 TOPICS  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

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## 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.