Complexification

Complexification is the process of extending scalars from the real numbers to the complex numbers for mathematical structures defined over the real field. Formally, 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.

The idea generalizes to many objects. For a real associative algebra A, its complexification is A_C = A ⊗_R C, a complex algebra with A embedded via a ↦ a ⊗ 1. For a real algebraic variety X defined over R, the complexification X_C is X ×_R C; its complex points correspond to solutions in C^n of the defining equations. For a real Lie algebra g, its complexification is g_C = g ⊗_R C. A real Lie group G has a complexification G_C, a complex Lie group with Lie algebra g_C, of which G is a real form (often realized as the fixed points of an appropriate anti-holomorphic involution on G_C).

Properties and uses. Complexification is functorial: a real-linear map f: V → W extends to f_C: V_C → W_C. It preserves many algebraic structures and allows the use of complex-analytic or algebraic techniques to study real objects. It also helps in representation theory, differential geometry, and algebraic geometry by enabling holomorphic methods and base-change arguments.

Examples and caveats. The complexification of R^n is C^n, and R[x] becomes C[x]. Not every complex object arises as a complexification of a real one, and complexification can introduce new phenomena not present over R; some real-analytic or geometric structures may not extend nicely to the complex setting.