Quantum toroidal algebras Uq(g_tor) occur as the quantum affinizations of affine quantum groups. In particular, they contain (and are generated by) horizontal and vertical quantum affine subalgebras Uh and Uv.
After recalling the definition and basic properties of these algebras, our first main result will be constructing an action of the extended double affine braid group. (Here, a surprising finite presentation for Uq(g_tor) is a crucial ingredient in our proof.)
We shall then exploit this action to exhibit horizontal-vertical symmetries of Uq(g_tor) in untwisted types, in the form of anti-involutions and 'Miki automorphisms' which exchange Uh and Uv. We'll finish by discussing representation theoretic applications of these results (work in progress), and deducing congruence group actions of the central extension of SL(2,Z) on Uq(g_tor).