Related papers: Geometric Algebra and Algebraic Geometry of Loop a…
The symmetry of the Hamiltonian describing the asymmetric twin model was partially studied in earlier works, and our aim here is to generalize these results for the open transfer matrix. In this spirit we first prove, that the so called…
We introduce a diagram category, study its structure, and investigate some of its applications to the representation theory of Lie algebras and Lie superalgebras. The morphisms of the category, which contains a subcategory isomorphic to the…
The braid group appears in many scientific fields and its representations are instrumental in understanding topological quantum algorithms, topological entropy, classification of manifolds and so on. In this work, we study planer diagrams…
The affine Temperley-Lieb algebra $\mathsf{a}\hskip-1.8pt\mathsf{TL}_{N}(\beta)$ is an infinite-dimensional algebra parametrized by a number $\beta \in \mathbb{C}$ and an integer $N\in \mathbb{N}$. It naturally acts on…
We define and study the space of $q$-opers associated with Bethe equations for integrable models of XXZ type with quantum toroidal algebra symmetry. Our construction is suggested by the study of the enumerative geometry of cyclic quiver…
Guided by consideration of problems in 2 and 3 dimensional lattice model computation, we are led to define a number of new categories, and functors between these categories and the partition category, culminating in the introduction of two…
We consider the sl(2)_q-invariant open spin-1/2 XXZ quantum spin chain of finite length N. For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of…
We derive a novel multiple integral representation for a generating function of the $\s^z$-$\s^z$ correlation functions of the spin-$\2$ XXZ chain at finite temperature and finite, longitudinal magnetic field. Our work combines algebraic…
We study numerically the ODE/IM correspondence for untwisted affine Lie algebras associated with simple Lie algebras including exceptional type. We consider the linear problem obtained from the massless limit of that of the modified affine…
We construct new trigonometric solutions of the Yang-Baxter equation, using the Fuss-Catalan algebras, a set of multi-colored versions of the Temperley-Lieb algebra, recently introduced by Bisch and Jones. These lead to new two-dimensional…
The Temperley--Lieb algebra is a finite dimensional associative algebra that arose in the context of statistical mechanics and occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type $A$. It is often…
We introduce two new algebras that we call \emph{tied--boxed Hecke algebra} and \emph{tied--boxed Temperley--Lieb algebra}. The first one is a subalgebra of the algebra of braids and ties introduced by Aicardi and Juyumaya, and the second…
We present an algebraic method for solving the Bethe ansatz equations for the periodic totally asymmetric exclusion process (TASEP) with an arbitrary number of sites and particles. The Bethe ansatz equations are realized as an algebraic…
Using the Algebraic Bethe Ansatz we consider the correlation functions of the integrable higher spin chains. We apply a method recently developed for the spin $\frac 12$ Heisenberg chain, based on the solution of the quantum inverse…
Highest-weight representations of infinite dimensional Lie algebras and Hilbert schemes of points are considered, together with the applications of these concepts to partition functions, which are most useful in physics. Partition functions…
This work concerns to the studies of boundary integrability of the vertex models from representations of the Temperley-Lieb algebra associated with the quantum group ${\cal U}_{q}[X_{n}]$ for the affine Lie algebras $X_{n}$ = $A_{1}^{(1)}$,…
In this paper we propose a combinatorial framework to study a diagrammatic representation of the affine Temperley-Lieb algebra of type C introduced by Ernst. In doing this, we define two procedures, a decoration algorithm on diagrams and a…
In this paper, we discuss an interaction between complex geometry and integrable systems. Section 1 reviews the classical results on integrable systems. New examples of integrable systems, which have been discovered, are based on the Lax…
We develop a new approach to the study of supersymmetric gauge theories on ALE spaces using the theory of framed sheaves on root toric stacks, which illuminates relations with gauge theories on $\mathbb{R}^4$ and with two-dimensional…
We study the diagonalization problem of certain Hofstadter-type models through the algebraic Bethe ansatz equation by the algebraic geometry method. When the spectral variables lie on a rational curve, we obtain the complete and explicit…