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# Lie Algebra Representation

A representation of a Lie algebra is a linear transformation

where is the set of all linear transformations of a vector space . In particular, if , then is the set of square matrices. The map is required to be a map of Lie algebras so that

for all . Note that the expression only makes sense as a matrix product in a representation. For example, if and are antisymmetric matrices, then is skew-symmetric, but may not be antisymmetric.

The possible irreducible representations of complex Lie algebras are determined by the classification of the semisimple Lie algebras. Any irreducible representation of a complex Lie algebra is the tensor product , where is an irreducible representation of the quotient of the algebra and its Lie algebra radical, and is a one-dimensional representation.

A Lie algebra may be associated with a Lie group, in which case it reflects the local structure of the Lie group. Whenever a Lie group has a group representation on , its tangent space at the identity, which is a Lie algebra, has a Lie algebra representation on given by the differential at the identity. Conversely, if a connected Lie group corresponds to the Lie algebra , and has a Lie algebra representation on , then has a group representation on given by the matrix exponential.