adjoint representation

The adjoint representation of a Lie group G is the linearized version of the action of G on itself by conjugation. For each g in G, the inner automorphism x→gxg^-1 gives a linear transformation Ad(g) from the Lie algebra of G, i.e., the tangent space of G at the identity element, to itself. The map Ad(g) is called the adjoint endomorphism; the map g→Ad(g) is the adjoint representation. Any Lie group is a representation of itself (via

h\rightarrow ghg^{-1}

) and the tangent space is mapped to itself by the group action. This gives the linear adjoint representation.

Examples

If G is commutative of dimension n, the adjoint representation of G is the trivial n-dimensional representation.

The kernel of the adjoint representation of G is the center of G.

If G is SL₂(R) (real 2×2 matrices with determinant 1), the Lie algebra of G consists of real 2×2 matrices with trace 0. The representation is equivalent to that given by the action of G by linear substitution on the space of binary (i.e., 2 variable) quadratic forms.

Variants and analogues

The adjoint representation of a Lie algebra L sends x in L to ad(x), where

ad(x)(y) = y.

If L arises as the Lie algebra of a Lie group G, the usual method of passing from Lie group representations to Lie algebra representations sends the adjoint representation of G to the adjoint representation of L. The adjoint representation can also be defined for algebraic groups over any field. The co-adjoint representation is the contragradient representation of the adjoint representation. A. Kirillov observed that the orbit of any vector in a co-adjoint representation is a symplectic manifold. According to the philosophy in representation theory known as the orbit method, the irreducible representations of a Lie group G should be indexed in some way by its co-adjoint orbits. This relationship is closest in the case of nilpotent Lie groups.

Roots of a semisimple Lie group

If G is semisimple, the non-zero weights of the adjoint representation form a root system. To see how this works, consider the case G=SL_n(R). We can take the group of diagonal matrices diag(t₁,...,t_n) as our maximal torus T. Conjugation by an element of T sends

\begin{bmatrix} a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}\\ \end{bmatrix} \mapsto \begin{bmatrix} a_{11}&t_1t_2^{-1}a_{12}&\cdots&t_1t_n^{-1}a_{1n}\\ t_2t_1^{-1}a_{21}&a_{22}&\cdots&t_2t_n^{-1}a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ t_nt_1^{-1}a_{n1}&t_nt_2^{-1}a_{n2}&\cdots&a_{nn}\\ \end{bmatrix}.

Thus, T acts trivially on the diagonal part of the Lie algebra of G and with eigenvectors t_it_j^-1 on the various off-diagonal entries. The roots of G are the weights diag(t₁,...,t_n)→t_it_j^-1. This accounts for the standard description of the root system of G=SL_n(R) as the set of vectors of the form e_i-e_j.

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