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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.

See also

Group Representation, Irreducible Representation, Lie Algebra, Lie Group, Matrix Exponential, Simple Lie Algebra, Vector Space

This entry contributed by Todd Rowland

Cite this as:

Rowland, Todd. "Lie Algebra Representation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LieAlgebraRepresentation.html