Selfdual

Self-dual, or selfdual, is an adjective used in various fields to describe an object that is, in a suitable sense, equivalent to its dual. The meaning of duality depends on context, but common uses involve a dual object constructed from the original one, such as a dual vector space, a dual graph, or a dual code.

In linear algebra and representation theory, a finite-dimensional vector space V over a field with a nondegenerate bilinear form provides a canonical isomorphism with its dual space V*. A representation of a group G is called self-dual if V is isomorphic to V* as a G-module; such representations can be characterized as orthogonal or symplectic according to the invariant bilinear form that realizes the identification.

In differential geometry and gauge theory, duality often refers to the Hodge star operator on differential forms. In four dimensions, self-dual and anti-self-dual 2-forms decompose under the star operation. A connection whose curvature F satisfies F = *F is called self-dual; these instanton solutions minimize the Yang–Mills action and play a central role in gauge theory and topology.

In combinatorics and geometry, a graph is self-dual if its dual graph is isomorphic to the original. A polyhedron is self-dual if its dual has the same combinatorial structure (the tetrahedron is self-dual in this sense). In coding theory, a linear code is self-dual if it equals its dual code with respect to the standard inner product, typically implying length n and dimension n/2.

Self-duality thus denotes a deep symmetry between an object and its dual, across algebra, geometry, combinatorics, and information theory.