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A simple relation between inhomogeneous transfer matrices and boundary quantum KZ equations is exhibited for quantum integrable systems with reflecting boundary conditions, analogous to an observation by Gaudin for periodic systems. Thus…
We present here a complete description of the quantization of the baker's map. The method we use is quite different from that used in Balazs and Voros [BV] and Saraceno [S]. We use as the quantum algebra of observables the operators…
In a perturbative approach Einstein-Hilbert gravity is quantized about a flat background. In order to render the model power counting renormalizable, higher order curvature terms are added to the action. They serve as Pauli-Villars type…
The elliptic quantum Knizhnik-Zamolodchikov-Bernard (qKZB) difference equations associated to the elliptic quantum group $E_{\tau,\eta}(sl_2)$ is a system of difference equations with values in a tensor product of representations of the…
Multiparametric quantum deformations of $gl(2)$ are studied through a complete classification of $gl(2)$ Lie bialgebra structures. From them, the non-relativistic limit leading to harmonic oscillator Lie bialgebras is implemented by means…
We study solutions of the Yang-Baxter equation on a tensor product of an arbitrary finite-dimensional and an arbitrary infinite-dimensional representations of the rank one symmetry algebra. We consider the cases of the Lie algebra sl_2, the…
We consider the quantized Knizhnik-Zamolodchikov-Bernard difference equation (qKZB) with step $p$ and values in a tensor product of finite dimensional evaluation modules over the elliptic quantum group $E_{\tau,\eta}(sl_2)$, the equation…
We proceed to generalize the Yang-Baxter (YB) deformation of Wess-Zumino-Witten (WZW) model to the Lie supergroups case. This generalization enables us to utilize various kinds of solutions of the (modified) graded classical Yang-Baxter…
Observables of quantum or classical mechanics form algebras called quantum or classical Hamilton algebras respectively (Grgin E and Petersen A (1974) {\it J Math Phys} {\bf 15} 764\cite{grginpetersen}, Sahoo D (1977) {\it Pramana} {\bf 8}…
Using the Bethe ansatz solution, we analytically study expansionary, magnetic and interacting Gr\"uneisen parameters (GPs) for one-dimensional (1D) Lieb-Liniger and Yang-Gaudin models. These different GPs elegantly quantify the dependences…
Dynamical R-matrix relations are derived for the group-valued chiral vertex operators in the SU(n) WZNW model from the KZ equation for a general four-point function including two step operators. They fit the exchange relations for the…
A general form factor formula for the scaling Z(N)-Ising model is constructed. Exact expressions for matrix elements are obtained for several local operators. In addition, the commutation rules for order, disorder parameters and para-Fermi…
We study the solutions of the Yang-Baxter equation associated to nineteen vertex models invariant by the parity-time symmetry from the perspective of algebraic geometry. We determine the form of the algebraic curves constraining the…
The definitions of the main notions related to the quantum inverse scattering methods are given. The Yang-Baxter equation and reflection equations are derived as consistency conditions for the factorizable scattering on the whole line and…
We study the general solution of the Yang-Baxter equation with deformed $sl(2)$ symmetry. The universal R operator acting on tensor products of arbitrary representations is obtained in spectral decomposition and in integral forms. The…
For each simple euclidean Jordan algebra $V$ of rank $\rho$ and degree $\delta$, we introduce a family of classical dynamic problems. These dynamical problems all share the characteristic features of the Kepler problem for planetary…
We study a parametric version of the Kannan-Lipton Orbit Problem for linear dynamical systems. We show decidability in the case of one parameter and Skolem-hardness with two or more parameters. More precisely, consider a $d$-dimensional…
We consider the discrete Knizhnik-Zamolodchikov connection (qKZ) associated to $gl(N)$, defined in terms of rational R-matrices. We prove that under certain resonance conditions, the qKZ connection has a non-trivial invariant subbundle…
New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for $sl(2,\mathbb{C})$. These solutions are…
We extend basic properties of two dimensional integrable models within the Algebraic Bethe Ansatz approach to 2+1 dimensions and formulate the sufficient conditions for the commutativity of transfer matrices of different spectral…