Group and algebra representations

A group representation (AKA rep, linear representation) is a linear group action on a real or complex vector space $${V}$$, i.e. a homomorphism $${\rho\colon G\to GL(V)}$$ from $${G}$$ to the Lie group of linear invertible automorphisms of the manifold $${V=\mathbb{R}^{n}}$$ or $${\mathbb{C}^{n}}$$. We can choose a basis of $${V}$$ to get a isomorphism from $${GL(V)}$$ to $${GL(n,\mathbb{R})}$$ or $${GL(n,\mathbb{C})}$$, in which case the representation is called a matrix representation. For a matrix rep, the transpose switches left and right actions like the inverse, as does an action by the matrix on row vectors instead of column vectors; in particular, a left matrix rep on vector components $${v^{\prime\mu}=g^{\mu}{}_{\lambda}v^{\lambda}}$$ is equivalent to a left action

$$\displaystyle e_{\mu}^{\prime}=(g^{-1})^{\lambda}{}_{\mu}e_{\lambda}=\begin{bmatrix}e_{1} & \cdots & e_{n}\end{bmatrix}\left[g^{-1}\right]$$

on the basis, since the inverse of the matrix acts on a row vector. The $${G}$$-space $${V}$$ is called a representation space, and an equivariant linear map between representation spaces of the same group is called an intertwiner (AKA intertwining map).

 Δ It is common to use “representation” to refer to the representation space $${V}$$, with the group $${G}$$ and the mapping $${\rho}$$ inferred from context. Now, a group representation $${\rho\colon G\to GL(V)}$$, being a homomorphism, satisfies $${\rho\left(gh\right)=\rho\left(g\right)\rho\left(h\right)}$$. Similarly, an algebra representation of an associative algebra $${\mathfrak{a}}$$ is defined to be a linear homomorphism $${\rho\colon\mathfrak{a}\to gl\left(V\right)}$$, e.g. for scalar $${a}$$ and vectors $${A,B,C}$$ in $${\mathfrak{a}}$$, we require that $${\rho\left(aA+BC\right)=a\rho\left(A\right)+\rho\left(B\right)\rho\left(C\right)}$$. An algebra representation is also referred to as a G-module, or just a module, since if we ignore scalars in both $${V}$$ and $${\mathfrak{a}}$$, $${V}$$ acted on by $${\mathfrak{a}}$$ can be viewed as a module with vectors in the abelian group $${V}$$ and scalars in the ring $${\mathfrak{a}}$$.

We can now use the Lie commutator to define the related Lie algebra representation of a Lie algebra $${\mathfrak{g}}$$ as a smooth linear homomorphism $${\rho\colon\mathfrak{g}\to gl\left(V\right)}$$, i.e. we require $${\rho\left(\left[A,B\right]\right)=\rho\left(A\right)\rho\left(B\right)-\rho\left(B\right)\rho\left(A\right)}$$. Note that a Lie algebra derived from a real Lie group is by definition a real vector space, since it lies in the tangent space of a real manifold; thus the scalar field in such a Lie algebra, even if defined by complex matrices, is the field of reals.