Defined as the algebra of all G-equivariant meromorphic maps from a Riemann surface into a complex Lie algebra, the automorphic Lie algebra was introduced to provide Lax connections leading to integrable PDEs. Nowadays these Lie algebras are used in the context of integrable differential-difference equations and partial-difference equations as well.
Intrinsic interest in automorphic Lie algebras is largely due to the surprising observation that the structure of the finite reduction group G is irrelevant for the Lie structure of the automorphic Lie algebra. At least, this is true for all cases thus far computed. The computational classification project has been running for about a decade.
In this talk I will present the latest developments of the theoretical classification project, which is running alongside the computational project. I will show how to obtain three quotients of automorphic Lie algebras, which are largely independent of G, and explain what these three quotients tell us about the real thing. This provides the first significant information on automorphic Lie algebras which are computationally inaccessible, such as E_8.
The Dirac quantisation of the Ablowitz-Ladik chain yields the q-boson model. We discuss the quantisation of the classical Lax and Darboux matrices and single out a particular set of Bäcklund transformations which describe the discrete time evolution of the system in the presence of periodic boundary conditions. We explicitly relate their quantum counterparts to Baxter Q-operator and show that multi-Bäcklund transformations define a 2D TQFT which in the q=0 limit is the WZW fusion ring of type A.
Following the global (at all orders) classification of $1+1$-dimensional scalar polynomial homogeneous equations (Sanders, Wang) we extend this result to $2+1$-dimensions and classify scalar polynomial homogeneous equations with degenerate dispersion laws. We first classify the rational degenerate dispersion laws and show that the admissible cases are the dispersion relations of KP, BKP/CKP and Veselov-Novikov type only. In the framework of the Perturbative Symmetry Approach we then classify the homogeneous polynomial equations with these 3 classes of dispersion laws at all orders.
It is hard to overestimate the role of the quantum Yang-Baxter equation in the theory of integrable systems. Quantum groups, discovered independently by Drinfeld and Jimbo, provide a representation-theoretic framework for generating solutions of the quantum Yang-Baxter equation. On the other hand, many of the quantum groups can be described in terms of Dynkin diagrams and some combinatorial data assigned to them. In this talk I will present a generalization of the theory of quantum symmetric pairs as developed by Kolb and Letzter. I will introduce a class of generalized Satake diagrams that, along with some combinatorial data, describe quantum (symmetric) pair algebras. I will then explain how these in turn provide a representation-theoretic framework for generating solutions of the quantum reflection equation, the boundary analogue of the Yang-Baxter equation.
(Joint work with Bart Vlaar, arXiv:1602.08471)
The 1D system of N nonrelativistic bosons interacting via delta-function potential is arguably the simplest and very well studied example of a quantum integrable system. Its integrability can be established by a variety of methods, including the coordinate Bethe ansatz and Dunkl operators.
In the second quantisation, the corresponding quantum field theory in the Fock space, that is the orthogonal sum of the N-particle spaces, represents a quantisation of the Nonlinear Schroedinger equation.
We study the second quantised version of the Dunkl operators and obtain some new representations for the integrals of motion and creation/annihilation operators both in the quantum case and in the classical limits.
We suggest a systematic and algorithmic way to compute generalised symmetries of difference equations. Our approach exploits the theory of integrability conditions, employs Laurent and Taylor formal series of pseudo-difference operators and formulates algebraically the determining equations. We also present a method to solve certain classes of functional equations and discuss how this method can be implemented in Mathematica.
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Last updated 2nd May, 2017