# Other representations

For a Lie group $${G}$$, the inner automorphism $${\phi_{g}\colon G\to G}$$ induced by a fixed $${g\in G}$$ is defined by $${h\mapsto ghg^{-1}}$$, and can be viewed as an action of $${G}$$ on itself. At the identity $${e}$$ we then have the map $${\left(\mathrm{d}\phi_{g}\right)\left|_{e}\colon\mathfrak{g}\to\mathfrak{g}\right.}$$. The adjoint representation $${\textrm{Ad}\colon G\to GL\left(\mathfrak{\mathfrak{g}}\right)}$$ represents $${G}$$ on $${\mathfrak{g}}$$, and for $${A\in\mathfrak{g}}$$ is defined by $${g_{\textrm{Ad}}\left(A\right)\equiv\left(\mathrm{d}\phi_{g}\right)\left|_{e}\left(A\right)\right.}$$; it is often denoted $${\textrm{Ad}_{g}A}$$. Using the exponential map, one can show that $${\textrm{exp}\left(tg_{\textrm{Ad}}\left(A\right)\right)=g\textrm{exp}\left(tA\right)g^{-1}}$$. If $${G}$$ is a matrix group, so that $${g}$$ and $${A}$$ are both matrices, the adjoint representation is simply the similarity transformation $${g_{\textrm{Ad}}(A)=gAg^{-1}}$$.

The adjoint representation also sometimes refers to the representation of the Lie algebra $${\mathfrak{g}}$$ on itself defined by the differential of $${\textrm{Ad }}$$ at the identity: $${\textrm{ad}\equiv(\mathrm{d}\textrm{Ad})\left|_{e}\colon\mathfrak{g}\to gl\left(\mathfrak{g}\right)\right.}$$. It can easily be shown that for a given $${A\in\mathfrak{g}}$$, $${A_{\textrm{ad}}}$$ is just the Lie derivative $${A_{\textrm{ad}}\left(B\right)=L_{A}\left(B\right)=\left[A,B\right]}$$.

The trivial representation maps all of $${G}$$ to the identity on a one-dimensional vector space; this representation is irreducible, and the corresponding Lie algebra representation maps all of $${\mathfrak{g}}$$ to $${0}$$.

Closely related to linear representations are projective representations and affine representations, homomorphisms from $${G}$$ to $${\textrm{Aut}(X)}$$ where $${X}$$ is a projective or affine space. Recall that a projective space is obtained from a vector space by taking all lines through the origin, i.e. by identifying scalar multiples of vectors. We can then view a projective representation as mapping each group element to an automorphism of a vector space $${V}$$ “ignoring vector length.” An affine space is defined as a set on which a vector space acts freely and transitively as an additive group. Thus any two points in an affine space can be identified with the vector whose action relates them; i.e. an affine space “ignores the origin,” so that vectors are defined between any two points, even if one is not the origin. In particular, a group representation that maps each group element to an automorphism on $${V}$$ that is an affine map (AKA inhomogeneous transformation), the sum of a linear and constant map, is an affine representation.