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Adjoint Representation

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

ad(X)(Y)=[X,Y],
(1)

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

[ad(X_1),ad(X_2)](Y)=[X_1,[X_2,Y]]-[X_2,[X_1,Y]]
(2)
=[[X_1,X_2],Y]
(3)
=ad([X_1,X_2])(Y).
(4)

A Lie algebra representation is given by matrices. The simplest Lie algebra is gl_n the set of matrices. Consider the adjoint representation of gl_2, which has four dimensions and so will be a four-dimensional representation. The matrices

e_1=[1 0; 0 0]
(5)
e_2=[0 1; 0 0]
(6)
e_3=[0 0; 1 0]
(7)
e_4=[0 0; 0 1]
(8)

give a basis for gl_2. Using this basis, the adjoint representation is described by the following matrices:

ad(e_1)=[ 0 0 0 0; 0 1 0 0; 0 0 -1 0; 0 0 0 0]
(9)
ad(e_2)=[ 0 0 1 0; -1 0 0 1; 0 0 0 0; 0 0 -1 0]
(10)
ad(e_3)=[ 0 -1 0 0; 0 0 0 0; 1 0 0 -1; 0 1 0 0]
(11)
ad(e_4)=[ 0 0 0 0; 0 -1 0 0; 0 0 1 0; 0 0 0 0].
(12)

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.

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Adjoint Representation

Cite this as:

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

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