ENH: adjoint action is a Lie algebra morphism

NEWS.md

@@ -5,6 +5,11 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.1.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
+## [Unreleased]
+
+### Added
+
+- adjoint action is a Lie algebra morphism
 - `check_exp_ad` restricted to real groups
 
 ## [0.2.3]

src/Group.jl

@@ -109,6 +109,21 @@ The adjoint action of ``χ`` on ``ξ`` is an element of ``\mathrm{Alg}(G)``:
 """
 check_adjoint_action_in_alg(G, χ, ξ) = is_vector(G, identity_element(G), adjoint_action(G, χ, ξ))
 
+"""
+    check_adjoint_action_lie_bracket(G, χ, ξ1, ξ2)
+
+Adjoint action commutes with Lie bracket.
+
+```math
+χ [ξ_1, ξ_2] χ^{-1} = [χξ_1χ^{-1}, χξ_2χ^{-1}]
+```
+"""
+check_adjoint_action_lie_bracket(G, χ, ξ1, ξ2) = begin
+    left = adjoint_action(G, χ, lie_bracket(G, ξ1, ξ2))
+    right = lie_bracket(G, adjoint_action(G, χ, ξ1), adjoint_action(G, χ, ξ2))
+    return isapprox(algebra(G), left, right)
+end
+
 """
     check_grp_rep_Identity(G)
 

test/test_group.jl

@@ -23,6 +23,7 @@ alg_rep(::Any, x) = x
         @test GT.check_exp_ad(G, ξ1, ξ2)
     end
     @test GT.check_adjoint_action(G, grp_rep, alg_rep, χ1, ξ1)
+    @test GT.check_adjoint_action_lie_bracket(G, χ1, ξ1, ξ2)
     @test GT.check_inv_rep(G, grp_rep, χ1)
     @testset "$side" for side in [LeftSide(), RightSide()]
         @test GT.check_apply_diff_group_at_id(G, ξ1, side, Identity)