Hodge Dual - Dimensionalities and Algebra

Dimensionalities and Algebra

Suppose that n is the dimensionality of the oriented inner product space and k is an integer such that 0 ≤ k ≤ n, then the Hodge star operator establishes a one-to-one mapping from the space of k-vectors and the space of (n−k)-vectors. The image of a k-vector under this mapping is called the Hodge dual of the k-vector. The former space, of k-vectors, has dimensionality

while the latter has dimensionality

and by the symmetry of the binomial coefficients, these two dimensionalities are in fact equal. Two vector spaces over the same field with the same dimensionality are always isomorphic; but not necessarily in a natural or canonical way. The Hodge duality, however, in this case exploits the inner product and orientation of the vector space. It singles out a unique isomorphism, that reflects therefore the pattern of the binomial coefficients in algebra. This in turn induces an inner product on the space of k-vectors. The 'natural' definition means that this duality relationship can play a geometrical role in theories.

The first interesting case is on three-dimensional Euclidean space V. In this context the relevant row of Pascal's triangle reads

1, 3, 3, 1

and the Hodge dual sets up an isomorphism between the two three-dimensional spaces, which are V itself and the space of wedge products of two vectors from V. See the Examples section for details. In this case the content is just that of the cross product of traditional vector calculus. While the properties of the cross product are special to three dimensions, the Hodge dual applies to all dimensionalities.