Adjoint representation

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In mathematics, the adjoint representation (or adjoint action) of a Lie group G is the natural representation of G on its own Lie algebra. This representation is the linearized version of the action of G on itself by conjugation.

Formal definition

Let G be a Lie group and let be its Lie algebra (which we identify with TeG, the tangent space to the identity element in G). Define a map Ψ : G → Aut(G) by

For each g in G, Ψg is an automorphism of G. It follows that the derivative of Ψg at the identity is an automorphism of the Lie algebra . We denote this map by Adg:

To say that Adg is an Lie algebra automorphism is to say that Adg is a linear transformation of that preserves the Lie bracket. The map

which sends g to Adg is called the adjoint representation of G. This is indeed a representation of G since is a Lie subgroup of and the above adjoint map is a Lie group homomorphism. The dimension of the adjoint representation is the same as the dimension of the group G.

Adjoint representation of a Lie algebra

One may always pass from a representation of a Lie group G to a representation of its Lie algebra by taking the derivative at the identity. Taking the derivative of the adjoint map

gives the adjoint representation of the Lie algebra :

Here is the Lie algebra of which may be identified with the derivation algebra of . The adjoint representation of a Lie algebra is related in a fundamental way to the structure of that algebra. In particular, one can show that

for all . For more information see: adjoint representation of a Lie algebra.

Examples

  • If G is abelian of dimension n, the adjoint representation of G is the trivial n-dimensional representation.
  • If G is a matrix Lie group (i.e. a closed subgroup of ), then its Lie algebra is an algebra of n×n matrices with the commutator for a Lie bracket (i.e. a subalgebra of ). In this case, the adjoint map is given by Adg(x) = gxg−1.
  • 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.

Properties

The following table summarizes the properties of the various maps mentioned in the definition

Lie group homomorphism:
Lie group automorphism:
Lie group homomorphism:
Lie algebra automorphism:
  • is linear
Lie algebra homomorphism:
  • is linear
Lie algebra derivation:
  • is linear


The image of G under the adjoint representation is denoted by AdG. If G is connected, the kernel of the adjoint representation coincides with the kernel of Ψ which is just the center of G. Therefore the adjoint representation of a connected Lie group G is faithful if and only if G is centerless. More generally, if G is not connected, then the kernel of the adjoint map is the centralizer of the identity component G0 of G. By the first isomorphism theorem we have

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=SLn(R). We can take the group of diagonal matrices diag(t1,...,tn) 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 titj-1 on the various off-diagonal entries. The roots of G are the weights diag(t1,...,tn)→titj-1. This accounts for the standard description of the root system of G=SLn(R) as the set of vectors of the form ei-ej.

Variants and analogues

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. This relationship is closest in the case of nilpotent Lie groups.