Irreducible representation

A (linear) representation $\pi$ of a group (algebra, ring, semi-group) $X$ in a vector space (or topological vector space) $E$ with only two (closed) invariant subspaces, $(0)$ and $E$. Frequently, an irreducible representation in a topological vector space is called a topologically-irreducible representation. If $\pi$ is a representation in a topological vector space $E$ and if it is irreducible as a representation in $E$ considered as a vector space, then $\pi$ is called an algebraically-irreducible representation. An algebraically-irreducible representation is also topologically irreducible; the converse is not true, in general. There are a number of concepts close to that of an irreducible representation, among them an operator-irreducible representation, and a completely-irreducible representation (one for which the family of operators forms a completely-irreducible set, cf. also Completely-reducible set). A completely-irreducible representation is (topologically) irreducible and operator irreducible; the converse assertions are not true, in general.