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: |
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
- 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
- Fock space #15508
There might also be the following:
- su_{n}
- 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
- 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
- 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]
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 9 years ago by
- Commit changed from efecd11e40e7528972576cbcf9edabdf6ac449af to 4466b12a784901161b4700ebb06b4120022cf7d5
comment:18 Changed 9 years ago by
- Commit changed from 4466b12a784901161b4700ebb06b4120022cf7d5 to 84b241fc071155f21f29c7169b45a6505a02bb23
comment:19 Changed 9 years ago by
- Description modified (diff)
- Type changed from enhancement to task
comment:20 Changed 9 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 9 years ago by
- Description modified (diff)
Other (poosible) additions: