Related papers: Algebraic structure of classical integrability for…
We investigate the algebro-geometric structure of a novel two-parameter quantum deformation which exhibits the nature of a semidirect or cross-product algebra built upon GL(2) x GL(1), and is related to several other known examples of…
We study the canonical Poisson structure on the loop space of the super-double-twisted-torus and its quantization. As a consequence we obtain a rigorous construction of mirror symmetry as an intertwiner of the N=2 super-conformal structures…
In this paper we present a unified algebraic framework to discuss the reduction of classical and quantum systems. The underlying algebraic structure is a Lie-Jordan algebra supplemented, in the quantum case, with a Banach structure. We…
We introduce an integral structure in orbifold quantum cohomology associated to the K-group and the Gamma-class. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for…
The complete integrability of the hyperbolic Gaudin Hamiltonian and other related integrable systems is shown to be easily derived by taking into account their sl(2,R) coalgebra symmetry. By using the properties induced by such a coalgebra…
We provide a brief characterization of the main features of the homogeneous sine-Gordon models and discuss a general construction principle for colour valued S-matrices, associated to a pair of simply laced Lie algebras, which contain the…
The review paper presents generalization of d'Alembert's variational principle: the dynamics of a quantum system for an external observer is defined by the exact equilibrium of all acting in the system forces, including the random quantum…
The construction of superintegrable systems based on Lie algebras and their universal enveloping algebras has been widely studied over the past decades. However, most constructions rely on explicit differential operator realisations and…
The sine-Gordon Y-systems and the reduced sine-Gordon Y-systems were introduced by Tateo in the 90's in the study of the integrable deformation of conformal field theory by the thermodynamic Bethe ansatz method. The periodicity property and…
We present a new vorticity-raising transformation for the second integrable complexification of the sine-Gordon equation on the plane. The new transformation is a product of four Schlesinger maps of the Painlev\'{e}-V to itself, and allows…
We proceed to study Yang-Baxter deformations of the AdS$_5\times$S$^5$ superstring with the classical Yang-Baxter equation. We make a general argument on the supercoset construction and present the master formula to describe the dilaton in…
According to Etingof and Varchenko, the classical dynamical Yang-Baxter equation is a guarantee for the consistency of the Poisson bracket on certain Poisson-Lie groupoids. Here it is noticed that Dirac reductions of these Poisson manifolds…
Using the bicomplex approach we discuss a noncommutative system in two--dimensional Euclidean space. It is described by an equation of motion which reduces to the ordinary sine--Gordon equation when the noncommutation parameter is removed,…
Yang-Baxterising a braid group representation associated with multideformed version of $GL_{q}(N)$ quantum group and taking the corresponding $q\rightarrow 1$ limit, we obtain a rational $R$-matrix which depends on $\left ( 1+ {N(N-1) \over…
The classification of real Clifford algebras in terms of matrix algebras is well--known. Here we consider the real Clifford algebra ${\mathcal Cl}(r,s)$ not as a matrix algebra, but as a Clifford module over itself. We show that ${\mathcal…
It is shown how the complex sine-Gordon equation arises as a symmetry flow of the AKNS hierarchy. The AKNS hierarchy is extended by the ``negative'' symmetry flows forming the Borel loop algebra. The complex sine-Gordon and the vector…
The survey is devoted to algebraic structures related to integrable ODEs and evolution PDEs. A description of Lax representations is given in terms of vector space decomposition of loop algebras into a direct sum of Taylor series and a…
Motivated by a conjecture that doubled four-dimensional Chern-Simons produces new integrable models, we perform its Hamiltonian analysis and find the theory's Poisson algebra. This requires carefully accounting for a set of boundary…
We study integrable dynamical systems described by a Lax pair involving a spectral parameter. By solving the classical Yang-Baxter equation when the R-matrix has two poles we show that they can be interpreted as natural motions on a twisted…
Among other results we show that near the equilibrium point, the Hamiltonian of the sine-Gordon (SG) equation on the circle can be viewed as an element in the Poisson algebra of the modified Korteweg-de Vries (mKdV) equation and hence by…