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A Lie algebra is a vector space with a Lie bracket , satisfying the Jacobi identity. Hence any element gives a linear transformation given by

 (1)

which is called the adjoint representation of . It is a Lie algebra representation because of the Jacobi identity,

 (2) (3) (4)

A Lie algebra representation is given by matrices. The simplest Lie algebra is the set of matrices. Consider the adjoint representation of , which has four dimensions and so will be a four-dimensional representation. The matrices

 (5) (6) (7) (8)

give a basis for . Using this basis, the adjoint representation is described by the following matrices:

 (9) (10) (11) (12)

Commutator, Group Representation, Lie Algebra, Lie Group, Lie Bracket, Nilpotent Lie Algebra, Semisimple Lie Algebra

This entry contributed by Todd Rowland

## References

Fulton, W. and Harris, J. Representation Theory. New York: Springer-Verlag, 1991.Jacobson, N. Lie Algebras. New York: Dover, 1979.Knapp, A. Lie Groups Beyond an Introduction. Boston, MA: Birkhäuser, 1996.