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 g→ad(g) is the adjoint representation.

Any Lie group is a representation of itself (via ) and the tangent space is mapped to itself by the group action. This gives the linear adjoint representation.

Table of contents

1 Examples 2 Variants and analogues 3 Roots of a semisimple Lie group

Examples

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

• 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) = [x 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 contragredient 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.

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

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 desciption of the root system of G=SL_n(R) as the set of vectors of the form e_i-e_j.