Starts a bit towards multiplication groups.

docs/make.jl

@@ -122,17 +122,17 @@ makedocs(;
         "Home" => "index.md",
         "About" => "about.md",
         (tutorials_in_menu ? [tutorials_menu] : [])...,
+        "Lie groups" => [
+            "List of Lie Groups" => "groups/index.md",
+            "General Linear" => "groups/general_linear.md",
+            "Translation group" => "groups/translation.md",
+        ],
         "Interfaces" => [
             "Lie group" => "interface/group.md",
             "Lie algebra" => "interface/algebra.md",
             "Group operation" => "interface/operations.md",
             "Group action" => "interface/actions.md",
         ],
-        "Lie groups" => [
-            "List of Lie Groups" => "groups/index.md",
-            "General Linear" => "groups/general_linear.md",
-            "Translation group" => "groups/translation.md",
-        ],
         "Contributing to LieGroups.jl" => "contributing.md",
         "Notation" => "notation.md",
         "Changelog" => "news.md",

src/documentation_glossary.jl

@@ -51,6 +51,7 @@ end
 # LaTeX
 # Define LaTeX shortcuts
 _tex(args...; kwargs...) = glossary(:LaTeX, args...; kwargs...)
+define!(:LaTeX, :big, raw"\big")
 define!(:LaTeX, :bigl, raw"\bigl")
 define!(:LaTeX, :bigr, raw"\bigr")
 define!(:LaTeX, :Big, raw"\Big")
@@ -60,10 +61,28 @@ define!(:LaTeX, :def, raw"\coloneqq")
 define!(:LaTeX, :Cal, (letter) -> raw"\mathcal " * "$letter")
 define!(:LaTeX, :exp, raw"\exp")
 define!(:LaTeX, :Frak, (letter) -> raw"\mathfrak " * "$letter")
+define!(:LaTeX, :l, "") # lazy fallback for sets
+define!(:LaTeX, :r, "") # lazy fallback for sets
 define!(:LaTeX, :log, raw"\log")
 define!(:LaTeX, :qquad, raw"\qquad")
 define!(:LaTeX, :quad, raw"\quad")
-define!(:LaTeX, :rm, (letter) -> raw"\mathrm " * "$letter")
+define!(:LaTeX, :rm, (letter) -> raw"\mathrm" * "{$letter}")
+define!(
+    :LaTeX,
+    :Set,
+    (content, size="") ->
+        _tex(Symbol("$(size)l")) *
+        raw"\{ " *
+        "$(content)" *
+        _tex(Symbol("$(size)r")) *
+        raw" \}",
+)
+define!(
+    :LaTeX,
+    :SetDef,
+    (elem, cond, size="") ->
+        _tex(:Set, elem * raw"\ " * _tex(Symbol("$(size)")) * raw"|\ " * "$(cond)", size),
+)
 #
 # ---
 # Mathematics and semantic symbols

