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 $\circ$ 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_{e}G\rightarrow T_{e}G

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_{e}G\rightarrow \operatorname {End}(T_{e}G).

The 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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