数理解析研究所講究録
第 1714 巻 2010 年 80-89

80

Dynamical braided monoids
anddynamical Yang-Baxter maps
北海道大学理学部数学 澁川 陽一 (Youichi Shibukawa)
Department of Mathematics, Faculty of Science,
Hokkaido University, Sapporo 060-0810, Japan

Abstract
By means of torsors (principal homogeneous spaces), we prove thatdynamical braided monoids can produce dynamical Yang-Baxter maps.

1

Introduction

Finding solutions to the quantum Yang-Baxter equation [1, 21] is essential in
the study of integrable systems [2, 8]. This quantum Yang-Baxter equation
is exactly the braid relation in a suitable tensor category; for example, theusual quantum Yang-Baxter equation is the braid relation in the tensor
category of vector spaces, and the quantum group [3, 7] is useful for theconstruction of solutions.
Lu, Yan, and Zhu [12] constructed Yang-Baxter maps [4, 20], solutions
to the braid relation in the tensor category of sets, by means of braided
groups [19]. Let and be groups whose unit elements are respectively
denoted byand , and let be a map from $S\cross B$ to $B\cross S$ .
$B$

$S$

$1_{S}$$1_{B}$$\sigma$

Definition 1.1. A triple $(S, B, \sigma)$ is a matched pair of groups [18], iff themap : $S\cross B\ni(s, b)\mapsto(sarrow b, sarrow b)\in B\cross S$ satisfies:$\sigma$

(1.1)
(1.2)

$sarrow(tarrow b)=(st)arrow b$ ;$(st)arrow b=(sarrow(tarrow b))(tarrow b)$ ;

(1.3)
(1.4)
(1.5)
(1.6)

$(sarrow b)arrow c=sarrow(bc)$ ;$sarrow(bc)=(sarrow b)((sarrow b)arrow c)$ ;$1_{S}arrow b=b$;$sarrow 1_{B}=s$$(\forall s, t\in S,\forall b, c\in B)$.

The Cartesian product $B\cross S$ is a group with the multiplication$(b, s)(c, t)=(b(sarrow c), (sarrow c)t)$$((b, s), (c, t)\in B\cross S)$.

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To be more precise, the unit element is, and the inverse of the$(s^{-1}arrowb^{-1},s^{-1}arrowb^{-1})$$(b,s)\inB\crossS$
elementis.
$(1_{B}, 1_{S})$Definition 1.2. A pairis a braided group, iff:$(G, \sigma)$of a group$G$and a map$\sigma$: $G\cross Garrow G\cross G$(1) $(G, G, \sigma)$ is a matched pair of groups;(2) if $(y’, x’)=\sigma(x, y)$ , then $y’x’=xy(x, y, x’, y’\in G)$ .

In [12], Lu, Yan, and Zhu showed
Theorem 1.3. If
tion.

$(G, \sigma)$

is a bmided group, then$\sigma$satisfies the braid rela.

$(\sigma\cross id_{G})\circ(id_{G}\cross\sigma)\circ(\sigma\cross id_{G})=(id_{G}\cross\sigma)\circ(\sigma\cross id_{G})\circ(id_{G}\cross\sigma)$

We can rephrase the definition of the matched pair of groups by using
category theory.
Letdenote the set
andforof one element. We write$B$
the multiplications of the groups and , respectively. We define the maps$\eta s:I_{Set}arrow S$ and $\eta_{B}:I_{Set}arrow B$ by$I_{Set}$$\{e\}$$m_{B}$$m_{S}$$S$$\eta_{S}(e)=1_{S};\eta_{B}(e)=1_{B}$.

The above equations $(1.1)-(1.6)$ are equivalent to:$(id_{B}\cross m_{S})\circ(\sigma\cross id_{S})\circ(id_{S}\cross\sigma)=\sigma\circ(ms\cross id_{B})$;$(m_{B}\cross id_{S})\circ(id_{B}\cross\sigma)\circ(\sigma\cross id_{B})=\sigma\circ(id_{S}\cross m_{B})$$(id_{B}\cross ms)\circ(\sigma\cross id_{S})\circ(\eta s\cross id_{B\cross S})=l_{B\cross S}$;$(m_{B}\cross id_{S})\circ(id_{B}\cross\sigma)\circ(id_{B\cross S}\cross\eta_{B})=r_{B\cross S}$

Here, the maps
are defined by$l_{B\cross S}$: $I_{Set}\cross B\cross Sarrow B\cross S$ and$l_{B\cross S}(e, b, s)=(b, s);r_{B\cross S}(b, s, e)=(b, s)$$r_{B\cross S}$

.

;

(1.7)
(1.8)
(1.9)
(1.10): $B\cross S\cross I_{Set}arrow B\cross S$$(I_{Set}=\{e\}, b\in B, s\in S)$.

