Generalized duality for k-forms

TL;DR

This paper introduces a new duality operator for arbitrary k-forms based on a fixed form Omega, extending classical Hodge duality and exploring its properties under invariance and symmetry conditions.

Contribution

It defines a generalized duality for k-forms dependent on a fixed form Omega, broadening the scope of classical dualities in differential geometry.

Findings

The duality extends Hodge duality to arbitrary k-forms.

Properties are analyzed under invariance with respect to subalgebras of so(V).

Examples include cases with invariant forms and discrete symmetries.

Abstract

We give the definition of a duality that is applicable to arbitrary $k$-forms. The operator that defines the duality depends on a fixed form $\Omega$. Our definition extends in a very natural way the Hodge duality of $n$-forms in $2n$ dimensional spaces and the generalized duality of two-forms. We discuss the properties of the duality in the case where $\Omega$ is invariant with respect to a subalgebra of $\mathfrak{so}(V)$. Furthermore, we give examples for the invariant case as well as for the case of discrete symmetry.