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^{1}Center for Mathematical Sciences, University of Aizu, Aizuwakamatsu, Fukushima 965-8580, Japan, **E-mail:** [email protected]

^{2}Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, Hokkaido 060-0810, Japan

**Received Date:** 30 August 2010; **Accepted Date:** 26 January 2011

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

81R50, 20N05, 20N10, 53C30, 53C35

The quantum dynamical Yang-Baxter equation (QDYBE for short) [9,10], a generalization of the quantum Yang- Baxter equation (QYBE for short) [2,3,40,41], has been studied extensively in recent years (see [7] and the references therein). Dynamical Yang-Baxter maps [31,32,34] are set-theoretical solutions to a version of the QDYBE.

Let *H* and *X* be nonempty sets with a map . A map is a dynamical Yang-Baxter map associated with *H*, *X* and (·), if and only if, for every satisfies the following equation on *X* × *X* × *X*:

Here , and others are the maps from *X* × *X* × *X* to itself defined as follows:
for *u*, *v*,*w* ∈ *X*,

Set-theoretical solutions to the QYBE [6], also known as Yang-Baxter maps [39], are dynamical Yang-Baxter
maps constant for the parameter λ of any set *H*; indeed, the dynamical Yang-Baxter map is a generalization of the
set-theoretical solution to the QYBE.

This dynamical Yang-Baxter map yields a bialgebroid [4]. Every dynamical Yang-Baxter map with some conditions
gives birth to an (*H*,*X*)-bialgebroid [35], a generalization of the quantum group [5,11], through the Faddeev-
Reshetikhin-Takhtajan construction [8].

It is worth pointing out that a ternary system (Definition 1(3)) can produce the dynamical Yang-Baxter map [32].
Each triple (*L*,*M*, π) consisting of a left quasigroup *L* = (*L*, ·) (Definition 1(1)), a ternary system *M* satisfying (2.2)
and (2.3), and a (set-theoretic) bijection π : *L* → *M* gives a dynamical Yang-Baxter map *R*(λ) associated with *L*, *L*
and (·) (see Section 2 for more details).

Homogeneous systems [18-23] are algebraic features of the reductive homogeneous space [24,28]
satisfying suitable conditions. Let *A* be a group with its subgroup *K*. We assume that a subset *G* of the group *A*
satisfies the following:

(1) the group *A* is uniquely factorized as *A* = *GK*,

(2) *G*^{−1} = *G*,

(3) *kGk*^{−1} = *G*, for all *k* ∈ *K*.

Let *p* : *A* → *G* denote the canonical projection with respect to the factorization *A* = *GK*, and (·) the binary
operation on *G* defined by *x* · *y* = *p*(*xy*) (*x*, *y* ∈ *G*). Because the map is bijective, we define
the ternary operation *η* on *G* by

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 concerning the ternary systems, which is equivalent to a category 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 , isomorphic to the category , by means of the homogeneous pre-systems with (4.1). Because the category is equivalent to the category , each object of 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 concerning the left quasigroups satisfying (5.1) and an essentially surjective functor 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.

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*).

(2) A quasigroup (*Q*, ·) is a left and right quasigroup (see [30, Definition I.1.1] and [38, Section I.2]).

(3) A ternary system (*M*,*μ*) is a pair of a nonempty set *M* and a ternary operation *μ* : *M* ×*M* ×*M* → *M*.

By this definition, the left quasigroup *L* = (*L*, ·) has another binary operation \* _{L}* called a left division [38,
Section I.2.2]. For

*u*\* _{L}w* =

The binary operation on the quasigroup is not always associative.

**Example 2.** We define the binary operation (∗) on the set *Q* = {1, 2, 3, 4, 5} of five elements by Table 1. Here
1 ∗ 2 = 3. This *Q* = (*Q*, ∗) is a quasigroup, because each element in *Q* appears once and only once in each row and
in each column of Table 1 [30, Theorem I.1.3]. The binary operation (∗) is not associative, since (1∗2)∗3 = 1∗(2∗3).
This quasigroup *Q* is due to Nobusawa [27, Section 6, type 1]. However, the order of the binary operation (∗) in
**Table 1** is reversed.

Each ternary system *M* = (*M*,*μ*) satisfying

(2.2)

(2.3)

for any *a*, *b*, *c*, *d* ∈ *M*, can provide a dynamical Yang-Baxter map [32, Theorem 3.2]. Let *L* = (*L*, ·) be a left
quasigroup isomorphic to *M* as sets, and π : *L* → *M* a (set-theoretic) bijection. For *λ*, *u* ∈ *L*, we define the maps and as follows: for *v* ∈ *L*,

(2.4)

(2.5)

Let *R*^{(L,M,π)}(λ) (λ ∈ *L*) denote the map from *L* × *L* to itself defined by

(2.6)

**Theorem 3.** *The map* *R*^{(L,M,π)}(λ) (2.6) *is a dynamical Yang-Baxter map* (1.1) *associated with L, L and* (·).

We now introduce two categories and 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 (cf. the category _{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

*μ*(*a*, *b*, *b*) = *a*, ∀*a*, *b* ∈ *M*, (2.7)

(2.8)

and *π* : *L* → *M* a bijection. The object of is, by definition, a triple (*L*,*M*, *π*).

