A Lie algebra is a vector space 
with a Lie bracket ![[X,Y]](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FAdjointRepresentation%2FInline2.svg){kind=small}
, satisfying the Jacobi
identity. Hence any element 
gives a linear transformation given by
![ad(X)(Y)=[X,Y],](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FAdjointRepresentation%2FNumberedEquation1.svg){kind=small} |
(1) |
which is called the adjoint representation of 
. It is a Lie algebra
representation because of the Jacobi identity,
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
give a basis for 
. Using this basis, the adjoint representation is described
by the following matrices:
See also
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.
Referenced on Wolfram|Alpha
Cite this as:
Rowland, Todd. "Adjoint Representation." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AdjointRepresentation.html