Dynamical Yang-Baxter Maps Associated with Homogeneous Pre-Systems

Dynamical Yang-Baxter maps and Hopf algebroids associated with s-sets

“…Definition 2.3. A pair (S, η) of a nonempty set S and a ternary operation η : S × S × S → S is called a homogeneous pre-system [13], iff the ternary operation η satisfies: for any x, y, u, v, w ∈ S, η(x, y, x) = y; η(x, y, η(u, v, w)) = η(η(x, y, u), η(x, y, v), η(x, y, w)).…”

“…Several studies clarified how to construct the dynamical Yang-Baxter maps [19,20,26,27,29]. In [13], suitable homogeneous pre-systems produce the dynamical Yang-Baxter maps. The homogeneous pre-system is a generalization of the homogeneous system [14] in differential geometry, an algebraic feature of the reductive homogeneous space with suitable conditions.…”

“…The organization of this paper is as follows. In Section 2, we give a brief exposition of the homogeneous pre-system, the dynamical Yang-Baxter map, the Hopf algebroid, and the rigid tensor category [13,30]. Section 3 discusses the construction of ternary operations by means of the s-sets.…”

“…In this section, we summarize without proofs the relevant material on homogeneous pre-systems, dynamical Yang-Baxter maps, Hopf algebroids, and rigid tensor categories [13,30], to render this paper as self-contained as possible.…”

Quantum Yang-Baxter Equation, Braided Semigroups, and Dynamical Yang-Baxter Maps

“…This dynamical Yang-Baxter map yields bialgebroids [25] and discrete integrable systems through 3D compatible ternary systems [14]. Furthermore, suitable homogeneous pre-systems [12], related to reductive homogeneous spaces, can produce the dynamical Yang-Baxter map.…”