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
• Fock space #15508

There might also be the following:

• su_n
• recovery of Lie group from Lie algebra
• 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).

[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.