Quantum many-body integrable systems and related algebraic structures

This thesis deals with various many-body quantum integrable Hamiltonian systems and algebraic structures related to them. More specifically, it discusses generalisations of Calogero–Moser–Sutherland (CMS) and Macdonald–Ruijsenaars (MR) type systems and their connections with the theory of double affine Hecke and related algebras. Firstly, we consider the generalised CMS operators associated with the deformed root systems BC(l, 1) and a CMS type operator associated with a planar configuration of vectors called AG2, which is a union of the root systems A2 and G2. We construct suitably-defined (multidimensional) Baker–Akhiezer eigenfunctions for these operators, and we use this to prove a bispectral duality for each of these generalised CMS systems. In the case of AG2, we give two corresponding dual difference operators of rational MR type in an explicit form, which we generalise to the trigonometric case as well by using the theory of double affine Hecke algebras (DAHAs). In the case of BC(l, 1), the bispectral dual is a rational difference operator introduced by Sergeev and Veselov. Secondly, we study systems with spin degrees of freedom. Quantum integrable spin CMS type systems with non-symmetric configurations of the singularities of the potential appeared in the rational case in the work of Chalykh, Goncharenko, and Veselov in 1999. In this thesis, we obtain various trigonometric spin CMS type systems by making use of the representation theory of degenerate DAHAs. Particular cases of our construction reproduce in the rational limit the examples discovered by Chalykh, Goncharenko, and Veselov. Finally, inside the DAHA of type GLn, which depends on two parameters q and τ , we define a subalgebra Hgln that may be thought of as a q-analogue of the degree zero part of the corresponding rational Cherednik algebra. We prove that the algebra Hgln is a flat τ -deformation of the crossed product of the group algebra of the symmetric group with the image of the Drinfeld–Jimbo quantum group Uq(gln) under the q-oscillator (Jordan–Schwinger) representation. We find all the defining relations and an explicit PBW basis for the algebra Hgln . We describe its centre and establish a double centraliser property. As an application, we obtain new integrable generalisations of Van Diejen’s MR system in an external field.