Symmetries of equivariant Khovanov-Rozansky homology

We construct an ${\mathfrak{s}\mathfrak{l}}_2$-action on equivariant ${\mathfrak{g}\mathfrak{l}}_N$-link homologies. As a consequence we obtain an action of ${\mathfrak{s}\mathfrak{l}}_2$ on these homologies as well as a $p$-DG structures for $p$ a prime number. We explore topological applications of these structures.