Grassmann extensions of Yang-Baxter maps

Abstract

In this paper we show that there are explicit Yang-Baxter maps with Darboux-Lax representation between Grassmann extensions of algebraic varieties. Motivated by some recent results on noncommutative extensions of Darboux transformations, we first derive a Darboux matrix associated with the Grassmann-extended derivative Nonlinear Schrodinger (DNLS) equation, and then we deduce novel endomorphisms of Grassmann varieties, which possess the Yang-Baxter property. In particular, we present ten-dimensional maps which can be restricted to eight-dimensional Yang-Baxter maps on invariant leaves, related to the Grassmann-extended NLS and DNLS equations. We consider their vector generalisations.

18 pages, LaTeX

References in corpus (16)

• Classification of integrable equations on quad-graphs. The consistency approach
• Yang-Baxter maps and integrable dynamics
• Reduction Groups and Automorphic Lie Algebras
• Lax matrices for Yang-Baxter maps
• Integrable systems on quad-graphs
• Yang-Baxter maps: dynamical point of view
• Darboux transformations, finite reduction groups and related Yang-Baxter maps
• Integrable Discretisations for a Class of Nonlinear Schrodinger Equations on Grassmann Algebras
• An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions
• Reduction groups and related integrable difference systems of NLS type
• Non-commutative rational Yang-Baxter maps
• Yang Baxter maps with first degree polynomial 2 by 2 Lax matrices
• Yang-Baxter maps and multi-field integrable lattice equations
• Bäcklund-Darboux Transformations and Discretizations of Super KdV Equation
• Poisson Yang-Baxter maps with binomial Lax matrices
• A supersymmetric AKNS problem and its Darboux-Bäcklund transformations and discrete systems

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• Entwining Yang-Baxter maps related to NLS type equations
• The Coxeter relations and KP map for non-commuting symbols
• Non-Abelian hierarchies of compatible maps, associated integrable difference systems and Yang-Baxter maps
• Bäcklund-Darboux Transformations and Discretizations of $N=2a=-2$ Supersymmetric KdV Equation
• Matrix factorizations and pentagon maps
• Birational solutions to the set-theoretical 4-simplex equation
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Darboux transformations, discrete integrable systems and related Yang-Baxter maps
• Discrete Lax pairs and hierarchies of integrable difference systems
• Integrable multi-component difference systems of equations
• Discrete integrable systems, Darboux transformations, and Yang-Baxter maps
• Integrable extensions of Adler's map via Grassmann algebras