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The diffeomorphism of a Lie group given by multiplying on the right by a fixed element.
Let be a Lie group and fix . The right translation by is the map
Since multiplication is smooth, is a diffeomorphism with inverse .
Conjugation by can be written as a composition of translations:
which underlies the adjoint action.
Right translations are used to define right-invariant vector fields.
A Lie group is a group equipped with the structure of a smooth manifold such that the group operations are smooth maps:
For each , the left translation and the right translation are diffeomorphisms of , with inverses and . The tangent space at the identity carries a canonical Lie algebra structure, called the Lie algebra of , and the exponential map relates this infinitesimal structure to local group behavior near .
Smooth group homomorphisms between Lie groups are Lie group homomorphisms.
Let and be smooth manifolds of dimensions and . A function is smooth (or ) if for every there exist charts on with and on with such that the coordinate expression is a smooth map between open subsets of and in the usual multivariable sense.
Because the transition maps in a smooth atlas are smooth, this definition is independent of the particular charts chosen: if it holds for one pair of charts around and , then it holds for any other such pair. Smooth maps are closed under composition, and the identity map on any smooth manifold is smooth.
A smooth map has a well-defined differential (pushforward) on tangent spaces at each point; dually, it induces the pullback of covectors and the pullback of differential forms.
Let be a smooth manifold and . The tangent space at , denoted , is a vector space capturing first-order directions through .
Let be a Lie group. The conjugation map by is
This defines a smooth action of on itself by group automorphisms.