Quantum toroidal algebras, quantum affine algebras, and their representation theory

Thesis

In this thesis we investigate the structure and representation theory of quantum algebras. After gathering the necessary preliminaries, we begin by focusing in particular on quantum toroidal algebras. We first construct an action of the extended double affine braid group B̈ on the quantum toroidal algebra Uq(g_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 untwisted case, using our action and certain involutions of B̈ we produce automorphisms and anti-involutions of Uq(g_tor) that exchange its horizontal and vertical subalgebras. Moreover, they switch the central elements C and k0^a0...kn^an up to inverse. This can be regarded as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups utilized by Cherednik to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures. Our work generalises existing results in type An(1) due to Miki, which have been instrumental in the study of Uq(sl_n+1,tor).

We proceed by proving certain compatibilities between (anti-)automorphisms on either side of our braid group action. From these identities, we deduce that the central extension of SL(2,Z) – which is isomorphic to the braid group on three strands – acts on Uq(g_tor) in untwisted types. This provides a counterpart within the quantum setting to the congruence group actions on double affine braid groups and Hecke algebras established by Cherednik and Ion-Sahi.

We conclude our treatment of Uq(g_tor) by discussing ongoing research related to its representation theory. In particular, we show that twisting Drinfeld's topological coproduct with our anti-involution ψ yields a tensor product structure that is well-defined for loop highest weight modules. Additionally, in the simply laced case, we construct vector representations for Uq(g_tor) that are explicitly described in terms of Young column bases.

In the latter sections of this thesis, we consider crystal bases for representations of quantum affine algebras in types E6(1), E7(1) and E8(1). Specifically, we construct Young wall models for the level 1 irreducible highest weight crystals B(λ) and Fock space crystals B(F(λ)). In both instances the starting point is a perfect crystal of level 1, which we represent in terms of equivalence classes of Young columns stacked within a Young column pattern.

In conjunction with the theory of perfect crystals and energy functions, we then realise the crystals B(λ) and B(F(λ)) in terms of reduced and proper Young walls respectively. These consist of coloured blocks stacked inside a relevant Young wall pattern, satisfying certain combinatorial conditions. Moreover, the crystal structure in each case is described entirely in terms of adding and removing blocks.

Authors: Laurie, D

• DOI: 10.5287/ora-9ovpag5bo
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: quantum affine algebras; quantum toroidal algebras; quantum affinizations; extended double affine braid groups; Fock space representations; crystal bases; Young wall realizations
• Subjects: Representations of algebras; Mathematics; Algebra; Quantum groups; Representations of quantum groups