arXiv subjects
Publications and source records attributed to Martin Vrabec.
Integrable spin Calogero-Moser type systems with non-symmetric configurations of the singularities of the potential appeared in the work of Chalykh, Goncharenko, and Veselov in 1999. We obtain various generalisations of these examples by making use of the representation theory of Cherednik algebras.
We show that a Sergeev-Veselov difference operator of rational Macdonald-Ruijsenaars (MR) type for the deformed root system $BC(l,1)$ preserves a ring of quasi-invariants in the case of non-negative integer values of the multiplicity parameters. We prove that in this case the operator admits a (multidimensional) Baker-Akhiezer eigenfunction, which depends on spectral parameters and which is, moreover, as a function of the spectral variables an eigenfunction for the (trigonometric) generalised Calogero-Moser-Sutherland (CMS) Hamiltonian for $BC(l,1)$. By an analytic continuation argument, we generalise this eigenfunction also to the case of more general complex values of the multiplicities. This leads to a bispectral duality statement for the corresponding MR and CMS systems of type $BC(l,1)$.
Inside the double affine Hecke algebra of type $GL_n$, which depends on two parameters $q$ and $\tau$, we define a subalgebra $\mathbb{H}^{\mathfrak{gl}_n}$ 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 $\mathbb{H}^{\mathfrak{gl}_n}$ is a flat $\tau$-deformation of the crossed product of the group algebra of the symmetric group with the image of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{gl}_n)$ under the $q$-oscillator (Jordan-Schwinger) representation. We find all the defining relations and an explicit PBW basis for the algebra $\mathbb{H}^{\mathfrak{gl}_n}$. We describe its centre and establish a double centraliser property. As an application, we also obtain new integrable generalisations of a Macdonald-Ruijsenaars system with an external Morse (exponential) potential introduced by Van Diejen. In particular, we extend Van Diejen's Hamiltonian to the case of a system with two different types of interacting particles, and thus generalise the deformed Macdonald-Ruijsenaars operators of Chalykh-Sergeev-Veselov to the setting with an external field.
We consider the generalised Calogero-Moser-Sutherland quantum integrable system associated to the configuration of vectors $AG_2$, which is a union of the root systems $A_2$ and $G_2$. We establish the existence of and construct a suitably defined Baker-Akhiezer function for the system, and we show that it satisfies bispectrality. We also find two corresponding dual difference operators of rational Macdonald-Ruijsenaars type in an explicit form.
We prove that unital extensions of Kirchberg algebras by separable stable AF algebras have nuclear dimension one. The title follows.
We consider generalised Calogero-Moser-Sutherland quantum Hamiltonian $H$ associated with a configuration of vectors $AG_2$ on the plane which is a union of $A_2$ and $G_2$ root systems. The Hamiltonian $H$ depends on one parameter. We find an intertwining operator between $H$ and the Calogero-Moser-Sutherland Hamiltonian for the root system $G_2$. This gives a quantum integral for $H$ of order 6 in an explicit form thus establishing integrability of $H$.