Non-commutative rational Yang-Baxter maps

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.

7 pages, 2 figures; Remark on p. 6 corrected (v2)

References in corpus (2)

• Non-commutative lattice modified Gel'fand-Dikii systems
• Desargues maps and their reductions

Cited by in corpus (18)

• Grassmann extensions of Yang-Baxter maps
• A noncommutative discrete potential KdV lift
• On non-abelian quadrirational Yang-Baxter maps
• Variations of $q$-Garnier system
• Entwining Yang-Baxter maps related to NLS type equations
• Quantum groups, Yang-Baxter maps and quasi-determinants
• The Coxeter relations and KP map for non-commuting symbols
• Non-Abelian hierarchies of compatible maps, associated integrable difference systems and Yang-Baxter maps
• Matrix factorizations and pentagon maps
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Non-commutative double-sided continued fractions
• Two-component Yang-Baxter maps and star-triangle relations
• Discrete Lax pairs and hierarchies of integrable difference systems
• Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables
• Non-commutative q-Painleve VI equation
• Noncommutative solutions to Zamolodchikov's tetrahedron equation and matrix six-factorisation problems
• ${\mathbb{Z}}_N$ graded discrete Lax pairs and Yang-Baxter maps
• On refactorization problems and rational Lax matrices of quadrirational Yang-Baxter maps