Adjoint representation of a Lie algebra

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In mathematics, the adjoint endomorphism or adjoint action is a homomorphism of Lie algebras that plays a fundamental role in the development of the theory of Lie algebras.

Given an element $x$ of a Lie algebra $\mathfrak{g}$ , one defines the adjoint action of $x$ on $\mathfrak{g}$ as the map

\operatorname{ad}_x :\mathfrak{g}\to \mathfrak{g} \qquad \text{with} \qquad \operatorname{ad}_x (y) = [x,y]

for all $y$ in $\mathfrak{g}$ .

The concept generates the adjoint representation of a Lie group Ad. In fact, ad is the differential of Ad at the identity element of the group.

Adjoint representation

Let $\mathfrak{g}$ be a Lie algebra over a field $k$ . Then the linear mapping

\operatorname{ad}:\mathfrak{g} \to \operatorname{End}(\mathfrak{g})

given by $x \mapsto ad x$ is a representation of a Lie algebra and is called the adjoint representation of the algebra. (Its image actually lies in Der $(\mathfrak{g})$ . See below.)

Within End $(\mathfrak{g})$ , the Lie bracket is, by definition, given by the commutator of the two operators:

[\operatorname{ad}_x,\operatorname{ad}_y]=\operatorname{ad}_x \circ \operatorname{ad}_y - \operatorname{ad}_y \circ \operatorname{ad}_x

where ○ denotes composition of linear maps.

If $\mathfrak{g}$ is finite-dimensional, then End $(\mathfrak{g})$ is isomorphic to $\mathfrak{gl}(\mathfrak{g})$ , the Lie algebra of the general linear group over the vector space $\mathfrak{g}$ and if a basis for it is chosen, the composition corresponds to matrix multiplication.

Using the above definition of the Lie bracket, the Jacobi identity

[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0

takes the form

\left([\operatorname{ad}_x,\operatorname{ad}_y]\right)(z) = \left(\operatorname{ad}_{[x,y]}\right)(z)

where $x$ , $y$ , and $z$ are arbitrary elements of $\mathfrak{g}$ .

This last identity says that ad really is a Lie algebra homomorphism; i.e., a linear mapping that takes brackets to brackets.

In a more module-theoretic language, the construction simply says that $\mathfrak{g}$ is a module over itself.

The kernel of ad is, by definition, the center of $\mathfrak{g}$ . Next, we consider the image of ad. Recall that a derivation on a Lie algebra is a linear map $\delta:\mathfrak{g}\rightarrow \mathfrak{g}$ that obeys the Leibniz' law, that is,

\delta ([x,y]) = [\delta(x),y] + [x, \delta(y)]

for all $x$ and $y$ in the algebra.

That ad_x is a derivation is a consequence of the Jacobi identity. This implies that the image of $\mathfrak{g}$ under ad is a subalgebra of Der $(\mathfrak{g})$ , the space of all derivations of $\mathfrak{g}$ .

Structure constants

The explicit matrix elements of the adjoint representation are given by the structure constants of the algebra. That is, let {eⁱ} be a set of basis vectors for the algebra, with

[e^i,e^j]=\sum_k{c^{ij}}_k e^k.

Then the matrix elements for ad_eⁱ are given by

{\left[ \operatorname{ad}_{e^i}\right]_k}^j = {c^{ij}}_k ~.

Thus, for example, the adjoint representation of su(2) is the defining rep of so(3).

Relation to Ad

Ad and ad are related through the exponential map: crudely, Ad = exp ad, where Ad is the adjoint representation for a Lie group.

To be more precise, let $G$ be a Lie group, and let $Ψ: G \to Aut(G)$ be the mapping $g \mapsto Ψ g$ , with $Ψ g : G \to G$ given by the inner automorphism

\Psi_g(h)= ghg^{-1}~.

It is an example of a Lie group map. Define $Ad g$ to be the derivative of $Ψ g$ at the origin:

\operatorname{Ad}_g = (d\Psi_g)_e : T_eG \rightarrow T_eG

where $d$ is the differential and $T e G$ is the tangent space at the origin $e$ ( $e$ being the identity element of the group $G$ ).

The Lie algebra of $G$ is $\mathfrak{g}$ = $T e G$ . Since Ad_g ∈ Aut $(\mathfrak{g})$ , $Ad: g \mapsto Ad g$ is a map from $G$ to $Aut(T e G)$ which will have a derivative from $T e G$ to $End(T e G)$ (the Lie algebra of $Aut(V)$ being $End(V)$ ).

Then we have

\operatorname{ad} = d(\operatorname{Ad})_e:T_eG\rightarrow \operatorname{End} (T_eG).

The use of upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector $x$ in the algebra $\mathfrak{g}$ generates a vector field $X$ in the group $G$ . Similarly, the adjoint map $ad x y = [x, y]$ of vectors in $\mathfrak{g}$ is homomorphic to the Lie derivative $L X Y = [X, Y]$ of vector fields on the group $G$ considered as a manifold.

Further see the derivative of the exponential map.

References

Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6

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