ab5c | Putting out more fires. (Untwisted) Affine Lie algebras working agian.
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comment:35 Changed 8 years ago by
- Commit changed from 850ab5c582dbaa6f47aebc6ecdbb900bb57c186b to b8bcd7d206444e25d892d1821a1501851357be1b
Branch pushed to git repo; I updated commit sha1. Last 10 new commits:
ecdc00f | Fixed doctests because of different processing of input.
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909196e | Merge branch 'develop' into public/monoids/15289-indexed
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1db7c1d | Merge branch 'develop' into public/monoids/15289-indexed
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73a4e48 | Implemented other changes/improvements Nicolas suggested.
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b9f6f18 | Merge branch 'develop' into public/monoids/15289-indexed
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45ee958 | Merge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901
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399bb1e | Rebased over changes from indexed generators.
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ab4fa91 | Removed duplicate code and fixed ideals.
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469a5b2 | Putting out more fires.
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b8bcd7d | Merge branch 'public/algebras/lie_algebras-14901' of trac.sagemath.org:sage into public/algebras/lie_algebras-14901
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comment:36 Changed 8 years ago by
- Milestone changed from sage-6.3 to sage-6.4
comment:37 Changed 8 years ago by
- 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 8 years ago by
- Description modified (diff)
FYI - #16819 has been split off and is ready for review.
comment:39 Changed 8 years ago by
- Description modified (diff)
comment:40 Changed 7 years ago by
- Description modified (diff)
comment:41 Changed 7 years ago by
More references/information/future work:
Atlas of Lie groups and representations - http://www.liegroups.org/
comment:42 Changed 7 years ago by
- 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 5 years ago by
- Description modified (diff)
comment:44 Changed 5 years ago by
- 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 5 years ago by
- Description modified (diff)
- Milestone changed from sage-6.4 to sage-8.0
comment:46 Changed 5 years ago by
- Description modified (diff)
comment:47 Changed 5 years ago by
- 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.
comment:48 Changed 5 years ago by
- Description modified (diff)
- Milestone changed from sage-8.0 to sage-8.1
comment:49 Changed 5 years ago by
- Dependencies changed from #10963 #14898 #15151 #15289 #15384 #15827 #16820 #16821 #16822 #16823 #16824 #16825 #16826 #22883 #23037 to #10963 #14898 #15151 #15289 #15384 #15827 #16820 #16821 #16822 #16823 #16824 #16825 #16826 #22883 #23037 #23517
comment:50 Changed 4 years ago by
- Description modified (diff)
- Milestone changed from sage-8.1 to sage-8.2
comment:51 Changed 4 years ago by
- Description modified (diff)
comment:52 Changed 4 years ago by
- Description modified (diff)
- Milestone changed from sage-8.2 to sage-8.3
comment:53 Changed 4 years ago by
- Milestone changed from sage-8.3 to sage-8.4
update milestone 8.3 -> 8.4
comment:54 Changed 4 years ago by
- Description modified (diff)
comment:55 Changed 4 years ago by
- Description modified (diff)
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.
comment:56 Changed 4 years ago by
- Description modified (diff)
comment:57 Changed 4 years ago by
- Description modified (diff)
comment:58 Changed 3 years ago by
- Description modified (diff)
- Keywords fpsac2019 added
- Milestone changed from sage-8.4 to sage-feature
Other (poosible) additions: