Related papers: Open Spin Chains and Quadratic Algebras
We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integral of motion. We construct the most general associative cubic algebra and we present specific…
We present a recursion relation for the explicit construction of integrable spin chain Hamiltonians with long-range interactions. Based on arbitrary short-range (e.g. nearest-neighbor) integrable spin chains, it allows to construct an…
Potential algebras can be used effectively in the analysis of the quantum systems. In the article, we focus on the systems described by a separable, 2x2 matrix Hamiltonian of the first order in derivatives. We find integrals of motion of…
Affiliated and normal operators in octonion Hilbert spaces are studied. Theorems about their properties and of related algebras are demonstrated. Spectra of unbounded normal operators are investigated.
In this contribution we briefly review recent developments in the theory of long-range integrable spin chains. These spin chains constitute a natural generalisation of the well-studied integrable nearest-neighbour chains and are of…
We study a system of partial differential equations defined by commuting family of differential operators with regular singularities. We construct ideally analytic solutions depending on a holomorphic parameter. We give some explicit…
At present an algebra of strongly interacting fields is unknown. In this paper it is assumed that the operators of strongly nonlinear field can form a non-associative algebra. It is shown that such algebra can be described as an algebra of…
Spin networks, essentially labeled graphs, are ``good quantum numbers'' for the quantum theory of geometry. These structures encompass a diverse range of techniques which may be used in the quantum mechanics of finite dimensional systems,…
We present an algebraic framework for interacting extended quantum systems to study complex phenomena characterized by the coexistence and competition of different states of matter. We start by showing how to connect different…
We construct a Q-operator for the open XXZ Heisenberg quantum spin chain with diagonal boundary conditions and give a rigorous derivation of Baxter's TQ relation. Key roles in the theory are played by a particular infinite-dimensional…
Circuit algebras are a symmetric analogue of Jones's planar algebras introduced to study finite-type invariants of virtual knotted objects. Circuit algebra structures appear, in different forms, across mathematics. This paper provides a…
We introduce smooth sequences of integral domains as well-ordered ascending chains that behave well at limit ordinals. Subsequently, we use this notion to give some conditions on the freeness of kernels of extension maps between groups of…
We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank $N$. It combines and unifies the ideas of Duistermaat-Gr\"unbaum and Wilson. Our construction is completely…
In a configuration space whose boundary can be identified with a subset of its interior, a boundary condition can relate the behaviour of a function on the boundary and in the interior. Additionally, boundary values can appear as additive…
We obtain the symmetry algebra of multi-matrix models in the planar large N limit. We use this algebra to associate these matrix models with quantum spin chains. In particular, certain multi-matrix models are exactly solved by using known…
We explicitly construct two classes of infinitly many commutative operators in terms of the deformed W-algebra $W_{qt}(sl_N^)$, and give proofs of the commutation relations of these operators. We call one of them local integrals of motion…
We study connections between closure operators on an algebra $(A,\Om)$ and congruences on the extended power algebra defined on the same algebra. We use these connections to give an alternative description of the lattice of all subvarieties…
A key notion bridging the gap between {\it quantum operator algebras} \cite{LZ10} and {\it vertex operator algebras} \cite{Bor}\cite{FLM} is the definition of the commutativity of a pair of quantum operators (see section 2 below). This is…
We provide a theoretical set up for studying the dynamics in quantum spin chain models with inhomogeneous two-body interaction. We frame in our formalism models that can be mapped into a fermion system with a quadratic Hamiltonian. Local…
We construct a one-parameter family of algebras consisting of Fourier integral operators. We derive boundedness results, composition rules, and the spectral invariance of this class of operators. The operator algebra is defined by the decay…