Related papers: Quantum Geometry, Integrability, and Opers
The theory of representations of quivers and of their preprojective algebras are reviewed. In particular, moduli spaces of representations of these algebras, quiver varieties and reflection functor are described. The proof that the…
We study the higher spin anologs of the six vertex model on the basis of its symmetry under the quantum affine algebra $U_q(\slth)$. Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer…
We prove a system of relations in the Grothendieck ring of the category O of representations of the Borel subalgebra of an untwisted quantum affine algebra U_q(g^) introduced in [HJ]. This system was discovered in [MRV1, MRV2], where it was…
We discuss the category $\cal I$ of level zero integrable representations of loop algebras and their generalizations. The category is not semisimple and so one is interested in its homological properties. We begin by looking at some…
Various infinite-dimensional versions of Calogero-Moser operator are discussed in relation with the theory of symmetric functions and representation theory of basic classical Lie superlagebras. This is a version of invited talk given by the…
Quantum integrable systems have very strong mathematical properties that allow an exact description of their energetic spectrum. From the Bethe equations, I formulate the Baxter "T-Q" relation, that is the starting point of two…
We develop our method to prove quantum superintegrability of an integrable 2D system, based on recurrence relations obeyed by the eigenfunctions of the system with respect to separable coordinates. We show that the method provides rigorous…
This is a survey article for "Handbook of Linear Algebra", 2nd ed., Chapman & Hall/CRC, 2014. An informal introduction to representations of quivers and finite dimensional algebras from a linear algebraist's point of view is given. The…
This is a review article on the quantum toroidal algebras, focusing on their roles in various solvable structures of 2d conformal field theory, supersymmetric gauge theory, and string theory. Using $\mathcal{W}$-algebras as our starting…
The past few years have seen a revived interest in quantum geometrical characterizations of band structures due to the rapid development of topological insulators and semi-metals. Although the metric tensor has been connected to many…
The following work is an exploration into certain topics in the broad world of integrable models, both classical and quantum, and consists of two main parts of roughly equal length. The first part, consisting of chapters 1-3, concerns…
Higher structures - infinity algebras and other objects up to homotopy, categorified algebras, `oidified' concepts, operads, higher categories, higher Lie theory, higher gauge theory... - are currently intensively investigated in…
This paper develops a geometric model for coupled two-state quantum systems (qubits), which is formulated using geometric (aka Clifford) algebra. It begins by showing how Euclidean spinors can be interpreted as entities in the geometric…
Lie algebroids provide a natural medium to discuss classical systems, however, quantum systems have not been considered. In aim of this paper is to attempt to rectify this situation. Lie algebroids are reviewed and their use in classical…
Applying a unifying Lax operator approach to statistical systems a new class of integrable vertex models based on quantum algebra is proposed, which exhibits a rich variety for generic q, q roots of unity and q -> 1. Exact solutions are…
The representation theory of symmetric Lie superalgebras and corresponding spherical functions are studied in relation with the theory of the deformed quantum Calogero-Moser systems. In the special case of symmetric pair g=gl(n,2m),…
The paper is devoted to integral quantization, a procedure based on operator-valued measure and resolution of the identity. We insist on covariance properties in the important case where group representation theory is involved. We also…
An algebraic approach to the study of quantum mechanics on configuration spaces with a finite fundamental group is presented. It uses, in an essential way, the Gelfand-Naimark and Serre-Swan equivalences and thus allows one to represent…
We investigate the symmetry algebras of integrable spin Calogero systems constructed from Dunkl operators associated to finite Coxeter groups. Based on two explicit examples, we show that the common view of associating one symmetry algebra…
The relation between special connections on the projective line, called Miura opers, and the spectra of integrable models of Gaudin type provides an important example of the geometric Langlands correspondence. The possible generalization of…