Duality
The Hodge star defines a dual in that when it is applied twice, the result is an identity on the exterior algebra, up to sign. Given a k-vector in an n-dimensional space V, one has
where s is related to the signature of the inner product on V. Specifically, s is the sign of the determinant of the inner product tensor. Thus, for example, if n=4 and the signature of the inner product is either (+,−,−,−) or (−,+,+,+) then s=−1. For ordinary Euclidean spaces, the signature is always positive, and so s=+1. When the Hodge star is extended to pseudo-Riemannian manifolds, then the above inner product is understood to be the metric in diagonal form.
Note that the above identity implies that the inverse of can be given as
Note that if n is odd k(n−k) is even for any k whereas if n is even k(n−k) has the parity of k.
Therefore, if n is odd, it holds for any k that
whereas, if n is even, it holds that
where k is the degree of the forms operated on.
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