English
The Lie algebra of a Lie group is the tangent space at the identity; for a notational convenience, call it GroupLieAlgebra I G.
Русский
Лиальная алгебра группы — это касательный пространства в единице; обозначается как GroupLieAlgebra I G.
LaTeX
$$\\text{GroupLieAlgebra} : = \\text{TangentSpace} I (1 : G)$$
Lean4
/-- The invariant vector field associated to a vector `v` in the Lie algebra. At a point `g`, it
is given by the image of `v` under left-multiplication by `g`. -/
@[to_additive /-- The invariant vector field associated to a vector `v` in the Lie algebra. At a
point `g`, it is given by the image of `v` under left-addition by `g`. -/
]
noncomputable def mulInvariantVectorField (v : GroupLieAlgebra I G) (g : G) : TangentSpace I g :=
mfderiv I I (g * ·) (1 : G) v
