involutions

An involution is an operation or element that is its own inverse. In a set X with a self-map f: X → X, f is an involution if applying it twice yields the identity, f(f(x)) = x for every x ∈ X. In algebraic structures, the term takes slightly different but related meanings: an element g of a group is an involution if g^2 = e (the identity); an involution on an algebra A is an anti-automorphism of order 2, satisfying (ab)^* = b^* a^*, (a^*)^* = a, and (a + b)^* = a^* + b^* (and λa)^* = λ̄ a^* for scalars).

Examples abound. Complex conjugation z ↦ z̄ is an involution on the complex numbers, since z̄̄ = z and (zw)̄ = z̄ w̄. The transpose map A ↦ A^T is an involution on the ring of matrices; the conjugate transpose A ↦ A* is a star-involution on complex matrix algebras. In the broader group sense, any element with square equal to the identity, such as −I or a reflection, is an involution. The negation map x ↦ −x is an involution on any vector space, and more generally a linear transformation with T^2 = I is an involution.

In topology and geometry, an involution is a self-map f with f^2 = id, often representing a symmetry. Involutions help split objects into fixed-point sets and pairs swapped by the map, feeding into fixed-point theory, representation theory, and the structure theory of algebras. In physics, time-reversal and certain parity operations are modeled as involutions in suitable mathematical frameworks.