The morphism of is a homomorphism of left quasigroups such that is a homomorphism of ternary systems; that is, the map satisfies

(2.9)

The identity id, the source *s*(*f*) and the target *b*(*f*) of a morphism and the
composition *g *◦ *f* for morphisms and are defined as
follows: for an object (*L*,*M*, *π*) ∈ *A*,

the composition *g* ◦ *f* is the usual one of the maps

**Proposition 4. ** * is a category.*

The next task is to describe the category , which is exactly the category _{2} in [32, Section 6]. The object of this
category is a pair (*L*,*R*) of a left quasigroup *L* = (*L*, ·) and a dynamical Yang-Baxter map *R*(λ) : *L*×*L* → *L*×*L*
(λ ∈ *L*) satisfying

Here, (*ηλ*(*v*)(*u*), *ξλ*(*u*)(*v*)) := *R*(*λ*)(*u*, *v*) (*λ*, *u*, *v* ∈ *L*).

The morphism of is a homomorphism of left quasigroups satisfying

**Proposition 5.** *is a category; the definitions of the identity, the source, the target and the composition are similar
to those of the category* .

We can construct functors which establish an equivalence of the categories and (cf. [32, Proposition 6.15]): for (*L*,*M*, *π*) ∈ , set *S*(*L*,*M*, *π*) = (*L*,*R*^{(L,M, π)}). Here, *R*^{(L,M, π)}(λ) is
defined by (2.4), (2.5) and (2.6); for a morphism *f* of , write denotes the triple (*L*, (*M*, *μ*), id* _{L}*), where

These functors *S* and *T* satisfy , and gives a natural isomorphism . Thus, the following theorem holds.

**Theorem 6.** *is an equivalence of categories*.

This section is devoted to introducing homogeneous pre-systems.

**Definition 7.** (1) A ternary system *G* = (*G*, *η*) (Definition 1(3)) is a homogeneous pre-system if and only if the
ternary operation *η* satisfies

*η*(*x*, *y*, *x*) = *y*, ∀*x*, *y* ∈ *G*, (3.1)

(3.2)

for all *x*, *y*, *u*, *v*,*w* ∈ *G*.

(2) A homogeneous system *G* = (*G*, *η*) [18] is a homogeneous pre-system satisfying

(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

(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

(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.

In this section, we construct dynamical Yang-Baxter maps (2.6) by means of homogeneous pre-systems *G* = (*G*, *η*)
satisfying

(4.1)

In fact, we present a category concerning the homogeneous pre-systems with (4.1); this is isomorphic to the category in Section 2, and, on account of Theorem 6, every object of consequently gives a dynamical Yang- Baxter map.

Let *L* = (*L*, ·) be a left quasigroup, *G* = (*G*, *η*) a homogeneous pre-system satisfying (4.1) and *π* : *L* → *G* a
(set-theoretic) bijection. The object of is a triple (*L*, *G*, *π*).

The morphism of is a homomorphism of left quasigroups such that is a homomorphism of ternary systems; that is, the map satisfies (2.9) and

**Proposition 9.** is a category; the definitions of the identity, the source, the target and the composition are similar
to those of the category

In order to prove that the category is isomorphic to the category , we construct functors and .

We first introduce the functor . Let . Define the ternary system *G* = (*G*, *η*) by
*G* = *M* as sets and

(4.2)

**Proposition 10.**

*Proof*. We need only show that *G* is a homogeneous pre-system satisfying (4.1).

An easy computation shows (3.1) and (4.1).

By virtue of (4.2), the left-hand side of (3.2) is *μ*(*y*, *x*, *μ*(*v*, *u*,*w*)), and, with the aid of (2.2), (2.7) and (2.8),

which is the right-hand side of (3.2). This proves the proposition.

By setting *F*(*L*, (*M*,*μ*), *π*) = (*L*, *G*, *π*) and *F*(*f*) = *f* for a morphism *f* of , the following proposition holds.

**Proposition 11.** *is a functor.*

The next task is to construct a functor Let We define the ternary system as sets and

(4.3)

**Proposition 12.**

*Proof*. It suffices to prove that *M _{G}* satisfies (2.2), (2.7) and (2.8).

A trivial verification shows (2.7) and (2.8).

