Non-commutative birational 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 is derived from its affine (non-homogeneous) description. We also provide an interpretation of the maps within the local Yang–Baxter equation approach. We exploit the decomposition of the second map into two simpler maps, which, as we show, satisfy the pentagonal condition. We also provide geometric meaning of the matching ten-term condition between the pentagonal maps. The generic description of Desargues lattices in homogeneous coordinates allows us to define another solution of the Zamolodchikov equation, but with a functional parameter that should be adjusted in a particular way. Its ultra-local reduction produces a birational quantum map (with two central parameters) with the Zamolodchikov property, which preserves Weyl commutation relations. In the classical limit, our construction gives the corresponding Poisson map, satisfying the Zamolodchikov condition.