Complexify

Complexify is a verb meaning to make something more complex, or in a mathematical context, to extend a real structure to incorporate complex numbers. The term derives from complex and the suffix -ify, and is used both in general language and in specialized mathematical settings.

In mathematics, complexification refers to extending scalars from the real numbers to the complex numbers. For a real vector space V, its complexification is V_C = V ⊗R C, a complex vector space whose complex dimension equals the real dimension of V. There is a natural embedding V → V_C given by v ↦ v ⊗ 1, and V_C can be identified with V ⊕ iV. For a real Lie algebra g, the complexification g_C = g ⊗R C carries a natural complex Lie algebra structure; representations of g can be studied by complexifying representations of g.

Similar constructions occur for other real objects. A real algebraic variety X_R complexifies to a complex variety X_C, and a real analytic manifold M can be complexified to a complex manifold M^C under suitable conditions. Complexification thus serves as a bridge to apply complex-analytic and geometric methods to real objects, enabling techniques from complex geometry, representation theory, and algebraic geometry.

In broader usage, complexify can also mean to render something more intricate or nuanced, beyond the strict mathematical sense. The noun form complexification is used in contexts such as discussions of scalar extension, or the process of making a structure capable of admitting complex numbers.

See also: complexification (mathematics), scalar extension.