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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 smooth charts on with and on with such that the coordinate expression is a class map between open subsets of and in the usual multivariable sense.
Let be a smooth manifold, , and the algebra of germs of smooth real-valued functions at . The tangent space is the vector space of linear maps
that satisfy the Leibniz rule
Such maps are called derivations, or tangent vectors at .
Let be a Lie group. The conjugation map by is
This defines a smooth action of on itself by group automorphisms.