Matrix representation of a cross product and related curl-based differential operators in all space dimensions

TL;DR

This paper introduces a matrix-based generalization of the cross product for all space dimensions, providing new insights into algebraic structures and higher-dimensional differential operators like the Helmholtz decomposition.

Contribution

It presents an index-free matrix representation of the cross product in arbitrary dimensions, extending classical identities and decompositions to higher-dimensional spaces.

Findings

Generalized cross product via matrix multiplication

Extended algebraic identities to higher dimensions

Explicit Helmholtz decomposition in all space dimensions

Abstract

A higher dimensional generalization of the cross product is associated with an adequate matrix multiplication. This index-free view allows for a better understanding of the underlying algebraic structures, among which are generalizations of Grassmann's, Jacobi's and Room's identities. Moreover, such a view provides a higher dimensional analogue of the decomposition of the vector Laplacian which itself gives an explicit index-free Helmholtz decomposition in arbitrary dimensions $n\ge2$.