In mathematical physics, nonlinear realization of a Lie groupG possessing a Cartan subgroupH is a particular induced representation of G. In fact, it is a representation of a Lie algebra of G in a neighborhood of its origin.
A nonlinear realization, when restricted to the subgroup H reduces to a linear representation.
Let G be a Lie group and H its Cartan subgroup which admits a linear representation in a vector space V. A Lie
algebra of G splits into the sum of the Cartan subalgebra of H and its supplement , such that
(In physics, for instance, amount to vector generators and to axial ones.)
There exists an open neighborhood U of the unit of G such
that any element is uniquely brought into the form
Let be an open neighborhood of the unit of G such that
, and let be an open neighborhood of the
H-invariant center of the quotient G/H which consists of elements
Then there is a local section of
With this local section, one can define the induced representation, called the nonlinear realization, of elements on given by the expressions
The corresponding nonlinear realization of a Lie algebra
of G takes the following form.
Let , be the bases for and , respectively, together with the commutation relations