International Journal of Modern Physics B

We discuss quantum orthogonal groups and their real forms. We review the construction of inhomogeneous orthogonal q-groups and their q-Lie algebras. The geometry of the q-Poincaré group naturally induces a well defined q-deformed Heisenberg algebra of hermitian q-Minkowski coordinates x^a and momenta p_a.

The notion of coalgebra symmetry in Hamiltonian systems is analysed. It is shown how the complete integrability of some long-range interacting Hamiltonians can be extracted from their associated coalgebra structure with no use of a quantum R-matrix. Within this framework, integrable deformations can be considered as direct consequences of the introduction of coalgebra deformations (quantum algebras). As an example, the Gaudin magnet is derived from a sl(2) coalgebra, and a completely integrable deformation of this Hamiltonian is obtained through a twisted gl(2) quantum algebra.

A recipe is given to reproduce algebraically the Bose prescription in Quantum Statistics. Bosons are thus defined as coherent states of the representation of the Lie-Hopf algebra su(1, 1), while h(1) is shown to be related to Boltzmann statistics. The technical power of group theory is used to show that black-body radiation formula as well as all other experimental results obtained in continuum limit do not require Bose distribution but are compatible with many other less symmetric ones. This may have dramatical effects on the predictions on Bose condensation.

We discuss applications of the Sine-Gordon model to quasi-one-dimensional quantum magnets.

We construct a topological invariant of the renormalization group trajectories of a large class of 2D quantum integrable models, described by the thermodynamic Bethe ansatz approach. A geometrical description of this invariant in terms of triangulations of three-dimensional manifolds is proposed and associated dilogarithm identities are proven.

We study exactly solvable models of quantum statistical mechanics. Our main example is the Quantum nonlinear Schrödinger equation. This model can be solved by algebraic Bethe Ansatz. We are interested in the expression for a form factor of the local field in the finite volume . We found a determinant representation for this form factor. We think that this is the universal feature: finite volume form factor of any exactly solvable model can be represented a determinant.

Some of main achievements obtained recently in investigations of the symmetry properties of systems solved by the quantum inverse scattering methods and connected with the Yang–Baxter equation and reflection equations are reviewed. Special attention is devoted to the twist procedure. Twist relates three constant R-matrices corresponding to Lie super-algebra gl(1|1).

We propose a constructive method to prove the integrability of a given physical Hamiltonian in one dimension, which amounts to looking for appropriate polynomial R-matrices, solutions of GYBE, whose first coefficient in the power expansion with respect to the spectral parameter is the Hamiltonian itself. The method is applied to the extended Hubbard Hamiltonian, in particular to the cases in which it exhibits so(4) or gl(2,1) symmetries. We show that in the latter case the R-matrices are at most polynomial of second degree, whose coefficients are nothing but the Hamiltonian, the identity and the permutation operator. In this way, all known completely integrable cases are recovered. Also, the method allows to recognize that the possible integrability of the most general gl(2,1) invariant Hamiltonian depends on the existence of a non-additive R-matrix.

We discuss the determination of the lowest Form Factors relative to the trace operators of N = 1 Super Sinh-Gordon Model. Analytic continuations of these Form Factors as functions of the coupling constant allows the study of interesting models in a uniform way, among these the latest model of the Roaming Series and a class of minimal supersymmetric models.

This is a very elementary introduction to the Heisenberg (XXX) quantum spin chain, the Yang–Baxter equation, and the algebraic Bethe Ansatz.

We give a brief review of some group and quantum group theoretical methods used for the construction of photon states, generalisations of coherent and squeezed states. We finally describe a more general approach, exemplified by a new generalized coherent state, a generalization of the Kerr state.

We discuss properties of a recently proposed SO(5) symmetric ladder model. Key features of the single particle spectral function that are emerging from the symmetry are numerically identified in the ladder model and in the photoemission spectrum of the two-dimensional t–J model.