A noncommutative discrete potential KdV lift

Abstract

In this paper, we construct a Grassmann extension of a Yang-Baxter map which first appeared in [16] and can be considered as a lift of the discrete potential Korteweg-de Vries (dpKdV) equation. This noncommutative extension satisfies the Yang-Baxter equation, and it admits a 3 × 3 Lax matrix. Moreover, we show that it can be squeezed down to a system of lattice equations which possesses a Lax representation and whose bosonic limit is the dpKdV equation. Finally, we consider commutative analogues of the constructed Yang-Baxter map and its associated quad-graph system, and we discuss their integrability. partial differential equations through Darboux transformations [14,20]. Recently, a "lifting" method for constructing two-field equations, which can be viewed as Yang-Baxter maps, via equations on quadgraphs, was presented in [26].The relation between the Yang-Baxter property of maps and the multi-dimensional consistency property of integrable equations on quad-graphs is well established. The main approach is based on the symmetry analysis of the corresponding equations [25,27]. From another point of view, under some conditions 3D consistent equations can be lifted to four dimensional Yang-Baxter maps. This lifting procedure has been described in [26]. Conversely, Yang-Baxter maps of a specific form can be squeezed down to lattice equations and systems that admit a Lax representation derived from the Lax matrix of the corresponding Yang-Baxter map. Motivated by these results and working in the direction of extending the theory of Yang-Baxter maps to the noncommutative case, in this paper, we construct a Grassmann extension of a Yang-Baxter map, first derived in [16], and we show that it possesses the Yang-Baxter property. Furthermore, we use this map to construct a noncommutative extension of the well-celebrated dpKdV equation [12,22,24], also referred to as H 1 equation in the ABS list [1]. Both the Yang-Baxter map and its associated quad-graph system, which we construct, admit Lax representations with 3 × 3 Lax matrices and their bosonic limit leads to the original discrete systems.Finally, we consider all the variables of the constructed 3 × 3 Lax matrices to be commutative, and we derive a higher-dimensional commutative extension of the original Yang-Baxter map. This extension constitutes a symplectic map with respect to the Sklyanin bracket, and completely integrable in the Liouville sense. As in the noncommutative case, this map can be squeezed down to a corresponding integrable quad-graph system.The paper is organised as follows. In the next section, we begin with some preliminaries on Grassmann algebras, Yang-Baxter maps, quadrilateral equations and Lax representations. In section 3, we present the construction of the Grassmann extension of the lift of the dpKdV equation, and we show that it satisfies the Yang-Baxter equation. In section 4, we show that the constructed Yang-Baxter map can be reduced to a discrete quad-graph system, which constitutes a noncommutative extension of the dpKdV equation. The Lax ...

Casting light on shadow Somos sequences

“…Many of our considerations here extend naturally to the dual number analogues of other discrete integrable systems, such as the family of maps in [13], which are connected with Gale-Robinson sequences and cluster algebras, or the higher genus analogues of (3.4) in [12]. There are already versions of Yang-Baxter maps that include Grassmann variables, together with associated integrable lattice equations that satisfy the multidimensional consistency property [16]. Hopefully some of these techniques could also shed more light on superfriezes and cluster superalgebras, as in [17,22], which are relevant to Ptolemy relations in super-Teichmüller theory [18].…”

Высшие Разностные Уравнения Хироты И Их Редукции

“…раздел 7). Зависимость оператора K от переменных m вводится посредством той же ∂-задачи: 19), но с нулевой асимптотикой, так что эта разность обращается в ноль в силу предположения об однозначной разрешимости. Рассмотрим следствия уравнений (2.15) для оператора K. Заметим, что этот оператор, как и любой оператор из рассматриваемого класса, удовлетворяет равенству (2.11),…”

“…Оно было выведено в работах [17], [18] и подробно обсуждалось в литературе вместе со своими неабелевыми обобщениями (см. работу [19] и цитированную в ней литературу). Мы привели здесь вывод этого уравнения как пример размерной редукции в рамках нашего подхода.…”

Hopf Algebra Structure of Generalized Quasi-Symmetric Functions in Partially Commutative Variables

“…For example, the Schur functions when expressed in suitably scaled power sum functions (times of the KP hierarchy) provide polynomial τ -function solutions of the equations. Non-commutative extensions of integrable systems are of growing interest in mathematical physics [49,7,63,28,15,46,18,20,22,21].…”