Dynamical Yang-Baxter Maps Associated with Homogeneous Pre-Systems (cid:2)

We construct dynamical Yang-Baxter maps, which are set-theoretical solutions to a version of the quantum dynamical Yang-Baxter equation, by means of homogeneous pre-systems, that is, ternary systems encoded in the reductive homogeneous space satisfying suitable conditions. Moreover, a characterization of these dynamical Yang-Baxter maps is presented.

This ternary system G = (G, η) is a homogeneous system [23, Proposition 1] (see also Definition 7), and every homogeneous system is constructed in such a way. If G is a connected and second countable C ∞ -manifold with a C ∞ -map η : G × G × G → G, then the homogeneous system G = (G, η) is isomorphic to a reductive homogeneous space A/K for a connected Lie group A with its closed subgroup K [19, Theorem 1]. The homogeneous system is a ternary system, an algebraic structure, encoded in the reductive homogeneous space (for ternary systems in differential geometry and mathematical physics, see [13,14,15,29]).
It is natural to relate this homogeneous system to the dynamical Yang-Baxter map through the ternary system. The aim of this paper is to produce the dynamical Yang-Baxter maps by means of homogeneous pre-systems, which generalize the homogeneous system. Furthermore, we characterize such dynamical Yang-Baxter maps.
The organization of this paper is as follows. Section 2 contains a brief summary of the dynamical Yang-Baxter map. We focus on its construction by means of the ternary system. This construction yields a category A concerning the ternary systems, which is equivalent to a category D consisting of the dynamical Yang-Baxter maps.
Section 3 presents the notion of a homogeneous pre-system, together with examples. In Section 4, our main results are stated and proved. Every homogeneous pre-system satisfying (4.1) can produce a dynamical Yang-Baxter map via the ternary system. More precisely, we construct a category H, isomorphic to the category A, by means of the homogeneous pre-systems with (4.1). Because the category A is equivalent to the category D, each object of H gives a dynamical Yang-Baxter map; in particular, we demonstrate dynamical Yang-Baxter maps provided by a certain left quasigroup and the examples in Section 3.
The last section, Section 5, deals with a relation between the homogeneous pre-system satisfying (4.1) and the left quasigroup with (5.1), which is due to the work in [32,Section 6]. We introduce a category B concerning the left quasigroups satisfying (5.1) and an essentially surjective functor J : B → H to construct the dynamical Yang-Baxter maps by means of quasigroups of reflection [17,27].
Our viewpoint sheds some light on the relation between geometry and the dynamical Yang-Baxter map.

Dynamical Yang-Baxter maps
In this section, we briefly summarize without proofs the relevant material in [32] on the construction of the dynamical Yang-Baxter map.

Definition 1.
(1) (L, ·) is a left quasigroup (resp. right quasigroup [38,Section I.4.3]), if and only if L is a nonempty set, together with a binary operation (·) on L having the property that, for all u, w ∈ L, there uniquely exists v ∈ L such that u · v = w (resp. v · u = w). For the simplicity, one uses the notation uv instead of u · v (u, v ∈ L). By this definition, the left quasigroup L = (L, ·) has another binary operation \ L called a left division [38,Section I.2.2]. For u, w ∈ L, we denote by u\ L w the unique element v ∈ L satisfying uv = w, The binary operation on the quasigroup is not always associative.

Journal of Generalized Lie
(2.6) We now introduce two categories A and D concerning a special class of the dynamical Yang-Baxter maps, which play a central role in this article.
The first task is to explain the category A (cf. the category A 2 in [32, Section 6]). We follow the notation of [16,Chapter XI]. Let L = (L, ·) be a left quasigroup, M = (M, μ) a ternary system satisfying (2.2) and (2.8) and π : L → M a bijection. The object of A is, by definition, a triple (L, M, π).
The morphism f : The next task is to describe the category D, which is exactly the category D 2 in [32, Section 6]. The object of this category D is a pair (L, R) of a left quasigroup L = (L, ·) and a dynamical Yang-Baxter map R(λ) :

Homogeneous pre-systems
This section is devoted to introducing homogeneous pre-systems.  for all x, y, u, v, w ∈ G.

