Abstract: We introduce a new topological coproduct Δuψ for quantum toroidal algebras Uq(gtor) in all 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. We moreover show that the q-character of a tensor product is equal to the product of q-characters for its factors. 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.

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.

Abstract: 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(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 untwisted case, using our action and certain involutions of B̈ we produce automorphisms and anti-involutions of Uq(gtor) that exchange its horizontal and vertical subalgebras. Moreover, they switch the central elements C and k0a0… knan 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(sln+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 SL2(Z) – which is isomorphic to the braid group on three strands – acts on Uq(gtor) 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(gtor) 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(gtor) 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.

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).

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.