Related papers: On a New Class of Integrable Models
This talk gives an introduction into the subject of Seiberg-Witten curves and their relation to integrable systems. We discuss some motivations and origins of this relation and consider explicit construction of various families of…
For any integral lattice $Q$, one can construct a vertex algebra $V_Q$ called a lattice vertex algebra. If $\sigma$ is an automorphism of $Q$ of finite order, it can be lifted to an automorphism of $V_Q$. In this paper we classify the…
A class of quasi two and three dimensional quantum lattice spin models with nearest and next nearest neighbour interactions is proposed. The basic idea of construction is to introduce interactions in an array of XXZ spin chains through…
It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter $(q+q^{-1})/2$ is evaluated at roots of unity $q^{N}=1$. These new…
Specializing properly the parameters contained in the maximal cyclic representation of the non-restricted A-type quantum algebra at roots of unity, we find the unique primitive vector in it. We show that the submodule generated by the…
In this part one of a series of papers, we introduce a new version of quantum covering and super groups with no isotropic odd simple root, which is suitable for the studies of integrable modules, integral forms and bar-involution. A quantum…
We define a new mathematical structure ( graph quantum group) which combines the tower of algebras associated with a graph ${\cal G}$ and the structure of a Hopf algebra {\cal A}. In this structure Ocneanu's string operators become Hopf…
In this article we give a systematic definition of the recently introduced spin foam models for four dimensional quantum gravity reviewing the main results on their semiclassical limit on fixed discretizations.
We present a construction of integrable hierarchies without or with boundary, starting from a single R-matrix, or equivalently from a ZF algebra. We give explicit expressions for the Hamiltonians and the integrals of motion of the hierarchy…
Group representations play a central role in theoretical physics. In particular, in quantum mechanics unitary --- or, in general, projective unitary --- representations implement the action of an abstract symmetry group on physical states…
In this paper we present the novel qualities of entanglement of formation for general (so also infinite dimensional) quantum systems and we introduce the notion of coefficient of quantum correlations. Our presentation stems from rigorous…
Given a complex of groups, we construct a new class of complex of groups that records its local data and offer a functorial perspective on the statement that complexes of groups are locally developable. We also construct a new notion of an…
The main goal of this note is to suggest an algebraic approach to the quasi-isometric classification of partially commutative groups (alias right-angled Artin groups). More precisely, we conjecture that if the partially commutative groups…
We describe a new class of scattering matrices for quantum graphs in which back-scattering is prohibited. We discuss some properties of quantum graphs with these scattering matrices and explain the advantages and interest in their study. We…
Inspired by classical results in integrable boundary quantum field theory, we propose a definition of integrable initial states for quantum quenches in lattice models. They are defined as the states which are annihilated by all local…
Results obtained by us are overviewed from a general set up. The universal $R$-matrix is exploited to obtain various important relations and structures involved in quantum group algebra, which are used subsequently for generating different…
The formulation and resolution of integrable lattice statistical models in a quantum group covariant way is the subject of this review. The Bethe Ansatz turns to be remarkably useful to implement quantum group symmetries and to provide…
In this short note we introduce a notion called "quantum injectivity" of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. Particularly, this provides a new characterization of amenability of…
We construct a new invariant metric for a compact subgroup of the automorphism group of a domain in complex space. Applications are provided.
We give a very brief introduction to the group field theory approach to quantum gravity, a generalisation of matrix models for 2-dimensional quantum gravity to higher dimension, that has emerged recently from research in spin foam models.