In quantum physics we have to work a lot with Clebsch-Gordan coefficients and generalizations like the Wigner 3j,6j, and 9j symbols. In our coursework we are taught that the coefficients are coupling constants between angular momenta, or more specifically, transformation constants between one tensor space basis (m,m' - individual spin as basis vectors) and another tensor space basis (j,m total spin as basis vectors).
What I don't understand is what this "coupling" represents in an abstract mathematical setting. We are taught to think of spin as representation of some abstract Lie Group. How does one "couple" two Lie groups together? From whence comes more complicated constructs like 6j or 9j coupling coefficients?
I'm sorry if the question is a little broad and/or unclear, I have (what I believe) to be a good intuitive understanding of spin, however I know almost nothing of the true formalism behind the concept so any resources to remedy this would be greatly appreciated.