id,summary,reporter,owner,description,type,status,priority,milestone,component,resolution,keywords,cc,merged,author,reviewer,upstream,work_issues,branch,commit,dependencies,stopgaps
14901,Lie algebras,tscrim,sage-combinat,"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
- Lie algebra (co)homology
- Chargeless and Verma modules for the Virasoro algebra
- Fock space
There might also be the following:
- '''su''',,n,,
- 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 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):
* #16820 - the categories and the basic framework and finite dimensional given by structure coefficients
* #17416 - subalgebras
* #16821 - classical Lie algebras
* #16822 - PBW basis for the universal enveloping algebra
* #16823 - the free Lie algebra (follow-up #25166)
* #16824 - ideals, quotients, 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
Bug tickets: #25007, #25008
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 A,,n,,^(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 '''gl''',,n,,
* #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",task,new,major,sage-8.4,algebra,,"Lie algebras, Kac Moody, days54",sage-combinat aschilling nthiery darij bsalisbury1 bump SimonKing,,Travis Scrimshaw,,N/A,,public/algebras/lie_algebras-14901,b8bcd7d206444e25d892d1821a1501851357be1b,#10963 #14898 #15151 #15289 #15384 #15827 #16820 #16821 #16822 #16823 #16824 #16825 #16826 #22883 #23037 #23517,