From b3f16df1ffa79480e4f03d1a12d89365335e96f0 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Lars=20G=C3=B6ttgens?= <lars.goettgens@rwth-aachen.de>
Date: Thu, 13 Jul 2023 17:05:12 +0200
Subject: [PATCH] Add `centralizer` and `normalizer`

---
 experimental/LieAlgebras/src/LieAlgebra.jl    | 24 +++++++--
 .../LieAlgebras/src/LieAlgebraIdeal.jl        | 20 ++++++++
 experimental/LieAlgebras/src/LieAlgebras.jl   |  4 ++
 experimental/LieAlgebras/src/LieSubalgebra.jl | 49 ++++++++++++++++++-
 .../LieAlgebras/src/LinearLieAlgebra.jl       |  2 +-
 5 files changed, 92 insertions(+), 7 deletions(-)

diff --git a/experimental/LieAlgebras/src/LieAlgebra.jl b/experimental/LieAlgebras/src/LieAlgebra.jl
index 63ca2517921d..b6fc09f0f7ba 100644
--- a/experimental/LieAlgebras/src/LieAlgebra.jl
+++ b/experimental/LieAlgebras/src/LieAlgebra.jl
@@ -270,7 +270,7 @@ end
 
 ###############################################################################
 #
-#   Important ideals
+#   Important ideals and subalgebras
 #
 ###############################################################################
 
@@ -280,9 +280,6 @@ function derived_algebra(L::LieAlgebra)
 end
 
 function center(L::LieAlgebra)
-  characteristic(coefficient_ring(L)) == 2 &&
-    error("GAP does something else for char 2 which is not implemented here.")
-
   dim(L) == 0 && return ideal(L, [])
 
   mat = zero_matrix(coefficient_ring(L), dim(L), dim(L)^2)
@@ -296,6 +293,24 @@ function center(L::LieAlgebra)
   return ideal(L, [L(c_basis[i, :]) for i in 1:c_dim]; is_basis=true)
 end
 
+function centralizer(L::LieAlgebra, xs::AbstractVector{<:LieAlgebraElem})
+  @req all(x -> parent(x) == L, xs) "Incompatible Lie algebras."
+
+  mat = zero_matrix(coefficient_ring(L), dim(L), dim(L) * length(xs))
+  for (i, bi) in enumerate(basis(L))
+    for (j, xj) in enumerate(xs)
+      mat[i, ((j - 1) * dim(L) + 1):(j * dim(L))] = _matrix(bracket(bi, xj))
+    end
+  end
+
+  c_dim, c_basis = left_kernel(mat)
+  return sub(L, [L(c_basis[i, :]) for i in 1:c_dim]; is_basis=true)
+end
+
+function centralizer(L::LieAlgebra, x::LieAlgebraElem)
+  return centralizer(L, [x])
+end
+
 ###############################################################################
 #
 #   Properties
@@ -307,6 +322,7 @@ function is_abelian(L::LieAlgebra)
 end
 
 function is_simple(L::LieAlgebra)
+  is_abelian(L) && return false
   error("Not implemented.") # TODO
 end
 
diff --git a/experimental/LieAlgebras/src/LieAlgebraIdeal.jl b/experimental/LieAlgebras/src/LieAlgebraIdeal.jl
index a500815fe934..829b156a6b74 100644
--- a/experimental/LieAlgebras/src/LieAlgebraIdeal.jl
+++ b/experimental/LieAlgebras/src/LieAlgebraIdeal.jl
@@ -158,6 +158,21 @@ function bracket(
   return ideal(base_lie_algebra(I1), [x * y for x in gens(I1) for y in gens(I2)])
 end
 
+###############################################################################
+#
+#   Important ideals and subalgebras
+#
+###############################################################################
+
+function normalizer(L::LieAlgebra, I::LieAlgebraIdeal)
+  @req parent(I) == L "Incompatible Lie algebras."
+  return normalizer(L, sub(L, I))
+end
+
+function centralizer(L::LieAlgebra, I::LieAlgebraIdeal)
+  return centralizer(L, basis(I))
+end
+
 ###############################################################################
 #
 #   Conversion
@@ -170,6 +185,11 @@ function lie_algebra(
   return lie_algebra(basis(I))
 end
 
