Hodge dual

In mathematics, the Hodge star operator or Hodge dual is an important linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form.

Suppose that n is the dimension 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 to 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 dimension

while the latter has dimension

and by the symmetry of the binomial coefficients, these two dimensions are equal. Two vector spaces over the same field with the same dimension 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, which 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

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