#14901: Lie algebras
-------------------------------------+-------------------------------------
Reporter: tscrim | Owner: sage-combinat
Type: task | Status: new
Priority: major | Milestone: sage-6.4
Component: algebra | Resolution:
Keywords: Lie algebras, Kac | Merged in:
Moody, days54 | Reviewers:
Authors: Travis Scrimshaw | Work issues:
Report Upstream: N/A | Commit:
Branch: | b8bcd7d206444e25d892d1821a1501851357be1b
public/algebras/lie_algebras-14901 | Stopgaps:
Dependencies: #10963 #14898 |
#15151 #15289 #15384 #15827 |
#16819 #16820 #16821 #16822 |
#16823 #16824 #16825 #16826 |
-------------------------------------+-------------------------------------
Description changed by tscrim:
Old description:
> 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
>
> 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 F,,q,,)
>
> 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):
>
> * #16819 - the categories
> * #16820 - the basic framework and finite dimensional given by structure
> coefficients
> * #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
> * #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 A,,n,,^(1)^
> * #16055 - Jordan algebras
> * #15300 - Weyl and Clifford algebras
> * #16056 - Global crystal bases
> * #16851 - Koszul complexes
New description:
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
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 F,,q,,)
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):
* #16819 - the categories
* #16820 - the basic framework and finite dimensional given by structure
coefficients
* #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
* #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 A,,n,,^(1)^
* #16055 - Jordan algebras
* #15300 - Weyl and Clifford algebras
* #16056 - Global crystal bases
* #16851 - Koszul complexes
* #15484 - Yangians for '''gl''',,n,,
* #15485 - WZW fusion algebras
--
--
Ticket URL: <http://trac.sagemath.org/ticket/14901#comment:39>
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