Abstract
Hodge's star ∗ and the exterior derivative d generate an algebra A over C , which if regarded as an algebra over its centre is four-dimensional. The centre is the polynomial ring C [ k] over C generated by k = d∗ + ∗d. If it is assumed that k is invertible, then the centre is C[k] + C[ 1 k ] and it becomes possible to choose a basis for the algebra in such a way that the four elements with their negatives constitute the dihedral group D 4 under multiplication. The study of how the group arises provides considerable insight into the structures of both the algebra under consideration and D 4.
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