Lie algebras

[tscrim]

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
• su_n

There might also be the following:

• 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, #26078 - subalgebras
• #16821 - classical Lie algebras
• #16822 - PBW basis for the universal enveloping algebra
• #16823 - the free Lie algebra (follow-up #25166)
• #16824 - quotients (#26079 for quotients of finite-dimensional Lie algebras) 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
• #28149 - Compact real forms (including su_n)

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

[tscrim]

Other (poosible) additions:

• LLT basis and Fock spaces for U_q(sl_n) (arbitrary g?)
• implementation of the quantum group
• connection between the quantum group and the Hall algebra (coming from representations of a quiver over F_q)
• 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

[darij]

That's a hell of a project!

At some point, when linear algebra works in combinatorial free modules, I'll want (and probably make) the Lie algebra of primitives of a Hopf algebra.

[tscrim]

I'm currently setting #14898 as a dependency since that will likely be done before this patch and because they should play together.

[tscrim]

For defining the action on Fock space needs #15151.

[tscrim]

To do PBW(-type) bases, I'll need #15289.

[tscrim]

Okay, here's the current version of the patch for reference for those don't want to get/use the combinat queue. It does about 80% or so of the functionality I want, but it probably could use some restructuring. It's also missing a lot of documentation (most importantly doctests). Here's what's done and (mostly) working:

• Free Lie algebras in the Hall basis
• The Lyndon basis for the free Lie algebra
• 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 and the Lie derivative) [done]
• Universal enveloping algebras
• Kac-Moody algebras based only on a (generalized) Cartan matrix
• LLT basis and Fock spaces for U_q(sl_n) [done]
• PBW(-type) bases of universal enveloping algebra (quantum group) [done for Lie algebras up to abstracting internal structure]
• canonical bases of quantum groups [90% done]
• the Heisenberg Lie algebras and some other misc examples
• quotient, sub, and direct sum Lie algebras and Lie algebra ideals [doesn't always seem to reduce properly]

Everything else:

• twisted affine Lie and Kac-Moody algebras via loop groups/diagram automorphims [I want to do this, but haven't started and might push to a later ticket]
• the Goodman and Wenzl modified LLT algorithm [haven't started yet]
• polynomial representations for sl_n [doesn't give the correct results]
• implementation of quantum groups [65% done]
• connection between the quantum group and the Hall algebra (coming from representations of a quiver over F_q) [I would need to understand Hall algebras better to do this]
• su_n [possibly to be dropped as the Sage infrastructure does not seem to be there]
• recovery of Lie group from Lie algebra (where it makes sense) [0% done, likely to be dropped]

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​9d33268	#14901: Initial implementation of Lie algebras.
​db19b7c	Merge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901
​643bfcb	Merge branch 'public/combinat/root_systems/improvements' into public/algebras/lie_algebras-14901
​362fd5e	# Tue Oct 29 20:14:19 2013 +0100
​b2914f3	# Sun Oct 27 13:58:49 2013 +0100
​9d9cae3	# Sat Oct 19 11:50:04 2013 +0200
​0251a33	Trac #13394: Implement faster and safer WeakValueDictionary?
​c1b5afe	#15289: Implemented indexed monoids and groups.
​6fd33b2	#15169: Fix FreeAlgebra? element constructor from a base field.
​0fa598a	imported patch root_system_data-ts.patch

[kcrisman]

You might find ​this link interesting and relevant. Apparently this person implemented quite a bit...

[tscrim]

Replying to kcrisman:

You might find ​this link interesting and relevant. Apparently this person implemented quite a bit...

Thanks. That has the branching rules and operations on the algebras themselves, which is useful, than doing computations within them.

Also as a reference for myself: ​http://coropa.sourceforge.net/

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​658e41d

Merge branch 'master' into public/algebras/lie_algebras-14901

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​efecd11	Fixes to printing of PBW basis elements.
​eec85ad	Merge branch 'public/combinat/root_systems/improvements-15384' into public/algebras/lie_algebras-14901
​b4c7865	Merge branch 'master' into public/combinat/root_systems/improvements-15384
​e40f892	Changed horizontal to basic_untwisted.
​930782b	Fixed indentation error.
​f87789d	Merge branch 'master' into public/combinat/root_systems/improvements-15384

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​4466b12	Merge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901
​9dca526	Added comparison operations.
​a493bee	Merge branch 'master' into public/monoids/15289-indexed

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​84b241f	Merge branch 'public/functorial_constructions-10963' into public/algebras/lie_algebras-14901
​80d55fe	merging sage/categories/category.py

[tscrim]

I've separated out Fock space into #15508. More splittings to come.

I'd also like to get Lie cohomology and connections with the exterior algebra (#15300) and De Rahm cohomology done as well.

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​d79c483	Merge branch 'public/combinat/root_systems/improvements-15384' into public/algebras/lie_algebras-14901
​585729a	Merge branch 'public/modules/fock_space' into public/algebras/lie_algebras-14901
​702f96a	Merge branch 'u/tscrim/lie_algebras-14901' of trac.sagemath.org:sage into public/algebras/lie_algebras-14901
​4548a5b	Merge branch 'public/modules/fock_space' of trac.sagemath.org:sage into public/modules/fock_space
​31b1cf2	Finished GW algorithm and finalized files.
​0374b6f	Merge branch 'master' into public/algebras/lie_algebras-14901
​20af672	Merge branch 'master' into public/modules/fock_space
​1d8057a	Merge branch 'u/tscrim/lie_algebras-14901' of ​ssh://trac.sagemath.org:2222/sage into public/algebras/lie_algebras-14901
​63ac373	Tweaks to Virasoro algebra documentation.
​90424b3	More work on GW algorithm.

