Irreducible (mathematics)

In mathematics, the term irreducible is used in several ways.

• In abstract algebra, irreducible can be an abbreviation for irreducible element; for example an irreducible polynomial.

• In universal algebra, irreducible can refer to the inability to represent an algebraic structure as a composition of simpler structures using a product construction; for example subdirectly irreducible.

• In representation theory (group theory), an irreducible representation is a nontrivial representation with no nontrivial subrepresentations. Similarly, an irreducible module is another name for a simple module.

• In the theory of manifolds, an n-manifold is irreducible if any embedded (n-1) sphere bounds an embedded n-ball. Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds.

The notions of irreducibility in algebra and manifold theory are related. An n-manifold is called prime, if it cannot be written as a connected sum of two n-manifolds (neither of which is an n-sphere). An irreducible manifold is thus prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however, the topologist (in particular the 3-manifold topologist) finds the definition above more useful. The only compact, connected 3-manifolds that are prime but not irreducible are the trivial 2-sphere bundle over S^1 and the twisted 2-sphere bundle over S^1.

A theorem of 3-manifold theory is: every compact, connected 3-manifold has a prime decomposition, i.e. can be written as a connected sum with each summand being prime. This prime decomposition is also unique (up to homeomorphism of summand). [Again, we must be working in either the differentiable or piecewise-linear category]

4. In algebraic geometry, an irreducible algebraic variety W is one that cannot be written as a union of subvarieties U and V, except when one of those is contained in the other.