The irreducible representations of the complex simple Lie algebras have been fully classified, and as it turns out, these representations apply directly to their compact real forms. Using similar reasoning to above, the irreducible representations of the compact connected Lie algebras and Lie groups have also been fully classified.

An important example is that of the Lie algebra \({su(2)}\): up to similarity transformations there is one unique complex irreducible representation of \({su(2)}\) with dimension \({m}\) for every \({m\geq1}\). These representations are associated with angular momentum in quantum physics.

One can also classify the representations of finite groups, which are always completely reducible. In particular, **Young tableaux** are combinatorial diagrams used to enumerate the representations of the symmetric group \({S_{n}}\).