Non-linear realizations and invariant action principles in higher gauge theory

We propose an extension of the formalism developed by Stelle-West and Grignani-Nardelli to the case of FDAs. We first consider the case of FDAs carrying one $p$-form extension and no non-trivial cohomology. We show that it is possible to define large gauge transformations as a direct extension of the large transformations induced by their Lie subalgebras and study the resulting non-linear realizations. Furthermore, we extend the results to the case FDAs with non-trivial cohomology by introducing large gauge transformations that carry the information about the FDA cocycle structure constants. We consider two examples of this type of gauge algebra, namely, FDA extensions of the bosonic Poincar\'{e} and Maxwell algebras, write down their dual $L_{\infty}$ algebras and study their non-linear realizations and possible invariant action principles.