Non-Commutative Rational Yang–Baxter Maps

Adam Doliwa

doi:10.1007/s11005-013-0669-7

Abstract

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 maps arising from geometric crystals. Our approach works under additional condition of centrality of certain products of non-commuting variables. Then we apply such a restriction on the level of the Gel'fand-Dikii systems what allows to obtain non-autonomous (but with central non-autonomous factors) versions of the equations. In particular we recover known non-commutative version of Hirota's lattice sine-Gordon equation, and we present an integrable non-commutative and non-autonomous lattice modified Boussinesq equation.

Highlights

Let X be any set, a map R : X × X satisfying in X × X × X the relationR12 ◦ R13 ◦ R23 = R23 ◦ R13 ◦ R12, (1.1)where Ri j acts as R on the ith and jth factors and as identity on the third, is called Yang–Baxter map [12,29]
In this paper we study properties of a non-commutative version of the maps arising from geometric crystals [13,18,28]
In recent studies on discrete integrable systems, the property of multidimensional consistency [1,24] is considered as the main concept of the theory

Summary

Introduction

Where Ri j acts as R on the ith and jth factors and as identity on the third, is called Yang–Baxter map [12,29]. In recent studies on discrete integrable systems, the property of multidimensional consistency [1,24] is considered as the main concept of the theory Speaking, it is the possibility of extending the number of independent variables of a given nonlinear system by adding its copies in different directions without creating this way inconsistency or multivaluedness. It is the possibility of extending the number of independent variables of a given nonlinear system by adding its copies in different directions without creating this way inconsistency or multivaluedness It is known [2,27] how to relate threedimensional consistency of integrable discrete systems with Yang–Baxter maps. This approach is intuitively accessible, see [5] for formal definitions

KP MAPS

REALIZATION OF COXETER RELATIONS UNDER THE CENTRALITY ASSUMPTION

Non-Commutative Gel’fand–Dikii Systems with the Centrality Condition

Concluding Remarks