Indecomposable representation

A representation of a group (or algebra, ring, semi-group, etc.) that is not equivalent to a direct sum of non-zero representations of the same group (or algebra, etc.). Thus, the indecomposable representations must be regarded the simplest representations of the relevant algebraic system. With the aid of these representations one can study the structure of the algebraic system, its representation theory and harmonic analysis on the system. A representation of a topological group (or algebra, etc.) in a topological vector space is called indecomposable if it is not equivalent to a topological direct sum of non-zero representations of the same algebraic system.

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.

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