Automorphisms of quantum toroidal algebras from an action of the extended double affine braid group
Abstract
We first construct an action of the extended double affine braid group $\mathcal{\ddot{B}}$ on the quantum toroidal algebra $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ 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 $\mathcal{\ddot{B}}$ we produce automorphisms and antiinvolutions of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements $C$ and $k_{0}^{a_{0}}\dots k_{n}^{a_{n}}$ 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 $U_{q}(\mathfrak{sl}_{n+1,\mathrm{tor}})$.
 Publication:

arXiv eprints
 Pub Date:
 April 2023
 DOI:
 10.48550/arXiv.2304.06773
 arXiv:
 arXiv:2304.06773
 Bibcode:
 2023arXiv230406773L
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Rings and Algebras;
 Mathematics  Representation Theory;
 17B37;
 17B67;
 20F36;
 81R50
 EPrint:
 36 pages, comments very welcome