Opened 9 years ago

Last modified 2 months ago

#14901 new task

Lie algebras — at Version 4

Reported by: tscrim Owned by: sage-combinat
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:

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Description (last modified by tscrim)

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
  • gln
  • 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
  • sun
  • 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).

Change History (4)

comment:1 Changed 9 years ago by tscrim

  • Description modified (diff)

comment:2 Changed 9 years ago by darij

  • Cc darij added

comment:3 Changed 9 years ago by tscrim

  • Cc bsalisbury1 added

comment:4 Changed 9 years ago by tscrim

  • Description modified (diff)

Other (poosible) additions:

  • LLT basis and Fock spaces for Uq(sln) (arbitrary g?)
  • implementation of the quantum group
  • connection between the quantum group and the Hall algebra (coming from representations of a quiver over Fq)
  • recovery of Lie group from Lie algebra (where it makes sense)
  • PBW(-type) bases of universal enveloping algebra (quantum group)
  • canonical bases of quantum groups
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