tscrim commented 11 years ago

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, #33836, #34006

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
30744 - Symplectic derivation Lie algebra
30236 - Universal commutative 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

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

tscrim commented 10 years ago

Changed keywords from Lie algebras Kac Moody, days54 to Lie algebras, Kac Moody, days54

tscrim commented 10 years ago

Changed dependencies from #10963 #14898 #15151 #15289 #15384 #15827 to #10963 #14898 #15151 #15289 #15384 #15827 #16819 #16820 #16821 #16822 #16823 #16824 #16825 #16826

tscrim commented 10 years ago

Description changed:

--- 
+++ 
@@ -34,3 +34,15 @@
 - 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):
+
+* #16819 - the categories
+* #16820 - the basic framework and finite dimensional given by structure coefficients
+* #16821 - classical Lie algebras
+* #16822 - PBW basis for the universal enveloping algebra
+* #16823 - the free Lie algebra
+* #16824 - ideals, quotients, and finitely presented Lie algebras
+* #16825 - untwisted affine Lie algebras
+* #16826 - Kac-Moody algebras
+

tscrim commented 10 years ago

comment:38

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

tscrim commented 10 years ago

Description changed:

--- 
+++ 
@@ -25,13 +25,13 @@
   * Virasoro algebra
   * Some nilpotent Lie algebras
 - Kac-Moody algebras based only on a (generalized) Cartan matrix
-- Fock space #15508
+- Fock space

 There might also be the following:

 - **su**<sub>n</sub>
 - recovery of Lie group from Lie algebra
-- connection between the quantum group and the Hall algebra (coming from representations of a quiver over Fq)
+- connection between the quantum group and the Hall algebra (coming from representations of a quiver over F<sub>q</sub>)

 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).

@@ -45,4 +45,14 @@
 * #16824 - ideals, quotients, and finitely presented Lie algebras
 * #16825 - untwisted affine Lie algebras
 * #16826 - Kac-Moody algebras
+* #16860 - Interface with Gap (not part of the current branch)
+* #16861 - Interface with LiE (not part of the current branch)

+Related tickets:
+
+* #15508 - Fock space and LLT algorithm for type A<sub>n</sub><sup>(1)</sup>
+* #16055 - Jordan algebras
+* #15300 - Weyl and Clifford algebras
+* #16056 - Global crystal bases
+* #16851 - Koszul complexes
+

tscrim commented 10 years ago

Description changed:

--- 
+++ 
@@ -55,4 +55,6 @@
 * #15300 - Weyl and Clifford algebras
 * #16056 - Global crystal bases
 * #16851 - Koszul complexes
+* #15484 - Yangians for **gl**<sub>n</sub>
+* #15485 - WZW fusion algebras

tscrim commented 10 years ago

Description changed:

--- 
+++ 
@@ -39,6 +39,7 @@

 * #16819 - the categories
 * #16820 - the basic framework and finite dimensional given by structure coefficients
+* #17416 - subalgebras
 * #16821 - classical Lie algebras
 * #16822 - PBW basis for the universal enveloping algebra
 * #16823 - the free Lie algebra

tscrim commented 9 years ago

comment:41

More references/information/future work:

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

tscrim commented 9 years ago

Changed dependencies from #10963 #14898 #15151 #15289 #15384 #15827 #16819 #16820 #16821 #16822 #16823 #16824 #16825 #16826 to #10963 #14898 #15151 #15289 #15384 #15827 #16820 #16821 #16822 #16823 #16824 #16825 #16826

tscrim commented 9 years ago

Description changed:

--- 
+++ 
@@ -37,8 +37,7 @@

 Current ticket breakdown (more might follow):

-* #16819 - the categories
-* #16820 - the basic framework and finite dimensional given by structure coefficients
+* #16820 - the categories and the basic framework and finite dimensional given by structure coefficients
 * #17416 - subalgebras
 * #16821 - classical Lie algebras
 * #16822 - PBW basis for the universal enveloping algebra