Tensor products and R-matrices for quantum toroidal algebras

Duncan Laurie (2025)

arXiv:2503.08839 

Abstract: We introduce a new topological coproduct Δuψ for quantum toroidal algebras Uq(gtor) in untwisted types, leading to a well-defined tensor product on the category Ôint of integrable representations. This is defined by twisting the Drinfeld coproduct Δu with an anti-involution ψ of Uq(gtor) that swaps its horizontal and vertical quantum affine subalgebras. Other applications of ψ include generalising the celebrated Miki automorphism from type A, and an action of the universal cover of SL(2,Z).

Next, we investigate the ensuing tensor representations of Uq(gtor), and prove quantum toroidal analogues for a series of influential results by Chari-Pressley on the affine level. In particular, there is a compatibility with Drinfeld polynomials, and the product of irreducibles is generically irreducible. Furthermore, we obtain R-matrices with spectral parameter which provide solutions to the (trigonometric, quantum) Yang-Baxter equation, and thus endow Ôint with a meromorphic braiding. These moreover give rise to a commuting family of transfer matrices for each module.

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

Duncan Laurie (2024)

Published in 'Algebras and Representation Theory' -- open access journal link 

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 k0a0… knan 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 (2024)

Published in 'Journal of Algebra' -- open access journal link

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 E6(1), E7(1) and E8(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 Uq(E6(2)) and Uq(F4(1))

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

Published in 'Letters in Mathematical Physics' -- journal link 

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 E6(2) and F4(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.