Non-commutative bi-rational maps satisfying Zamolodchikov equation, and Desargues lattices

Abstract

We present new solutions of the functional Zamolodchikov tetrahedron equation in terms of birational maps in totally non-commutative variables. All the maps originate from Desargues lattices, which provide geometric realization of solutions to the non-Abelian Hirota-Miwa system. The first map is derived using the original Hirota's gauge for the corresponding linear problem, and the second one from its affine (non-homogeneous) description. We provide also an interpretation of the maps within the local Yang-Baxter equation approach. We exploit decomposition of the second map into two simpler maps which, as we show, satisfy the pentagonal condition. We provide also geometric meaning of the matching ten-term condition between the pentagonal maps. The generic description of Desargues lattices in homogeneous coordinates allows to define another solution of the Zamolodchikov equation, but with functional parameter which should be adjusted in a particular way. Its ultra-local reduction produces a birational quantum map (with two central parameters) with Zamolodchikov property, which preserves Weyl commutation relations. In the classical limit our construction gives the corresponding Poisson map satisfying the Zamolodchikov condition.

arXiv comment25 pages, 9 figures

References in corpus (21)

• T-systems and Y-systems in integrable systems
• Multidimensional Quadrilateral Lattices are Integrable
• On Discrete 3-Dimensional Equations Associated with the Local Yang-Baxter Relation
• Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups
• Functional Tetrahedron Equation
• Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy
• Algebraic integrable dynamical systems, 2+1-dimensional models in wholly discrete space-time, and inhomogeneous models in 2-dimensional statistical physics
• Quantum geometry of 3-dimensional lattices
• Quantum 2+1 evolution model
• Desargues maps and the Hirota-Miwa equation
• The affine Weyl group symmetry of Desargues maps and of the non-commutative Hirota-Miwa system
• Tetrahedron maps and symmetries of three dimensional integrable discrete equations
• Affine Toda field theory as a 3-dimensional integrable system
• Non-commutative lattice modified Gel'fand-Dikii systems
• Tetrahedron Equation and Quantum R Matrices for Spin Representations of B^{(1)}_n, D^{(1)}_n and D^{(2)}_{n+1}
• The pentagon relation and incidence geometry
• From the Tetrahedron Equation to Universal R-Matrices
• The Coxeter relations and KP map for non-commuting symbols
• Quantization of three-wave equations
• On Pentagon And Tetrahedron Equations
• Symmetrically Factorizable Groups and Set-Theoretical Solutions of the Pentagon Equation

Cited by in corpus (8)

• Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Non-commutative Hermite--Padé approximation and integrability
• Matrix factorizations and pentagon maps
• Birational solutions to the set-theoretical 4-simplex equation
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Set-theoretical solutions to the Zamolodchikov tetrahedron equation on associative rings and Liouville integrability
• Noncommutative solutions to Zamolodchikov's tetrahedron equation and matrix six-factorisation problems