From c02a480f7ea228ebe51b88ddb10a2073aad18b29 Mon Sep 17 00:00:00 2001 From: Mateusz Baran Date: Sat, 30 Nov 2024 18:10:59 +0100 Subject: [PATCH] improve some docs --- src/groups/semidirect_product_group.jl | 16 +++++++++------- 1 file changed, 9 insertions(+), 7 deletions(-) diff --git a/src/groups/semidirect_product_group.jl b/src/groups/semidirect_product_group.jl index 686358a..43b2c1d 100644 --- a/src/groups/semidirect_product_group.jl +++ b/src/groups/semidirect_product_group.jl @@ -1,7 +1,7 @@ # # -# Semidirect product groups – model semidirect products of rwo Lie groups +# Semidirect product groups – model semidirect products of two Lie groups # """ LeftSemidirectProductGroupOperation{O1,O2,A} <: AbstractGroupOperation @@ -10,14 +10,15 @@ A struct to model a semidirect Lie group product. Let ``($(_tex(:Cal, "N")), ⋄)`` and ``($(_tex(:Cal, "H")), ⋆)`` be two Lie groups with group operations ``⋄`` and ``⋆``, respectively, as well as a group action -``σ: $(_tex(:Cal, "H"))×$(_tex(:Cal, "N")) → $(_tex(:Cal, "N"))``, cf [`AbstractLeftGroupActionType`](#ref). +``σ: $(_tex(:Cal, "H"))×$(_tex(:Cal, "N")) → $(_tex(:Cal, "N"))``, cf [`AbstractLeftGroupActionType`](@ref). -We use here as well use the notation ``σ_h: $(_tex(:Cal, "N")) → $(_tex(:Cal, "N"))`` as a family of maps on ``$(_tex(:Cal, "N"))`` +We use here as well use the notation ``σ_h: $(_tex(:Cal, "N")) → $(_tex(:Cal, "N"))`` +as a family of maps on ``$(_tex(:Cal, "N"))`` -Then we define a group operation ``∘`` on the product manifold $(_tex(:Cal, "N"))×$(_tex(:Cal, "H")) by +Then we define a group operation ``∘`` on the product manifold ``$(_tex(:Cal, "N"))×$(_tex(:Cal, "H"))`` by ```math - (h_1,n_1) ∘ (h_2,n_2) := (h_1 ⋆ h_2, τ_{h_2}(n_1) ⋄ n_1). + (h_1,n_1) ∘ (h_2,n_2) := (h_1 ⋆ h_2, σ_{h_2}(n_1) ⋄ n_1). ``` See [HilgertNeeb:2012; Definition 9.2.22](@cite), second definition for more details. @@ -61,9 +62,10 @@ Let ``($(_tex(:Cal, "N")), ⋄)`` and ``($(_tex(:Cal, "H")), ⋆)`` be two Lie g with group operations ``⋄`` and ``⋆``, respectively, as well as a group action ``σ: $(_tex(:Cal, "H"))×$(_tex(:Cal, "N")) → $(_tex(:Cal, "N"))``, cf [`AbstractGroupActionType`](#ref). -We use here as well use the notation ``σ_h: $(_tex(:Cal, "N")) → $(_tex(:Cal, "N"))`` as a family of maps on ``$(_tex(:Cal, "N"))`` +We use here as well use the notation ``σ_h: $(_tex(:Cal, "N")) → $(_tex(:Cal, "N"))`` +as a family of maps on ``$(_tex(:Cal, "N"))`` -Then we define a group operation ``∘`` on the product manifold $(_tex(:Cal, "N"))×$(_tex(:Cal, "H")) by +Then we define a group operation ``∘`` on the product manifold ``$(_tex(:Cal, "N"))×$(_tex(:Cal, "H"))`` by ```math (n_1,h_1) ∘ (n_2,h_2) := (n_1 ⋄ σ_{h_1}(n_2), h_1 ⋆ h_2)