Tetrahedron equation: algebra, topology, and integrability

Noncommutative solutions to Zamolodchikov's tetrahedron equation and matrix six-factorisation problems

“…In particular, using a Darboux transformation for the noncommutative coupled NLS system, we constructed a noncommutative tetrahedron map (29). We showed that the latter can be restricted to a noncommutative Sergeev's map, namely map (33).…”

“…as well as x = x, ŷ = y and r = r in view of which the above system implies ŝ = s, ẑ = z and t = t. Therefore, map (33) satisfies the tetrahedron equation.…”

“…Substituting to (33), we obtain the three-dimensional map (32). Now, choosing a = b = c = k ∈ Z(R) in (32), we obtain the fully noncommutative map (33).…”

“…Substituting to (33), we obtain the three-dimensional map (32). Now, choosing a = b = c = k ∈ Z(R) in (32), we obtain the fully noncommutative map (33). This is the noncommutative version of Sergeev's map, namely, if we assume that x, y, z ∈ Z(R), then map (32) becomes map (20) in [31].…”

“…Tetrahedron maps, namely solutions to the Zamolodchikov's functional tetrahedron equation, are of great significance in the the theory of integrable systems since they are strictly related to integrable three-dimensional lattice equations (see, e.g., [2,7,15] and the references therein) which also discretise nonlinear integrable PDEs, and at the same time have very interesting algebro-geometric properties (see, e.g., [1,2,10,11,16]). On the other hand, noncommutative versions or extensions of integrable systems have been a growing field over the past few decades, with many applications in mathematical physics, and have been in the centre of interest for many scientists (indicatively we refer to [3,4,5,8,24,25,29,33]).…”