Solving the Lie algebra of generators: path from algebra to matrix representation

Given the Lie algebra, what is the systematic way to construct the matrix representation of the generators of the desired dimension? I ask this question here because it is the physicists for whom representation of groups is more important than mathematicians.

Let us, for example, take $SU(2)$ for concreteness. Starting from the generic parametrization of a $3\times 3$ unitary matrix $U$ with $\det U=1$, and using the formula of generators

$$J^i=-i\Big(\frac{\partial U}{\partial \theta_i}\Big)_{\{\theta_i=0\}}$$ one can find the $3\times 3$ matrix representation of the generators.

However, I'm looking for something else.

1. 
   

   Given the Lie algebra $[J^i, J^j]=i\epsilon^{ijk}J^k$, is there a way that one can explicitly construct (not by guess or trial) the $3\times 3$ representations of $\{J^i\}$?

   
   


2. 
   

   Will the same procedure apply to solve other Lie algebras appearing in physics such as that of $SO(3,1)$ (or $SL(2,\mathbb{C})$)?

   
   

Answer

If you have the structure constants, i.e. the coefficients $f_{ab}^c$ in the commutation relations $[T_a,T_b]=\sum_c f_{ab}^cT_c$ with $a,b,c\in \{1,\ldots p\}$ then you can construct $p$ matrices $M_a$ (labelled by $a$) of size $p\times p$ with entries $(M_a)_{cb}$. These matrices will be a $p\times p$ representation of the algebra (in fact, the adjoint representation.)

This will work for every algebra. However, in the case of non-compact algebras, the resulting representation is obviously finite dimensional and thus cannot be make hermitian (i.e. it cannot exponentiate to a unitary, finite dimensional representation of the group.)

Be aware that in the case of $so(3,1)$, there is a subtle point that comes in going from the complex back to the real form. Over $\mathbb{C}$, the adjoint of $so(3,1)$ is reducible into $su(2)\oplus su(2)$ but over the reals the adjoint is irreducible. In other words, for non-compact real forms, there are issues with reducibility and unitary.