[tscrim]

Also for reference (and in case anyone can translate mathematica code into pseudocode or psuedo Sage) ​http://www.equaonline.com/math/SuperLie/.

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​6863447	Merge branch 'master' into public/algebras/lie_algebras-14901
​a970464	Work on finite dimensional Lie algebras with bases.
​b153849	Merge branch 'master' into public/algebras/lie_algebras-14901
​494ad67	Merge branch 'develop' into public/algebras/lie_algebras-14901
​4038bc8	Starting work.
​b09a669	Fixed implementation, now to documentation.
​2fdb307	Started new file for PBW algebras and ideals.
​6fb3c0d	Merge branch 'develop' into public/algebras/fp_algebras
​d2d1019	Refactoring PBW ideals and FP algebras.
​6213e71	Merge branch 'develop' into public/algebras/fp_algebras

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​a278afa	Added cardinality methods.
​f681606	Added cardinality to free abelian monoid for consistancy.
​3a0e50b	Merge branch 'develop' into public/monoids/15289-indexed
​d4606cc	Added more robustness to element creation.
​991953a	Merge branch 'develop' into public/monoids/15289-indexed
​480188e	Merge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901
​cc0c2a5	Merge branch 'develop' into public/algebras/lie_algebras-14901
​c3e7244	Added are_linearly_dependent.
​b8ef03b	Merge branch 'public/linear_algebra/linear_dep_check-15827' into public/algebras/lie_algebras-14901
​4b66ac3	Refactoring of Lie algebras with structure coefficients.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​f356239

Fixes from refactoring and (hopefully) a working version of finitely presented algebras.

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​7db4ef0	Trac 10963: Fixed repr for the symmetric group algebra by improving a bit the generic repr methods in the categories
​367d72e	Trac 10963: fixed trivial doctest failure
​8919180	Trac 10963: reverted change that was meant to be put in #15801
​d1b5374	Trac 10963: reverted change that required #15801 for no good reason
​4580db1	Trac 10963: fixed trivial doctest failures
​f44661f	Trac 15801: added repr in HeckeModule_free_module (the generic one in modules with basis can't handle it anymore)
​feab04a	manual merge with 6.2.beta8
​ce2193e	Merge branch 'public/ticket/10963-doc-distributive' of trac.sagemath.org:sage into public/ticket/10963-doc-distributive
​e5567c1	Merge branch 'public/ticket/10963-doc-distributive' into public/algebras/lie_algebras-14901
​11c9bf3	Cleaning up bad merging.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​6de0121	Tweaks to elements and struct coeffs.
​1c1b4eb	Changed monomial_cmp to generator_cmp and added free (static)method to monoids and groups category.
​47b5be7	Removed __contains__ and fix monomial_cmp in indexed_monoid.py
​4097a8c	Merge branch 'develop' into public/monoids/15289-indexed
​374ae3d	Merge branch 'public/monoids/15289-indexed' into public/algebras/base_lie_algebras
​211e1c7	Merge branch 'public/algebras/base_lie_algebras' into public/algebras/lie_algebras-14901
​23301a8	Misc fixes and tweaks.
​e30bf4e	Merge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​4f719e1	Reverted FreeLieAlgebra error.
​6993e66	Fixes for removing of ngens() for morphisms.
​c738b02	Merge branch 'develop' into public/algebras/lie_algebras-14901

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​f9f8b9f	Merge branch 'public/combinat/root_systems/improvements-15384' into public/algebras/lie_algebras-14901
​56e3891	Putting out some fires.
​329ea79	Merge branch 'develop' into public/algebras/base_lie_algebras
​6da634b	Merge branch 'develop' into public/algebras/base_lie_algebras
​dc6747b	Better repr/latex and actually use IndexedGenerators.
​2346150	Merge branch 'public/algebras/base_lie_algebras' into public/algebras/lie_algebras-14901
​4785df5	Some fixes from the merge.
​21e7b0b	Removed s from affine Lie algebras module name.
​1edf001	Merge branch 'public/algebras/base_lie_algebras' into public/algebras/lie_algebras-14901
​850ab5c	Putting out more fires. (Untwisted) Affine Lie algebras working agian.

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​ecdc00f	Fixed doctests because of different processing of input.
​909196e	Merge branch 'develop' into public/monoids/15289-indexed
​1db7c1d	Merge branch 'develop' into public/monoids/15289-indexed
​73a4e48	Implemented other changes/improvements Nicolas suggested.
​b9f6f18	Merge branch 'develop' into public/monoids/15289-indexed
​45ee958	Merge branch 'public/monoids/15289-indexed' into public/algebras/lie_algebras-14901
​399bb1e	Rebased over changes from indexed generators.
​ab4fa91	Removed duplicate code and fixed ideals.
​469a5b2	Putting out more fires.
​b8bcd7d	Merge branch 'public/algebras/lie_algebras-14901' of trac.sagemath.org:sage into public/algebras/lie_algebras-14901

[tscrim]

FYI - #16819 has been split off and is ready for review.

[tscrim]

More references/information/future work:

Atlas of Lie groups and representations - ​http://www.liegroups.org/

[tscrim]

Lie algebra (co)homology is done in #22883. It is not the most efficient implementation and doesn't seem work very well for Lie algebras of dimension larger than 10 due to the amount of information needed to be computed (i.e., large dimensional vector spaces).

[tscrim]

#23037 adds a little bit more to the representation theory in Sage. Good follow-ups will include trivial, adjoint, and tensor product representations for Lie algebras.

[vdelecroix]

update milestone 8.3 -> 8.4

[gh-ehaka]

Added a subsection of tickets related to various constructions related to nilpotent Lie algebras. I have some existing implementations for these in a codebase that needs to be cleaned up and refactored in order to be imported to Sage.