Hodge star
In mathematics, the Hodge isomorphism or Hodge star operator 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. The result when applied to an element is called the element's Hodge dual.
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 its Hodge dual. The former space, of k-vectors, has dimension , while the latter has dimension , which are equal by the symmetry of the binomial coefficients. Equal-dimensional vector spaces are always isomorphic, but not necessarily in a natural or canonical way. In this case, however, Hodge duality exploits the nondegenerate symmetric bilinear form, hereafter referred to as the inner product (though it might not be positive definite), and a choice of orientation to single out a unique isomorphism, which parallels the combinatorial symmetry of binomial coefficients. This in turn induces an inner product on the space of k-vectors. The naturalness of this definition means the duality can play a role in differential geometry.
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