Lie Algebra Representation

A representation of a Lie algebra g is a linear transformation


where M(V) is the set of all linear transformations of a vector space V. In particular, if V=R^n, then M(V) is the set of n×n square matrices. The map psi is required to be a map of Lie algebras so that


for all A,B in g. Note that the expression AB only makes sense as a matrix product in a representation. For example, if A and B are antisymmetric matrices, then AB-BA is skew-symmetric, but AB 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 V of a complex Lie algebra g is the tensor product V=V_0 tensor L, where V_0 is an irreducible representation of the quotient g_(ss)/Rad(g) of the algebra g and its Lie algebra radical, and L 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 G has a group representation on V, its tangent space at the identity, which is a Lie algebra, has a Lie algebra representation on V given by the differential at the identity. Conversely, if a connected Lie group G corresponds to the Lie algebra g, and g has a Lie algebra representation on V, then G has a group representation on V 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

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Cite this as:

Rowland, Todd. "Lie Algebra Representation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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