#14901: Lie algebras
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Reporter: tscrim | Owner: sage-
Type: enhancement | combinat
Priority: major | Status: new
Component: algebra | Milestone: sage-5.13
Keywords: Lie algebras Kac Moody | Resolution:
Authors: Travis Scrimshaw | Merged in:
Report Upstream: N/A | Reviewers:
Branch: | Work issues:
Dependencies: #10963 #14898 #15151 #15289 | Commit:
| Stopgaps:
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Comment (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
- '''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
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]
--
Ticket URL: <http://trac.sagemath.org/ticket/14901#comment:9>
Sage <http://www.sagemath.org>
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