Related papers: On the Structure of Set-Theoretic Polygon Equation…
Spherically symmetrical reductions of self-dual Yang-Mills and Einstein-Plebanski equations are constructed at the same manner. As in the first case we come back to known before solutions (under such kind of reduction but in some different…
The set of triangulations of a cyclic polytope possesses two a priori different partial orders, known as the higher Stasheff-Tamari orders. The first of these orders was introduced by Kapranov and Voevodsky, while the second order was…
We introduce a homomorphism from the quantum affine algebras $U_q(D^{(2)}_{n+1}), U_q(A^{(2)}_{2n}), U_q(C^{(1)}_{n})$ to the $n$-fold tensor product of the $q$-oscillator algebra ${\mathcal A}_q$. Their action commute with the solutions of…
This paper aims to deepen the theory of bijective non-degenerate set-theoretic solutions of the Yang-Baxter equation, not necessarily involutive, by means of q-cycle sets. We entirely focus on the finite indecomposable ones among which we…
We derive first-order and second-order field equations from ambitwistor spaces as phase spaces of massless particles. In particular, the second-order field equations of Yang-Mills theory and general relativity are formulated in a unified…
We find a new type of non-linear supersymmetries, called N-fold parasupersymmetry, which is a generalization of both N-fold supersymmetry and parasupersymmetry. We provide a general formulation of this new symmetry and then construct a…
Yang-Baxter system related to quantum doubles is introduced and large class of both continuous and discrete symmetries of the solution manifold are determined. Strategy for solution of the system based on the symmetries is suggested and…
We find the general solution to the twisting equation in the tensor bialgebra $T({\bf R})$ of an associative unital ring ${\bf R}$ viewed as that of fundamental representation for a universal enveloping Lie algebra and its quantum…
A complete classification of generalized symmetries of the Yang-Mills equations on Minkowski space with a semi-simple structure group is carried out. It is shown that any generalized symmetry, up to a generalized gauge symmetry, agrees with…
Yang-Baxter (YB) deformations of type IIB string theory have been well studied from the viewpoint of classical integrability. Most of the works, however, are focused upon the local structure of the deformed geometries and the global…
In this paper we study dually flat spaces arising from Delzant polytopes equipped with a symplectic potential together with their corresponding toric K\"ahler manifolds as their torifications.We introduce a dually flat structure and the…
The symmetry group method is applied to a generalized Korteweg-de Vries equation and several classes of group invarint solution for it are obtained by means of this technique. Polynomial, trigonometric and elliptic function solutions can be…
This paper presents a novel and straight formulation, and gives a complete insight towards the understanding of the complexity of the problems of the so called NP-Class. In particular, this paper focuses in the Searching of the Optimal…
The Poisson structure arising in the Hamiltonian approach to the rational Gaudin model looks very similar to the so-called modified Reflection Equation Algebra. Motivated by this analogy, we realize a braiding of the mentioned Poisson…
A generalization of the already studied transformations of the linear differential equation into a system of the first order equations is given. The proposed transformation gives possibility to get new forms of the N-dimensional system of…
Hexagon relations are combinatorial or algebraic realizations of four-dimensional Pachner moves. We introduce some simple set-theoretic hexagon relations and then `quantize' them using what we call `polynomial hexagon cohomologies'. Based…
We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the…
We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the…
Reductions for systems of ODEs integrable via the standard factorization method (the Adler-Kostant-Symes scheme) or the generalized factorization method, developed by the authors earlier, are considered. Relationships between such…
This survey aims to collect the main results of the theory of the set-theoretical solutions to the pentagon equation obtained up to now in the literature. In particular, we present some classes of solutions and raise some questions.