src/group_operations/multiplication_operation.jl

@@ -13,3 +13,113 @@ abstract type AbstractMultiplicationGroupOperation <: AbstractGroupOperation end
 A grou poperation that is realised by a matrix multiplication.
 """
 struct MatrixMultiplicationGroupOperation <: AbstractMultiplicationGroupOperation end
+
+Base.:*(e::Identity{AbstractMultiplicationGroupOperation}) = e
+Base.:*(::Identity{AbstractMultiplicationGroupOperation}, p) = p
+Base.:*(p, ::Identity{AbstractMultiplicationGroupOperation}) = p
+function Base.:*(
+    e::Identity{AbstractMultiplicationGroupOperation},
+    ::Identity{AbstractMultiplicationGroupOperation},
+)
+    return e
+end
+function Base.:*(
+    ::Identity{AbstractMultiplicationGroupOperation}, e::Identity{AdditionGroupOperation}
+)
+    return e
+end
+function Base.:*(
+    e::Identity{AbstractMultiplicationGroupOperation},
+    ::Identity{AbstractMultiplicationGroupOperation},
+)
+    return e
+end
+
+Base.:/(p, ::Identity{AbstractMultiplicationGroupOperation}) = p
+Base.:/(::Identity{AbstractMultiplicationGroupOperation}, p) = inv(p)
+function Base.:/(
+    e::Identity{AbstractMultiplicationGroupOperation},
+    ::Identity{AbstractMultiplicationGroupOperation},
+)
+    return e
+end
+
+Base.:\(p, ::Identity{AbstractMultiplicationGroupOperation}) = inv(p)
+Base.:\(::Identity{AbstractMultiplicationGroupOperation}, p) = p
+function Base.:\(
+    e::Identity{AbstractMultiplicationGroupOperation},
+    ::Identity{AbstractMultiplicationGroupOperation},
+)
+    return e
+end
+
+Base.inv(e::Identity{AbstractMultiplicationGroupOperation}) = e
+
+LinearAlgebra.det(::Identity{AbstractMultiplicationGroupOperation}) = true
+LinearAlgebra.adjoint(e::Identity{AbstractMultiplicationGroupOperation}) = e
+
+_doc_compose_mult = """
+    compose(G::LieGroup{𝔽,AbstractMultiplicationGroupOperation}, g, h)
+    compose!(G::LieGroup{𝔽,AbstractMultiplicationGroupOperation}, k, g, h)
+
+Compute the group operation composition of `g` and `h` with respect to
+an [`AbstractMultiplicationGroupOperation`](@ref) on `G`, which falls back to calling
+`g*h`, where `*` is assumed to be overloaded accordingly.
+
+This can be computed in-place of `k`.
+"""
+
+@doc "$(_doc_compose_mult)"
+compose(::LieGroup{𝔽,AbstractMultiplicationGroupOperation}, g, h) where {𝔽}
+
+@doc "$(_doc_compose_mult)"
+compose!(::LieGroup{𝔽,AbstractMultiplicationGroupOperation}, k, g, h) where {𝔽}
+
+function _compose!(::LieGroup{𝔽,AbstractMultiplicationGroupOperation}, k, g, h) where {𝔽}
+    k .= g * h
+    return k
+end
+
+_doc_identity_element_mult = """
+    identity_element(G::LieGroup{𝔽,AbstractMultiplicationGroupOperation})
+    identity_element!(G::LieGroup{𝔽,AbstractMultiplicationGroupOperation}, e)
+
+Return the a point representation of the [`Identity`](@ref),
+which for the [`AdditionGroupOperation`](@ref) is the one-element or identity array.
+"""
+
+@doc "$(_doc_identity_element_mult)"
+identity_element(::LieGroup{𝔽,AbstractMultiplicationGroupOperation}) where {𝔽}
+
+@doc "$(_doc_identity_element_mult)"
+identity_element!(::LieGroup{𝔽,AbstractMultiplicationGroupOperation}, e) where {𝔽}
+function identity_element!(
+    ::LieGroup{𝔽,AbstractMultiplicationGroupOperation}, e::AbstractMatrix
+) where {𝔽}
+    return copyto!(e, LinearAlgebra.I)
+end
+
+LinearAlgebra.mul!(q, ::Identity{AbstractMultiplicationGroupOperation}, p) = copyto!(q, p)
+LinearAlgebra.mul!(q, p, ::Identity{AbstractMultiplicationGroupOperation}) = copyto!(q, p)
+function LinearAlgebra.mul!(
+    q::AbstractMatrix,
+    ::Identity{AbstractMultiplicationGroupOperation},
+    ::Identity{AbstractMultiplicationGroupOperation},
+)
+    return copyto!(q, I)
+end
+function LinearAlgebra.mul!(
+    q,
+    ::Identity{AbstractMultiplicationGroupOperation},
+    ::Identity{AbstractMultiplicationGroupOperation},
+)
+    return copyto!(q, one(q))
+end
+function LinearAlgebra.mul!(
+    q::Identity{AbstractMultiplicationGroupOperation},
+    ::Identity{AbstractMultiplicationGroupOperation},
+    ::Identity{AbstractMultiplicationGroupOperation},
+)
+    return q
+end
+Base.one(e::Identity{AbstractMultiplicationGroupOperation}) = e

src/groups/general_linear_group.jl

@@ -1,8 +1,13 @@
 
-@doc raw"""
+@doc """
     GeneralLinearGroup{𝔽,T}
 
-The general linear group ``\operatorname{GL}(n)`` is the set of all invertible matrices in ``𝔽^{n×n}``
+The general linear group ``$(_tex(:rm,"GL"))(n)`` is the set of all invertible matrices
+
+```math
+$(_tex(:rm,"GL"))(n) = $(_tex(:SetDef, "g ∈ 𝔽^{n×n}", "$(_tex(:rm,"det"))(p) ≠ 0", "big")),
+$(_tex(:qquad)) 𝔽 ∈ $(_tex(:Set, "ℝ, ℂ")),
+```
 equipped with the [`MatrixMultiplicationGroupOperation`](@ref) as the group operation.
 
 # Constructor
@@ -13,9 +18,7 @@ Generate the general linear group  group on ``𝔽^{n×n}``.
 All keyword arguments in `kwargs...` are passed on to [`InvertibleMatrices`](@extref `Manifolds.InvertibleMatrices`).
 """
 const GeneralLinearGroup{𝔽,T} = LieGroup{
-    𝔽,
-    MatrixMultiplicationGroupOperation,
-    Manifolds.InvertibleMatrices{𝔽,T},
+    𝔽,MatrixMultiplicationGroupOperation,Manifolds.InvertibleMatrices{𝔽,T}
 }
 
 function GeneralLinearGroup(n::Int...; kwargs...)