Given a Lie Group (take for example the classical ones) we know that there are certain representations for it. For example in $SU(N)$ we can have an adjoint, fundamental, symmetric and antisymmetric representations.
On the other hand we can study the associated Lie Algebra and the root system, weight and root lattices etc (For example take the Appendix C of this). What's the relation between the aforementioned representations and the classification of the Algebra?