+function sub(L::LieAlgebra{C}, I::LieAlgebraIdeal{C}) where {C<:RingElement}
+  @req base_lie_algebra(I) === L "Incompatible Lie algebras."
+  return sub(L, basis(I); is_basis=true)
+end
+
 ###############################################################################
 #
 #   Constructor
diff --git a/experimental/LieAlgebras/src/LieAlgebras.jl b/experimental/LieAlgebras/src/LieAlgebras.jl
index 2573c5a6d263..368bef3308c5 100644
--- a/experimental/LieAlgebras/src/LieAlgebras.jl
+++ b/experimental/LieAlgebras/src/LieAlgebras.jl
@@ -16,6 +16,7 @@ import ..Oscar:
   basis,
   basis_matrix,
   center,
+  centralizer,
   coeff,
   coefficient_ring,
   coefficients,
@@ -31,6 +32,7 @@ import ..Oscar:
   is_abelian,
   is_simple,
   ngens,
+  normalizer,
   parent_type,
   sub,
   symbols,
@@ -62,6 +64,7 @@ export highest_weight_module
 export is_direct_sum
 export is_dual
 export is_exterior_power
+export is_self_normalizing
 export is_standard_module
 export is_symmetric_power
 export is_tensor_power
@@ -113,6 +116,7 @@ export highest_weight_module
 export is_direct_sum
 export is_dual
 export is_exterior_power
+export is_self_normalizing
 export is_standard_module
 export is_symmetric_power
 export is_tensor_power
diff --git a/experimental/LieAlgebras/src/LieSubalgebra.jl b/experimental/LieAlgebras/src/LieSubalgebra.jl
index 0dfb5b3e70ad..dc170755231f 100644
--- a/experimental/LieAlgebras/src/LieSubalgebra.jl
+++ b/experimental/LieAlgebras/src/LieSubalgebra.jl
@@ -150,6 +150,51 @@ function bracket(
   return ideal(base_lie_algebra(S1), [x * y for x in gens(S1) for y in gens(S2)])
 end
 
+###############################################################################
+#
+#   Important ideals and subalgebras
+#
+###############################################################################
+
+function normalizer(L::LieAlgebra, S::LieSubalgebra)
+  @req base_lie_algebra(S) == L "Incompatible Lie algebras."
+
+  mat = zero_matrix(coefficient_ring(L), dim(L) + dim(S)^2, dim(L) * dim(S))
+  for (i, bi) in enumerate(basis(L))
+    for (j, sj) in enumerate(basis(S))
+      mat[i, ((j - 1) * dim(L) + 1):(j * dim(L))] = _matrix(bracket(bi, sj))
+    end
+  end
+  for i in 1:dim(S)
+    mat[(dim(L) + (i - 1) * dim(S) + 1):(dim(L) + i * dim(S)), ((i - 1) * dim(L) + 1):(i * dim(L))] = basis_matrix(
+      S
+    )
+  end
+  show(stdout, MIME"text/plain"(), mat)
+  println("")
+  sol_dim, sol = left_kernel(mat)
+  sol = sol[:, 1:dim(L)]
+  c_dim, c_basis = rref(sol)
+  println(c_dim)
+  show(stdout, MIME"text/plain"(), c_basis)
+  println("")
+  return sub(L, [L(c_basis[i, :]) for i in 1:c_dim]; is_basis=true)
+end
+
+function centralizer(L::LieAlgebra, S::LieSubalgebra)
+  return centralizer(L, basis(S))
+end
+
+###############################################################################
+#
+#   Properties
+#
+###############################################################################
+
+function is_self_normalizing(S::LieSubalgebra)
+  return normalizer(base_lie_algebra(S), S) == S
+end
+
 ###############################################################################
 #
 #   Conversion
@@ -159,7 +204,7 @@ end
 function lie_algebra(
   S::LieSubalgebra{C,LieT}
 ) where {C<:RingElement,LieT<:LieAlgebraElem{C}}
-  return lie_algebra(basis(S))
+  return lie_algebra(basis(S)) #, embedding_hom   # TODO
 end
 
 ###############################################################################
@@ -169,7 +214,7 @@ end
 ###############################################################################
 
 function sub(L::LieAlgebra, gens::Vector; is_basis::Bool=false)
-  return LieSubalgebra{elem_type(coefficient_ring(L)),elem_type(L)}(L, gens; is_basis) #, embedding_hom   # TODO
+  return LieSubalgebra{elem_type(coefficient_ring(L)),elem_type(L)}(L, gens; is_basis)
 end
 
 function sub(L::LieAlgebra{C}, gen::LieAlgebraElem{C}) where {C<:RingElement}
diff --git a/experimental/LieAlgebras/src/LinearLieAlgebra.jl b/experimental/LieAlgebras/src/LinearLieAlgebra.jl
index ad8c81ff57df..3cc290430f0c 100644
--- a/experimental/LieAlgebras/src/LinearLieAlgebra.jl
+++ b/experimental/LieAlgebras/src/LinearLieAlgebra.jl
@@ -217,7 +217,7 @@ function lie_algebra(
   R = coefficient_ring(parent_L)
   n = parent_L.n
   s = map(AbstractAlgebra.obj_to_string, basis)
-  return lie_algebra(R, n, basis, s; check)
+  return lie_algebra(R, n, matrix_repr.(basis), s; check)
 end
 
 @doc raw"""