arXiv subjects
Publications and source records attributed to Pavlos Kassotakis.
We provide necessary and sufficient conditions for maps that satisfy associative-like conditions on families of n-ary magmas to be pentagon maps. We obtain parametric-pentagon maps and we propose a procedure that generates families of multicomponent pentagon and entwining pentagon maps from a given pentagon map.
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. The maps in question can be reinterpreted as systems of difference equations defined on the edges of the $\mathbb{Z}^2$ graph. Finally, we associate these compatible systems of difference equations with integrable difference equations defined on the triangular lattice $Q(A2)$.
We present rational Lax representations for one-component parametric quadrirational Yang-Baxter maps in both the abelian and non-abelian settings. We show that from the Lax matrices of a general class of non-abelian involutive Yang-Baxter maps ($\mathcal{K}$-list), by considering the symmetries of the $\mathcal{K}$-list maps, we obtain compatible refactorization problems with rational Lax matrices for other classes of non-abelian involutive Yang-Baxter maps ($Λ$, $\mathcal{H}$ and $\mathcal{F}$ lists). In the abelian setting, this procedure generates rational Lax representations for the abelian Yang-Baxter maps of the $F$ and $H$ lists. Additionally, we provide examples of non-involutive (abelian and non-abelian) multi-parametric Yang-Baxter maps, along with their Lax representations, which lie outside the preceding lists.
We extend the equations of motion that describe non-relativistic elastic collision of two particles in one dimension to an arbitrary associative algebra. Relativistic elastic collision equations turn out to be a particular case of these generic equations. Furthermore, we show that these equations can be reinterpreted as difference systems defined on the ${\mathbb Z}^2$ graph and this reinterpretation relates (unifies) the linear and the non-linear approach of discrete analytic functions.
We present three non-equivalent procedures to obtain entwining (non-constant) tetrahedron maps. Given a tetrahedron map, the first procedure incorporates its underlying symmetry group. With the second procedure we obtain several classes of entwining tetrahedron maps by considering certain compositions of pentagon with reverse-pentagon maps which satisfy certain compatibility relations the so-called ten-term relations. Using the third procedure, provided that a given tetrahedron map admits at least one companion map (partial inverse), we obtain entwining set theoretical solutions of the tetrahedron equation.
We classify rational solutions of a specific type of the set theoretical version of the pentagon equation. That is, we find all quadrirational maps $R:(x,y)\mapsto (u(x,y),v(x,y)),$ where $u, v$ are two rational functions on two arguments, that serve as solutions of the pentagon equation. Furthermore, provided a pentagon map that admits a partial inverse, we obtain genuine entwining pentagon set theoretical solutions. Finally, we show how to obtain Yang-Baxter maps from entwining pentagon maps.
We present two non-equivalent families of hierarchies of non-Abelian compatible maps and we provide their Lax pair formulation. These maps are associated with families of hierarchies of non-Abelian Yang-Baxter maps, which we provide explicitly. In addition, these hierarchies correspond to integrable difference systems with variables defined on edges of an elementary cell of the $\mathbb{Z}^2$ graph, that in turn lead to hierarchies of difference systems with variables defined on vertices of the same cell. In that respect we obtain the non-Abelian lattice-modified Gel'fand-Dikii hierarchy, together with the explicit form of a non-Abelian hierarchy that we refer to as the lattice-NQC (or lattice-$(Q3)_0$) Gel'fand-Dikii hierarchy.
We propose a specific class of matrices which participate in factorization problems that turn to be equivalent to constant and entwining (non-constant) pentagon, reverse-pentagon or Yang-Baxter maps, expressed in non-commutative variables. In detail, we show that factorizations of order $N=2$ matrices of this specific class are equivalent to the homogeneous normalization map. From order $N=3$ matrices, we obtain an extension of the homogeneous normalization map, as well as novel entwining pentagon, reverse-pentagon and Yang-Baxter maps.
We introduce four lists of families of non-abelian quadrirational Yang-Baxter maps.
The Kahan discretization of the Lotka-Volterra system, associated with any skew-symmetric graph $Γ$, leads to a family of rational maps, parametrized by the step size. When these maps are Poisson maps with respect to the quadratic Poisson structure of the Lotka-Volterra system, we say that the graph $Γ$ has the Kahan-Poisson property. We show that if $Γ$ is connected, it has the Kahan-Poisson property if and only if it is a cloning of a graph with vertices $1,2,\dots,n$, with an arc $i\to j$ precisely when $i<j$, and with all arcs having the same value. We also prove a similar result for augmented graphs, which correspond with deformed Lotka-Volterra systems and show that the obtained Lotka-Volterra systems and their Kahan discretizations are superintegrable as well as Liouville integrable.
