Irreducible representation

A (linear) representation of a group (algebra, ring, semi-group) in a vector space (or topological vector space) with only two (closed) invariant subspaces, and . Frequently, an irreducible representation in a topological vector space is called a topologically-irreducible representation. If is a representation in a topological vector space and if it is irreducible as a representation in considered as a vector space, then 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.