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DIMENSIONS AND ALGEBRA


The Hodge star operator establishes a correspondence between the space of ''k''-vectors and the space of (''n'' −''k'')-vectors. The image of a ''k''-vector under this isomorphism is called the ''Hodge dual'' of the ''k''-vector. The former space, of ''k''-vectors, has dimension

: {n \choose k}

while the latter has dimension

: {n \choose n - k},

and by the symmetry of the Binomial Coefficient s, these two dimensions are in fact equal.
Two Vector Space s 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, 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 spaces of dimension 3, which are ''V'' itself and the space of Wedge Product s 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 is available in all dimensions.


EXTENSIONS


Since the space of alternating linear forms in ''k'' arguments on a vector space is naturally isomorphic to the dual of the space of ''k''-vectors over that vector space, the Hodge dual can be defined for these spaces as well. As with most constructions from linear algebra, the Hodge dual can then be extended to a Vector Bundle . Thus a context in which the Hodge dual is very often seen is the exterior algebra of the cotangent bundle (i.e. the space of differential forms on a manifold) where it can be used to construct the codifferential from the Exterior Derivative , and thus the Laplace-de Rham Operator , which leads to the Hodge Decomposition of Differential Forms in the case of Compact Riemannian Manifold s.


FORMAL DEFINITION OF THE HODGE STAR OF ''K''-VECTORS

The Hodge star operator on an Oriented Inner Product 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''. It has the following property, which defines it completely: given an oriented orthonormal basis e_1,e_2,...,e_n we have

  • (e_1\wedge e_2\wedge ... \wedge e_k)= e_{k+1}\wedge e_{k+2}\wedge ... \wedge e_n.



TENSOR NOTATION FOR THE STAR OPERATOR


Using index notation, the Hodge dual is obtained by contracting the indices of a ''k''-form with the ''n''-dimensional completely antisymmetric Levi-Civita Symbol . Thus one writes

  • \eta)_{i_1,i_2,\ldots,i_{n-k}}=

    • rac{1}{k!} \eta^{j_1,\ldots,j_k}\epsilon_{j_1,\ldots,j_k,i_1,\ldots,i_{n-k}}