Related papers: Grassmann extensions of Yang-Baxter maps
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…
According to Shibukawa, ternary systems defined on quasigroups and satisfying certain conditions provide a way of constructing dynamical Yang-Baxter maps. After noticing that these conditions can be interpreted as 3-dimensional…
Starting from multidimensional consistency of non-commutative lattice modified Gel'fand-Dikii systems we present the corresponding solutions of the functional (set-theoretic) Yang-Baxter equation, which are non-commutative versions of the…
We propose a lattice action for two dimensional super Yang-Mills theory with a twisted N=2 supersymmetry. The extended supersymmetry is fully and exactly realized on the lattice. The method employed is quite general and its extension to the…
A first aim of this paper is to give sufficient conditions on left non-degenerate bijective set-theoretic solutions of the Yang-Baxter equation so that they are non-degenerate. In particular, we extend the results on involutive solutions…
We study a class of observables in four-dimensional superconformal Yang--Mills theories which, in the planar limit at finite 't Hooft coupling, can be expressed as determinants of semi-infinite matrices built from Bessel functions. This…
Every unitary solution of the Yang-Baxter equation (R-matrix) in dimension $d$ can be viewed as a unitary element of the Cuntz algebra ${\mathcal O}_d$ and as such defines an endomorphism of ${\mathcal O}_d$. These Yang-Baxter endomorphisms…
We investigate a class of non-involutive solutions of the Yang-Baxter equation which generalize self-distributive (derived) solutions. In particular, we study generalized multipermutation solutions in this class. We show that the…
We use the classification of the quadrirational maps given by Adler, Bobenko and Suris to describe when such maps satisfy the Yang-Baxter relation. We show that the corresponding maps can be characterized by certain singularity invariance…
We present three equivalence classes of rational non-invertible multidimensional compatible maps. These maps turns out to be idempotent and by construction they admit birational partial inverses (companion maps) which are Yang-Baxter maps.…
We study symplectic properties of the monodromy map of the Schr\"odinger equation on a Riemann surface with a meromorphic potential having second-order poles. At first, we discuss the conditions for the base projective connection, which…
We define B\"acklund--Darboux transformations in Sato's Grassmannian. They can be regarded as Darboux transformations on maximal algebras of commuting ordinary differential operators. We describe the action of these transformations on…
Enhanced Yang-Baxter operators give rise to invariants of oriented links. We expand the enhancing method to generalized Yang-Baxter operators. At present two examples of generalized Yang-Baxter operators are known and recently three types…
In this article, we introduce a method to extend involutive nondegenerate set-theoretic solutions to the Yang--Baxter equation by means of equivariant mappings to graded modules, thus leading to the notion of a twisted extension.…
The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given PDE. The use of the exotic cohomology of the symmetry algebras opens a way to formulate such…
We prove that the m-generated Grassmann algebra can be embedded into a 2^{m-1}x2^{m-1} matrix algebra over a factor of a commutative polynomial algebra in m indeterminates. Cayley-Hamilton and standard identities for nxn matrices over the…
The Gauss map of a projective variety $X \subset \mathbb{P}^N$ is a rational map from $X$ to a Grassmann variety. In positive characteristic, we show the following results. (1) For given projective varieties $F$ and $Y$, we construct a…
We construct rational and piecewise-linear Yang-Baxter maps for a general N-reduction of the discrete BKP equation.
Motivated by the problem of expanding single-trace tree-level amplitude of Einstein-Yang-Mills theory to the BCJ basis of Yang-Mills amplitudes, we present an alternative expansion formula in the gauge invariant vector space. Starting from…
Lax pairs with operator valued coefficients, which are explicitly connected by means of an additional condition, are considered. This condition is proved to be covariant with respect to the Darboux transformation of a general form.…