Opened 9 years ago

Last modified 2 months ago

## #14901 new task

# Lie algebras — at Version 1

Reported by: | tscrim | Owned by: | sage-combinat |
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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: | Commit: | ||

Dependencies: | #10963 | Stopgaps: |

### Description (last modified by )

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 + 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).

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