Selbstdual

Selbstdual is a term used in mathematics, particularly in abstract algebra and graph theory, to describe objects that are equivalent to their own dual. The concept of duality refers to a relationship where two mathematical structures can be transformed into each other in a systematic way, often preserving certain properties.

In the context of vector spaces, a space is considered selbstdual if there exists a non-degenerate bilinear form that is symmetric or skew-symmetric, establishing an isomorphism between the space and its dual space. This means that the space can be mapped to its own dual in a way that is both linear and preserves the bilinear form.

In graph theory, a graph is selbstdual if it is isomorphic to its own dual graph. The dual of a planar graph is another planar graph where vertices correspond to the regions (or faces) of the original graph, and edges connect vertices that correspond to adjacent regions. A graph that is selbstdual exhibits a symmetry where the graph and its corresponding dual representation are structurally identical.

The property of being selbstdual often implies a high degree of symmetry or a special kind of structure within the mathematical object. It is a concept that arises in various areas of mathematics and physics, highlighting fundamental relationships and equivalences.