2023/04/13 by Duncan Laurie, Laurie, Duncan
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2304.06773
We first construct an action of the extended double affine braid group \mathcalB on the quantum toroidal algebra Uq(\mathfrakgtor) 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 \mathcalB we produce automorphisms and anti-involutions of Uq(\mathfrakgtor) which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements C and k0^a0… kn^an 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 Uq(\mathfraksln+1,tor).