# Indecomposable representation

Every irreducible representation is indecomposable. The class of finite-dimensional indecomposable representations of the group $\mathbf R$ and the decomposition of a given finite-dimensional representation of $\mathbf R$ into indecomposable ones are directly connected with the Jordan normal form of a matrix and the theory of linear ordinary differential equations with constant coefficients. The classification of indecomposable representations of even such groups as $\mathbf R ^ {n}$ and $\mathbf Z ^ {n}$, $n> 1$, is (1982) far from complete. Indecomposable representations of semi-direct products of groups, in particular, of solvable Lie groups, can be reducible (even in the finite-dimensional case). On the other hand, finite-dimensional indecomposable representations of real semi-simple Lie groups are irreducible. However, these groups have reducible infinite-dimensional indecomposable representations, notably, the analytic continuation of the fundamental continuous series of representations of such groups.