In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge. For example, in an oriented 3-dimensional Euclidean space, an oriented plane can be represented by the exterior product of two basis vectors, and its Hodge dual is the normal vector given by their cross product; conversely, any vector is dual to the oriented plane perpendicular to it, endowed with a suitable bivector. Generalizing this to an n {\displaystyle n} -dimensional vector space, the Hodge star is a one-to-one mapping of k {\displaystyle k} -vectors to ( n − k ) {\displaystyle (n-k)} -vectors; the dimensions of these spaces are the binomial coefficients ( n k ) = ( n n − k ) {\displaystyle {\tbinom {n}{k}}={\tbinom {n}{n-k}}} . The naturality of the star operator means it can play a role in differential geometry when applied to the cotangent bundle of a pseudo-Riemannian manifold, and hence to differential k-forms. This allows the definition of the codifferential as the Hodge adjoint of the exterior derivative, leading to the Laplace–de Rham operator. This generalizes the case of 3-dimensional Euclidean space, in which divergence of a vector field may be realized as the codifferential opposite to the gradient operator, and the Laplace operator on a function is the divergence of its gradient. An important application is the Hodge decomposition of differential forms on a closed Riemannian manifold.
Formal definition Let V be an n-dimensional oriented vector space with a symmetric bilinear form ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } , referred to here as an inner product. (In more general contexts such as pseudo-Riemannian manifolds and Minkowski space, the bilinear form may not be positive-definite.) This induces an inner product on k-vectors α , β ∈ ⋀ k V {\textstyle \alpha ,\beta \in \bigwedge ^{\!k}V} , for 0 ≤ k ≤ n {\displaystyle 0\leq k\leq n} , by defining it on simple k-vectors α = α 1 ∧ ⋯ ∧ α k {\displaystyle \alpha =\alpha _{1}\wedge \cdots \wedge \alpha _{k}} and β = β 1 ∧ ⋯ ∧ β k {\displaystyle \beta =\beta _{1}\wedge \cdots \wedge \beta _{k}} to equal the Gram determinant
⟨ α , β ⟩ = det ( ⟨ α i , β j ⟩ i , j = 1 k ) {\displaystyle \langle \alpha ,\beta \rangle =\det \left(\left\langle \alpha _{i},\beta _{j}\right\rangle _{i,j=1}^{k}\right)}
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