Conjugation and the adjoint representation of a Lie group

Let $G$ be a finite-dimensional real Lie group with identity $e$ and Lie algebra $\mathfrak{g}=T_{e}G$. For $g\in G$, conjugation by $g$ is

$C_g:G⟶G,C_g(h)=gh{g}^{-1}\mathrm{.}$

It is a Lie-group automorphism. Indeed, associativity gives $C_g(hk)=C_g(h)C_g(k)$, its smoothness follows from the smooth multiplication and inversion of Lie group, and $C_{g^{-1}}$ is its smooth inverse. Thus its differential at the identity is the invertible linear map

${\mathrm{Ad}}_g:=d(C_g{)}_e:\mathfrak{g}⟶\mathfrak{g}\mathrm{.}$

Here $C_g(e)=e$, so the source and target tangent spaces are both $\mathfrak{g}$; invertibility follows from The differential of a diffeomorphism is an isomorphism. Write $\mathrm{GL}(\mathfrak{g})$ for the group of invertible real-linear maps of $\mathfrak{g}$ in the sense of Invertible linear maps, linear isomorphisms, and inverse linear maps. The adjoint map is

$\mathrm{Ad}:G⟶\mathrm{GL}(\mathfrak{g}),g⟼{\mathrm{Ad}}_g\mathrm{.}$

The target carries its standard smooth structure. Explicitly, if $n=\dim \mathfrak{g}\ge 1$, one fixed basis identifies $\mathcal{L}(\mathfrak{g},\mathfrak{g})$ with $M_n(\mathbb{R})\simeq {\mathbb{R}}^{n^2}$ by Coordinate columns $[v{]}_{\mathcal{B}}$ and matrices $[T{]}_{\mathcal{B}}^{\mathcal{C}}$ of linear maps relative to ordered bases. Under this identification $\mathrm{GL}(\mathfrak{g})$ is the open set $\det \mathrm{\ne }0$: determinant is a polynomial by For every fixed finite size at least one, the determinant of a real square matrix is a polynomial in its matrix entries, and invertibility is equivalent to nonzero determinant by A finite square real matrix is invertible if and only if its determinant is nonzero. It therefore has the restricted smooth structure from An open subset of a smooth manifold has a canonical restricted smooth structure. A second basis changes a matrix by $A\mapsto P^{-1}AP$, a linear diffeomorphism, so this smooth structure is independent of the fixed basis. If $n=0$, then $\mathrm{GL}(\mathfrak{g})$ is the singleton containing the unique endomorphism of the zero vector space, with its unique zero-dimensional smooth structure; no determinant criterion is needed.

The next proposition proves that this map is a smooth group representation; after that result it is called the adjoint representation of $G$.

A Lie group is nonempty and boundaryless. If $\dim G=0$, every ${\mathrm{Ad}}_g$ is the unique automorphism of the zero vector space, even when the discrete group itself is nonabelian; dimension one requires no change. No metric, nondegeneracy, interval, endpoint, choice principle, or biconditional is involved. Fixing one finite basis of the supplied finite-dimensional space is a single finite existential instantiation, not a choice from a family.