Set-theoretical solutions of the pentagon equation on groups

Abstract

Let M be a set. A set-theoretical solution of the pentagon equation on M is a map s : M × M −→ M × M such that s 23 s 13 s 12 = s 12 s 23 , where s 12 = s × id M , s 23 = id M ×s and s 13 = (id M ×τ )s 12 (id M ×τ ), and τ is the flip map, i.e., the permutation on M × M given by τ (x, y) = (y, x), for all x, y ∈ M .In this paper we give a complete description of the set-theoretical solutions of the form s(x, y) = (x · y, x * y) when either (M, ·) or (M, * ) is a group; moreover, we raise some questions.

Set-theoretical solutions of the Yang–Baxter and pentagon equations on semigroups

“…By (p5), we get α u (1 S ) = 1 S , for every u ∈ T and similarly, by (r5), β a (1 T ) = 1 T , for every a ∈ S. Moreover, by (9), with b = 1 S and v = 1 T we obtain…”

“…According to [9,Definition 1], such a map t is called a set-theoretical solution of the reversed pentagon equation, or briefly a reversed solution. Thus, every reversed solution will be written as t(x, y) = (θ y (x), yx).…”

“…In literature, there are few systematic constructions of PE solutions. In [25, Proposition 1] Kashaev and Sergeev provide a construction of PE solutions on a closed under multiplication subset S of a group G. Recently, a complete description of PE solutions s on groups G of the form s(x, y) = (xy, θ x (y)) is presented in [9,Theorem 15].…”

“…Set-theoretical solutions of the pentagon equation are used in the pioneering work of Baaj and Skandalis [1] to obtain multiplicative unitary operators on a Hilbert space. Later, there have been appeared several papers on this topic: Zakrzewski [38], Baaj and Skandalis [2], Kashaev and Sergeev [25], Jiang and Liu [22], Kashaev and Reshetikhin [24], Kashaev [23], and Catino, Mazzotta, and Miccoli [9]. On the other hand, a map s : S × S → S × S is a set-theoretical solution of the quantum Yang-Baxter equation [14] on a set S if the relation s 23 s 13 s 12 = s 12 s 13 s 23 is satisfied, with the same notation adopted for the pentagon equation.…”

“…To find other examples on groups, by conditions(7) and (8) one has to look at maps ᾱ and β whose images Im ᾱ and Im β are contained in the kernel K s of s and K t of t, respectively. Given a PE solution s(a, b) = (ab, θ a (b)) on a group S, we remind that the subset of S defined byK s := {a | a ∈ S, θ 1S (a) = 1 S }is a normal subgroup of S called the kernel of s (cf [9,. Lemma 13]).…”