(3.3)
We explain two examples in this section: one homogeneous pre-system and one homogeneous system, which imply dynamical Yang-Baxter maps in the next section.
Let G be an abelian group. We define the ternary operation η on G by η(x, y, z) = x + y − z, x, y, z ∈ G.

(3.4)
A trivial verification shows that G = (G, η) is a homogeneous pre-system, which is not always a homogeneous system because of (3.3) (cf. [18,Remark 4]). Another example is a homogeneous system on an arbitrary group G [18, Example in Section 1]. We define the ternary operation η on the group G by η(x, y, z) = yx −1 z, x, y, z ∈ G. (3.5) It is clear that this G = (G, η) is a homogeneous system. Remark 8. The homogeneous system (G, η) (3.5) is equivalent to the notion of a torsor [25,33,36], also known as the principal homogeneous space, up to the choice of the unit element. Hence, the principal homogeneous space provides a homogeneous system.

A relation between dynamical Yang-Baxter maps and homogeneous pre-systems
In this section, we construct dynamical Yang-Baxter maps (2.6) by means of homogeneous pre-systems G = (G, η) satisfying η(x, y, z) = η w, η(x, y, w), z , ∀x, y, z, w ∈ G. (4.1) In fact, we present a category H concerning the homogeneous pre-systems with (4.1); this H is isomorphic to the category A in Section 2, and, on account of Theorem 6, every object of H consequently gives a dynamical Yang-Baxter map.

Proposition 9. H is a category; the definitions of the identity, the source, the target and the composition are similar to those of the category A.
In order to prove that the category H is isomorphic to the category A, we construct functors F : A → H and As a consequence of Corollary 15 and Proposition 16, the homogeneous pre-system G (3.4) and the homogeneous system G (3.5) imply dynamical Yang-Baxter maps. Let L = (G, ·) denote the left quasigroup whose binary operation (·) is defined by (4.4) and let π : L(= G) → G be the identity map on G. The corresponding dynamical Yang-Baxter maps are as follows: if G is a homogeneous pre-system (3.4), then and if G is a homogeneous system (3.5), then

A relation between homogeneous pre-systems and left quasigroups
Because of the work in [32, Section 6] and the fact that the categories A and H are isomorphic, every homogeneous pre-system G (Definition 7(1)) in the object (L, G, π) ∈ H is a left quasigroup (Definition 1(1)) whose binary operation gives the ternary operation of G. This last section demonstrates it by constructing a category B concerning the left quasigroups with (5.1) and an essentially surjective functor J : B → H (see [32,Proposition 6.17]). The functors J : B → H, S : A → D in Section 2, and F : H → A in Section 4, together with quasigroups of reflection [17,27], provide examples of the dynamical Yang-Baxter map.
The first task is to introduce a category B. Let L 1 , L 2 = (L 2 , * ) be left quasigroups. We assume that the left quasigroup L 2 satisfies Here the symbol \ L2 is the left division (2.1) of L 2 . Let π : L 1 → L 2 be a (set-theoretic) bijection. An object of B is such a triple (L 1 , L 2 , π).
A morphism f : (L 1 , L 2 , π) → (L 1 , L 2 , π ) is a homomorphism f : L 1 → L 1 of left quasigroups such that π • f • π −1 : L 2 → L 2 is also a homomorphism of left quasigroups. Proposition 17. B is a category; the definitions of the identity, the source, the target and the composition are similar to those of the category A.
We give a proof only for (3.2) because the rest of the proof is straightforward. Let x, y, u, v, w ∈ G(= L 2 ). From (5.2) we have With the aid of (5.1), the right-hand side of (5.3) is y, w)). This is the desired conclusion.
(3) The quasigroup of reflection gives an involutory quandle [1,12,37] by reversing the order of the binary operation in Definition 23.
A straightforward computation shows that Nobusawa's quasigroup (Q, * ) in Example 2 is a quasigroup of reflection.
Let (G, * ) be a quasigroup of reflection, and L a left quasigroup isomorphic to G as sets. We denote by π a set-theoretic bijection from L to G. Because (5.