Hodge dual

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In mathematics, the Hodge star operator on a vector space V is a linear operator on the exterior algebra of V, interchanging the subspaces of k-vectors and n−k-vectors where n = dim V, for 0 ≤ k ≤ n. In rough terms it is defined by dividing into a volume element ω, thought of as n standard basis vectors wedged together, so that

α*α = ω

up to sign, whenever α is a wedge of some standard basis vectors.

A common example of the star operator is the case n = 3, when it can be taken as the correspondence between the vectors and the skew-symmetric matrices of that size.

Given a measure over an n-dimensional manifold expressible as an n-form μ (not all measures are of this form, for example, the Dirac delta function), the Hodge dual of a p-form A is defined as the contraction where is the dual n-vector. See sign convention.

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