#14901: Lie algebras
-------------------------------------+-------------------------------------
Reporter: tscrim | Owner: sage-combinat
Type: task | Status: new
Priority: major | Milestone: sage-5.13
Component: algebra | Resolution:
Keywords: Lie algebras Kac | Merged in:
Moody, days54 | Reviewers:
Authors: Travis Scrimshaw | Work issues:
Report Upstream: N/A | Commit:
Branch: | 84b241fc071155f21f29c7169b45a6505a02bb23
public/algebras/lie_algebras-14901 | Stopgaps:
Dependencies: #10963 #14898 |
#15151 #15289 #15384 |
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Changes (by tscrim):
* type: enhancement => task
Old description:
> Initial implementation of Lie algebras in sage.
>
> This will contain the following:
>
> - Free Lie algebras in the Hall 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,,
> - 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)
> - Universal enveloping algebras
>
> There might also be the following:
>
> - The Lyndon basis for the free Lie algebra
> - '''su''',,n,,
> - Kac-Moody algebras based only on a (generalized) Cartan matrix
>
> 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).
New description:
Initial implementation of Lie algebras in sage.
This will contain the following:
- Free Lie algebras in the Hall 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
- The Lyndon basis for the free Lie algebra
- 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).
--
Comment:
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.
--
Ticket URL: <http://trac.sagemath.org/ticket/14901#comment:19>
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