Abstract
In this paper, we formulate a “Grassmann extension” scheme for constructing noncommutative (Grassmann) extensions of Yang-Baxter maps together with their associated systems of PΔEs, based on the ideas presented in [15]. Using this scheme, we first construct a Grassmann extension of a Yang-Baxter map which constitutes a lift of a lattice Boussinesq system. The Grassmann-extended Yang-Baxter map can be squeezed down to a novel, integrable, Grassmann lattice Boussinesq system, and we derive its 3D-consistent limit. We show that some systems retain their 3D-consistency property in their Grassmann extension.
Highlights
Over the past few decades, there has been an increasing interest in the study of noncommutative extensions of integrable equations or systems of equations, due to their numerous applications in Physics
Due to the useful properties of 3D consistent quad-graph systems and the availability of simple algebraic schemes for contructing solutions to them, they can be used as good models for studying their continuous analogues, i.e. systems of
The importance of noncommutative extensions of integrable systems from a Physics perspective, and the innovating results that have already been obtained in the continuously developing field of Discrete Integrable Systems, motivates us to extend to the noncommutative case the already existing methods for constructing solutions to integrable systems in the commutative case
Summary
Over the past few decades, there has been an increasing interest in the study of noncommutative extensions of integrable equations or systems of equations (indicatively we refer to [7, 8, 11, 12, 18, 24]), due to their numerous applications in Physics. The importance of noncommutative extensions of integrable systems from a Physics perspective, and the innovating results that have already been obtained in the continuously developing field of Discrete Integrable Systems, motivates us to extend to the noncommutative case the already existing methods for constructing solutions to integrable systems in the commutative case. Towards this direction, a few steps have been made over the past few years. It owes its popularity to its quite interesting and, simple form, with a number of applications in Fluid Dynamics and in the theory of Integrable Systems
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