Lie algebras #14901
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
- gln
- 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
- sun
There might also be the following:
- recovery of Lie group from Lie algebra (Nilpotent Lie groups #26344 for nilpotent Lie groups)
- 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).
Current ticket breakdown (more might follow):
- Implement ABCs and categories for Lie algebras and finite dimensional Lie algebras given by structure coefficients #16820 - the categories and the basic framework and finite dimensional given by structure coefficients
- Implement subalgebras of a Lie algebra #17416, Ideals and subalgebras of finite dimensional Lie algebras #26078 - subalgebras
- Implement classical Lie algebras #16821 - classical Lie algebras
- Implement PBW bases for the universal enveloping algebra #16822 - PBW basis for the universal enveloping algebra
- Implement the free Lie algebra #16823 - the free Lie algebra (follow-up Hall basis of a free Lie algebra #25166)
- Implement ideals, quotients, and finitely presented Lie algebras #16824 - quotients (Quotients of finite dimensional Lie algebras #26079 for quotients of finite-dimensional Lie algebras) and finitely presented Lie algebras
- Implement untwisted affine Lie algebras #16825 - untwisted affine Lie algebras
- Implement Kac-Moody algebras #16826 - Kac-Moody algebras
- Implement Lie algebra (co)homology #22883 - Lie algebra (co)homology
- Implement representations for Virasoro algebra #23037 - Chargeless and Verma modules for the Virasoro algebra
- Add a method to contruct a finite-dimensional Lie algebra as a FiniteDimensionalAlgebra #22949 - Add method to construct a
FiniteDimensionalAlgebraisomorphic to the Lie algebra - Implement Verma modules #23517 - Verma modules
- Implement coercions between classical Lie algebras #25013 - Add coercions between classical Lie algebras
- Implement interface between Lie algebras and gap #16860 - Interface with Gap (not part of the current branch)
- Implement interface between Lie algebras and LiE #16861 - Interface with LiE (not part of the current branch)
- Lie algebra morphism defined by values on a Lie generating subset #26036 - Morphisms defined only on generators
- Implement compact real forms for matrix Lie algebras #28149 - Compact real forms (including sun)
Bug tickets: #25007, #25008, #33836, #34006
Quantum group tickets:
- Implement quantum group q-numbers #25067 -
q-analogs for quantum groups - Implement modules for quantum (affine) minuscule and adjoint representations #25068 - minuscule and adjoint representations for finite/affine quantum groups
- Implement tensor products of quantum group representations #25069 - tensor products of quantum group representations
- Implement Fock space #15508 - Fock space and LLT algorithm for type An(1)
- Implement quantum groups by wrapping GAP's QuaGroup package #22623 - Quantum groups using GAP's QuaGroup package
- Implement global bases from a crystal #16056 - Global crystal bases
Other related tickets:
- Implement Jordan algebras #16055 - Jordan algebras
- Clifford algebras and differential Weyl algebras #15300 - Weyl and Clifford algebras
- Implement direct sum and tensor products for chain complexes and Koszul complexes #16851 - Koszul complexes
- Implement Yangian for gl(n) #15484 - Yangians for gln
- Implement fusion algebras #15485 - WZW fusion algebras
- Implement derivation module basis for algebras #22900 - Methods to construct the derivation algebra
- Symplectic derivation Lie algebra #30744 - Symplectic derivation Lie algebra
- Implement universal commutative algebra of a finite-dimensional Lie algebra #30236 - Universal commutative algebra
Nilpotent Lie algebras tickets:
- Subcategories and base class for nilpotent Lie algebras #26074 - Subcategories and base class for nilpotent Lie algebras
- Free nilpotent Lie algebras #26076 - Free nilpotent Lie algebras
- Ideals and subalgebras of finite dimensional Lie algebras #26078 - Ideals of finite dimensional Lie algebras
- Quotients of finite dimensional Lie algebras #26079 - Quotients of finite dimensional Lie algebras
- The Baker-Campbell-Hausdorff formula for nilpotent Lie algebras #26080 - The Baker-Campbell-Hausdorff formula for nilpotent Lie algebras
- Zeroth Tanaka prolongation of free nilpotent Lie algebras #26081 - Zeroth Tanaka prolongation of free nilpotent Lie algebras
- Nilpotent Lie groups #26344 - Nilpotent Lie groups
Depends on #16824
Depends on #16826
CC: @sagetrac-sage-combinat @anneschilling @nthiery @darijgr @bsalisbury1 @dwbump @simon-king-jena
Component: algebra
Keywords: Lie algebras, Kac Moody, days54, fpsac2019
Author: Travis Scrimshaw
Branch/Commit: public/algebras/lie_algebras-14901 @ b8bcd7d
Issue created by migration from https://trac.sagemath.org/ticket/14901