Due to (4.1) and (4.3), the left-hand side of (2.2) is

From (3.1) and (3.2),

which is exactly the right-hand side of (2.2).

By setting and for a morphism *f* of , the following proposition
holds.

**Proposition 13.** *is a functor*.

Since the functors *F* and *F*′ satisfy , the following theorem holds.

**Theorem 14.** *The categories* *and* *are isomorphic.*

By taking account of Theorem 6, we have the following corollary.

**Corollary 15.** *Each object of* *provides a dynamical Yang-Baxter map.*

The proof of the following proposition is straightforward.

**Proposition 16.** *The ternary operations* (3.4) *and* (3.5) *satisfy* (4.1).

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

Because of the work in [32, Section 6] and the fact that the categories *and* are isomorphic, every homogeneous
pre-system *G* (Definition 7(1)) in the object 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 concerning
the left quasigroups with (5.1) and an essentially surjective functor (see [32, Proposition 6.17]). The
functors , in Section 2, and 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 . * L*_{1}, *L*_{2} = (*L*_{2}, ∗) be left quasigroups. We assume that the left
quasigroup *L*_{2} satisfies

(5.1)

Here the symbol is the left division (2.1) of *L*_{2}. Let π : *L*_{1} → *L*_{2} be a (set-theoretic) bijection. An object of is such a triple (*L*_{1}, *L*_{2}, π).

A morphism is a homomorphism of left quasigroups such that is also a homomorphism of left quasigroups.

**Proposition 17.** *is a category; the definitions of the identity, the source, the target and the composition are similar
to those of the category*

The next task is to construct a functor . We define the ternary system as sets and

(5.2)

**Proposition 18.**

*Proof*. It suffices to prove that is a homogeneous pre-system satisfying (4.1).

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

(5.3)

With the aid of (5.1), the right-hand side of (5.3) is

which is exactly . This is the desired conclusion.

Let be a morphism of the category . The map is a homomorphism of left quasigroups. Moreover, is a homomorphism of ternary systems from to, because is a homomorphism of left quasigroups. As a result, is a morphism of the category .

We set for and * J*(*f*) = *f* for a morphism *f* of .

**Proposition 19.** *is a functor.*

This functor *J* is essentially surjective. In fact, for any , we can construct a left quasigroup *L*_{2}
such that and *J*(*L*,*L*_{2}, *π*) = (*L*, *G*, *π*). We fix any element λ_{0} ∈ *G*. Set *L*_{2} = *G* as sets and

(5.4)

Due to (3.1) and (4.1), *L*_{2} is a left quasigroup; its left division is defined by

(5.5)

**Proposition 20.** .

*Proof*. We need only show (5.1). Let . With the aid of (5.4) and (5.5) we have

(5.6)

From (3.2) and (4.1),

By taking into account (4.1) again, the right-hand side of (5.6) is

which is exactly the right-hand side of (5.1) by virtue of (5.4) and (5.5). This proves the proposition.

It is immediate that *J*(*L*,*L*_{2}, *π*) = (*L*, *G*, *π*), and consequently, the following holds.

**Proposition 21.** *The functor is essentially surjective.*

**Corollary 22.** *The functor* *is essentially surjective.*

The final task of this section is to construct dynamical Yang-Baxter maps by means of the functor and quasigroups of reflection; see [17, Section 1].

**Definition 23.** A pair (*G*, ∗) of a nonempty set *G* and a binary operation (∗) on *G* is called a quasigroup of reflection
if and only if (*G*, ∗) is a left quasigroup (Definition 1(1)) satisfying

(5.7)

(5.8)

It follows from (5.7) that (*G*, ∗) is a quasigroup (Definition 1(2)).

**Remark 24.** (1) The above definition is slightly different from that in [17] (see also [26, II.1.1] and [27, Section 1]);
the order of the binary operation (∗) on *G* is reversed.

(2) The identity (5.8) is called a right distributive law [30, Section V.2].

(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 (5.8) immediately induces (5.1),

**Proposition 25.**.

The quasigroup *G* = (*G*, ∗) of reflection hence produces the dynamical Yang-Baxter map *R*(λ) defined by .

For example, let *L* = (*G*, ·) denote the left quasigroup (4.4) and *π* : *L*(= *G*) → *G* the identity map on *G*. The
above dynamical Yang-Baxter map *R*(λ) induced by is

(5.9)

For Nobusawa’s quasigroup *Q* = (*Q*, ∗), the corresponding dynamical Yang-Baxter map (5.9) is really dependent
on the parameter λ; in fact,

**Acknowledgments** The authors wish to thank organizers of the conference “Noncommutative Structures in Mathematics and Physics”
in July 2008, when this work started, for the invitation and hospitality.

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