Lie algebras: Add root system detection (#4084)

Project.toml

@@ -25,7 +25,7 @@ UUIDs = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"
 cohomCalg_jll = "5558cf25-a90e-53b0-b813-cadaa3ae7ade"
 
 [compat]
-AbstractAlgebra = "0.42.3"
+AbstractAlgebra = "0.42.5"
 AlgebraicSolving = "0.5.1"
 Distributed = "1.6"
 GAP = "0.11.3"

docs/oscar_references.bib

@@ -1090,6 +1090,17 @@ @InProceedings{GHJ16
   year          = {2016}
 }
 
+@Article{GIR96,
+  author        = {de Graaf, Willem A. and Ivanyos, Gábor and Rónyai, Lajos},
+  title         = {Computing Cartan subalgebras of Lie algebras},
+  journal       = {Appl. Algebra Eng. Commun. Comput.},
+  volume        = {7},
+  number        = {5},
+  pages         = {339--349},
+  year          = {1996},
+  doi           = {10.1007/BF01293593}
+}
+
 @InCollection{GJ00,
   author        = {Gawrilow, Ewgenij and Joswig, Michael},
   title         = {polymake: a Framework for Analyzing Convex Polytopes},
@@ -1222,6 +1233,17 @@ @Article{Gat96
   doi           = {10.1007/BF01191379}
 }
 
+@Book{Gra00,
+  author        = {de Graaf, Willem A.},
+  title         = {Lie algebras: theory and algorithms},
+  series        = {North-Holland Mathematical Library},
+  volume        = {56},
+  publisher     = {North-Holland Publishing Co., Amsterdam},
+  pages         = {xii+393},
+  year          = {2000},
+  url           = {https://www.sciencedirect.com/bookseries/north-holland-mathematical-library/vol/56/}
+}
+
 @PhDThesis{Gro20,
   author        = {Gromada, Daniel},
   title         = {Compact matrix quantum groups and their representation categories},

experimental/LieAlgebras/docs/src/ideals_and_subalgebras.md

@@ -17,7 +17,7 @@ They are used similarly in most cases.
 dim(::LieAlgebraIdeal)
 basis(::LieAlgebraIdeal)
 basis(::LieAlgebraIdeal, ::Int)
-Base.in(::LieAlgebraElem, ::LieAlgebraIdeal)
+Base.in(::LieAlgebraElem{C}, ::LieAlgebraIdeal{C}) where {C<:FieldElem}
 bracket(::LieAlgebraIdeal{C,LieT}, ::LieAlgebraIdeal{C,LieT}) where {C<:FieldElem,LieT<:LieAlgebraElem{C}}
 normalizer(::LieAlgebra, ::LieAlgebraIdeal)
 centralizer(::LieAlgebra, ::LieAlgebraIdeal)
@@ -29,7 +29,7 @@ centralizer(::LieAlgebra, ::LieAlgebraIdeal)
 dim(::LieSubalgebra)
 basis(::LieSubalgebra)
 basis(::LieSubalgebra, ::Int)
-Base.in(::LieAlgebraElem, ::LieSubalgebra)
+Base.in(::LieAlgebraElem{C}, ::LieSubalgebra{C}) where {C<:FieldElem}
 bracket(::LieSubalgebra{C,LieT}, ::LieSubalgebra{C,LieT}) where {C<:FieldElem,LieT<:LieAlgebraElem{C}}
 normalizer(::LieAlgebra, ::LieSubalgebra)
 centralizer(::LieAlgebra, ::LieSubalgebra)

experimental/LieAlgebras/src/AbstractLieAlgebra.jl

@@ -95,22 +95,20 @@ end
 has_root_system(L::AbstractLieAlgebra) = isdefined(L, :root_system)
 
 function root_system(L::AbstractLieAlgebra)
-  @req has_root_system(L) "no root system known."
+  assure_root_system(L)
   return L.root_system
 end
 
 function chevalley_basis(L::AbstractLieAlgebra)
-  @req has_root_system(L) "no root system known."
-  # TODO: once there is root system detection, this function needs to be updated to indeed return the Chevalley basis
-
-  npos = n_positive_roots(root_system(L))
-  b = basis(L)
-  # root vectors
-  r_plus = b[1:npos]
-  r_minus = b[(npos + 1):(2 * npos)]
-  # basis for cartan algebra
-  h = b[(2 * npos + 1):dim(L)]
-  return (r_plus, r_minus, h)
+  assure_root_system(L)
+  return L.chevalley_basis::NTuple{3,Vector{elem_type(L)}}
+end
+
+function set_root_system_and_chevalley_basis!(
+  L::AbstractLieAlgebra{C}, R::RootSystem, chev::NTuple{3,Vector{AbstractLieAlgebraElem{C}}}
+) where {C<:FieldElem}
+  L.root_system = R
+  L.chevalley_basis = chev
 end
 
 ###############################################################################
@@ -234,7 +232,7 @@ function lie_algebra(
     @req fl "Not closed under the bracket."
     struct_consts[i, j] = sparse_row(row)
   end
-  s = map(AbstractAlgebra.obj_to_string, basis)
+  s = map(AbstractAlgebra.obj_to_string_wrt_times, basis)
   return lie_algebra(R, struct_consts, s; check)
 end
 
@@ -264,7 +262,14 @@ function lie_algebra(
   ]
 
   L = lie_algebra(R, struct_consts, s; check=false)
-  L.root_system = rs
+
+  npos = n_positive_roots(rs)
+  # set Chevalley basis
+  r_plus = basis(L)[1:npos]
+  r_minus = basis(L)[(npos + 1):(2 * npos)]
+  h = basis(L)[(2 * npos + 1):end]
+  chev = (r_plus, r_minus, h)
+  set_root_system_and_chevalley_basis!(L, rs, chev)
   return L
 end
 

experimental/LieAlgebras/src/DirectSumLieAlgebra.jl

@@ -135,9 +135,26 @@ end
 #
 ###############################################################################
 
-# The following implementation needs direct sums of root systems, which
-# is not yet implemented.
-# has_root_system(D::DirectSumLieAlgebra) = all(has_root_system, D.summands)
+has_root_system(D::DirectSumLieAlgebra) = isdefined(D, :root_system)
+
+function root_system(D::DirectSumLieAlgebra)
+  assure_root_system(D)
+  return D.root_system
+end
+
+function chevalley_basis(D::DirectSumLieAlgebra)
+  assure_root_system(D)
+  return D.chevalley_basis::NTuple{3,Vector{elem_type(D)}}
+end
+
+function set_root_system_and_chevalley_basis!(
+  D::DirectSumLieAlgebra{C},
+  R::RootSystem,
+  chev::NTuple{3,Vector{DirectSumLieAlgebraElem{C}}},
+) where {C<:FieldElem}
+  D.root_system = R
+  D.chevalley_basis = chev
+end
 
 ###############################################################################
 #