Automorphisms of Quantum Toroidal Algebras from an Action of the Extended Double Affine Braid Group

We first construct an action of the extended double affine braid group \(\mathcal {\ddot{B}}\) $\ddot{\mathcal{B}}$ on the quantum toroidal algebra \(U_{q}(\mathfrak {g}_{\textrm{tor}})\) $U_q({\mathfrak{g}}_{\text{tor}})$ in untwisted and twisted types. As a crucial step in the proof, we obtain a finite Drinfeld new style presentation for a broad class of quantum affinizations. In the simply laced cases, using our action and certain involutions of \(\mathcal {\ddot{B}}\) $\ddot{\mathcal{B}}$ we produce automorphisms and anti-involutions of \(U_{q}(\mathfrak {g}_{\textrm{tor}})\) $U_q({\mathfrak{g}}_{\text{tor}})$ which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements C and \(k_{0}^{a_{0}}\dots k_{n}^{a_{n}}\) $k_0^{a_0}\cdots k_n^{a_n}$ up to inverse. This can be viewed as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups used by Cherednik to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures. Our work generalises existing results in type A due to Miki which have been instrumental in the study of the structure and representation theory of \(U_{q}(\mathfrak {sl}_{n+1,\textrm{tor}})\) $U_q({\mathfrak{sl}}_{n+1,\text{tor}})$ .