Automorphisms of quantum toroidal algebras from an action of the extended double affine braid group

Duncan Laurie (13 Apr 2023)

arXiv:2304.06773

Abstract: We first construct an action of the extended double affine braid group B̈ on the quantum toroidal algebra Uq(gtor) 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 B̈ we produce automorphisms and anti-involutions of Uq(gtor) 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(sln+1,tor).

Young wall realizations of level 1 irreducible highest weight and Fock space crystals of quantum affine algebras in type E

Duncan Laurie (7 Nov 2023)

arXiv:2311.03905 

Abstract: We construct Young wall models for the crystal bases of level 1 irreducible highest weight representations and Fock space representations of quantum affine algebras in types E_6^(1), E_7^(1) and E_8^(1). In each case, Young walls consist of coloured blocks stacked inside the relevant Young wall pattern which satisfy a certain combinatorial condition. Moreover the crystal structure is described entirely in terms of adding and removing blocks.

Young wall models for the level 1 highest weight and Fock space crystals of U_q(E_6^(2)) and U_q(F_4^(1))

Shaolong Han, Yuanfeng Jin, Seok-Jin Kang, and Duncan Laurie (27 Feb 2024)

arXiv:2402.15829 

Abstract: In this paper we construct Young wall models for the level 1 highest weight and Fock space crystals of quantum affine algebras in types E_6^(2) and F_4^(1). Our starting point in each case is a combinatorial realization for a certain level 1 perfect crystal in terms of Young columns. Then using energy functions and affine energy functions we define the notions of reduced and proper Young walls, which model the highest weight and Fock space crystals respectively.