Quantum toroidal algebras $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ are certain Drinfeld quantum affinizations of quantum groups associated to affine Lie algebras, and can therefore be thought of as `double affine quantum groups'.
In particular, they contain (and are generated by) a horizontal and vertical copy of the affine quantum group.
Utilising an extended double affine braid group action, Miki obtained in type $A$ an automorphism of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ which exchanges these subalgebras. This has since played a crucial role in the investigation of its structure and representation theory.
In this talk I shall present my recent work -- which extends the braid group action to all types and generalises Miki's automorphism to the ADE case -- as well as potential directions for future work in this area.