# 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.

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

- If
*G* is SL_{2}(**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.

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(*t*_{1},...,*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*_{i}*t*_{j}^{-1} on the various off-diagonal entries. The roots of *G* are the weights
diag(*t*_{1},...,*t*_{n})→*t*_{i}*t*_{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}.