We introduce a family of order $N\in \mathbb{N}$ Lax matrices that is indexed by the natural number $k\in \{1,\ldots,N-1\}.$ For each value of $k$ they serve as strong Lax matrices of a hierarchy of integrable difference systems in edge variables that in turn lead to hierarchies of integrable difference systems in vertex variables or in a combination of edge and vertex variables. Furthermore, the entries of the Lax matrices are considered as elements of a division ring, so we obtain hierarchies of discrete integrable systems extended in the non-commutative domain.
We study the basic relation between skew-symmetric Lotka-Volterra systems and graphs, both at the level of objects and morphisms, and derive a classification from it of skew-symmetric Lotka-Volterra systems in terms of graphs as well as in terms of irreducible weighted graphs. We also obtain a description of their automorphism groups and of the relations which exist between these groups. The central notion introduced and used is that of decloning of graphs and of Lotka-Volterra systems. We also give a functorial interpretation of the results which we obtain.
A relationship between the tetrahedron equation for maps and the consistency property of integrable discrete equations on $\mathbb{Z}^3$ is investigated. Our approach is a generalization of a method developed in the context of Yang-Baxter maps, based on the invariants of symmetry groups of the lattice equations. The method is demonstrated by a case-by-case analysis of the octahedron type lattice equations classified recently, leading to some new examples of tetrahedron maps and integrable coupled lattice equations.
We present two lists of multi-component systems of integrable difference equations defined on the edges of a $\mathbb{Z}^2$ graph. The integrability of these systems is manifested by their Lax formulation which is a consequence of the multi-dimensional compatibility of these systems. Imposing constraints consistent with the systems of difference equations, we recover known integrable quad-equations including the discrete version of the Krichever-Novikov equation. The systems of difference equations allow us for a straightforward reformulation as Yang-Baxter maps. Certain two-component systems of equation defined on the vertices of a $\mathbb{Z}^2$ lattice, their non-potential form and integrable equations defined on 5-point stencils, are also obtained.
For some involutive maps $Φ:{\mathbb C}P^1 \times {\mathbb C}P^1 \to {\mathbb C}P^1 \times {\mathbb C}P^1$ we find all invariants with separated variables. We investigate a link of the maps and their invariants with separated variables to discrete integrable systems. Maps correspond to integrable systems on edges (bond systems), while their invariants with separated variables yields potentials of the bond systems, that allows us to rewrite the integrable sytems as models on vertices. Among the latter ones one can find well known integrable difference equations as well as difference relations, which in contrast to the equations give non-single-valued evolution of the dependent variable. However, the non-single-valuedness can be resolved by the link with the bond system.
In this paper we construct a family of integrable reductions of the dressing chain, described in its Lotka-Volterra form. For each $k,n\in\mathbb N$ with $n\geqslant 2k+1$ we obtain a Lotka-Volterra system $\hbox{LV}_b(n,k)$ on $\mathbb R^n$ which is a deformation of the Lotka-Volterra system $\hbox{LV}(n,k)$, which is itself an integrable reduction of the $2m+1$-dimensional Bogoyavlenskij-Itoh system $\hbox{LV}(2m+1,m)$, where $m=n-k-1$. We prove that $\hbox{LV}_b(n,k)$ is both Liouville and non-commutative integrable, with rational first integrals which are deformations of the rational first integrals of $\hbox{LV}(n,k)$. We also construct a family of discretizations of $\hbox{LV}_b(n,0)$, including its Kahan discretization, and we show that these discretizations are also Liouville and superintegrable.
We present the explicit form of a family of Liouville integrable maps in 3 variables, the so-called triad family of maps and we propose a multi-field generalisation of the latter. We show that by imposing separability of variables to the invariants of this family of maps, the $H_{\rm I}$, $H_{\rm II}$ and $H_{\rm III}^A$ Yang-Baxter maps in general position of singularities emerge. Two different methods to obtain entwining Yang-Baxter maps are also presented. The outcomes of the first method are entwining maps associated with the $H_{\rm I}$, $H_{\rm II}$ and $H_{\rm III}^A$ Yang-Baxter maps, whereas by the second method we obtain non-periodic entwining maps associated with the whole $F$ and $H$-list of quadrirational Yang-Baxter maps. Finally, we show how the transfer maps associated with the $H$-list of Yang-Baxter maps can be considered as the $(k-1)$-iteration of some maps of simpler form. We refer to these maps as extended transfer maps and in turn they lead to $k$-point alternating recurrences which can be considered as alternating versions of some hierarchies of discrete Painlevé equations.
A QRT map is the composition of two involutions on a biquadratic curve: one switching the $x$-coordinates of two intersection points with a given horizontal line, and the other switching the $y$-coordinates of two intersections with a vertical line. Given a QRT map, a natural question is to ask whether it allows a decomposition into further involutions. Here we provide new answers to this question and show how they lead to a new class of maps, as well as known HKY maps and quadrirational Yang-Baxter maps.