It is natural to try to solve the braid relation in another tensor category
similarly.
The aim of this article is to make an analogy between the Yang-Baxter
maps and dynamical Yang-Baxter maps (Definition 2.1) [14], solutions tothe braid relation in a tensor category $Set_{H}[15]$ defined in the next section. We construct the dynamical Yang-Baxter maps by means of dynamical

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braided monoids in Definition 4.2. Torsors [9, 11], also known as the principal homogeneous spaces, are important in this construction.
The organization of this article is as follows.
In Section 2, we briefly sketch a tensor category Set . Section 3 explains monoids in $Set_{H}$ . After introducing dynamical braided monoids, ourmain results are stated and proved in Sections 4 and 5. The crucial fact isthat the dynamical braided monoid satisfying (3.1) is exactly a torsor (See$H$

Proposition 5.6).

2

Tensor category $Set_{H}$ and dynamical Yang-Baxter
mapsThis section explains the tensor category Set (cf. the tensor category りin [5, Section 3] , in which we will focus on the braid relation (For the tensor
category, see [10, Chapter XI] .
Let $H$ be a nonempty set. Set is a category whose object is a pair
: $H\cross X\ni(\lambda, x)\mapsto\lambda\cdot xx\in H$
of a nonempty set $X$ and a map
and whose morphism : $(X, \cdot x)arrow(Y, \cdot Y)$ is a map : $Harrow$ Map $(X, Y)$satisfying that$\mathcal{V}$

$H$

$)$

$)$

$H$

$(X, \cdot x)$$\cdot x$

$f$

$f$

$(\forall\lambda\in H,\forall x\in X)$$\lambda\cdot Yf(\lambda)(x)=\lambda\cdot xx$

.

(2.1)

.
instead of
To simplify notation, we will often use the symbol
objects
follows:for
The identity id and the composition are defined as$X,$and morphisms $f:Xarrow Y,$ $g:Yarrow Z$ ,$\lambda\cdot xx$$\lambda x$$0$$Y,$ $Z$$id_{X}(\lambda)(x)=x$$(\lambda\in H, x\in X);(gof)(\lambda)=g(\lambda)\circ f(\lambda)$$(\lambda\in H)$.

of the objects
is a tensor category: the tensor product$X=(X, \cdot x)$ and $Y=(Y, \cdot Y)$ is a pairof the Cartesian product$X\cross Y$ and the map . : $H\cross(X\cross Y)arrow H$ defined byThis Set$X\otimes Y$$H$$(X\cross Y, \cdot)$$\lambda\cdot(x, y)=(\lambda\cdot xx)\cdot Yy$$(\lambda\in H, (x,y)\in X\cross Y)$

;

(2.2)

the tensor product of the morphisms : $Xarrow X’$ and : $Yarrow Y’$ is definedby $(f\otimes g)(\lambda)(x, y)=(f(\lambda)(x),g(\lambda x)(y))(\lambda\in H, (x, y)\in X\cross Y)$.
The other ingredients of the tensor category Set are: the associativity
constraint $a_{XYZ}(\lambda)((x, y), z)=(x, (y, z))$ ; the unit $I=(\{e\}, \cdot I)$ , a pair of; the left andthe setof one element and the map defined by
the right unit constraints $l_{X}(\lambda)(e, x)=x=rx(\lambda)(x, e)$ .
In what follows, the associativity constraint will be omitted.
$f$

$g$

$H$

$\{e\}$$I$$\lambda_{I}e=\lambda$

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Definition 2.1. A morphism : $X\otimes Xarrow X\otimes X$ of Set is a dynamical
Yang-Baxter map [14, 15], iff satisfies the following braid relation in Set .
$\sigma$$H$$\sigma$$H$$(\sigma\otimes id_{X})\circ(id_{X}\otimes\sigma)\circ(\sigma\otimes id_{X})=(id_{X}\otimes\sigma)\circ(\sigma\otimes id_{X})\circ(id_{X}\otimes\sigma)$. $(2.3)$

Remark 2.2. (1) If $H$ is a set of one element, the tensor category Set isexactly the tensor category Set consisting of nonempty sets, and the$H$

dynamical Yang-Baxter map is a Yang-Baxter map.
(2) The dynamical Yang-Baxter maps satisfying suitable conditions can
produce bialgebroids, each of which gives birth to a tensor categoryof its dynamical representations [16]. Note that the definition of thetensor product in [16] is slightly different from that in this section.

3

Monoids in $Set_{H}$In this section, we introduce the monoid in $Set_{H}$ (See [13, VII.3]).
Let $X$ be an object of the tensor category $Set_{H}$ and let: $X\otimes Xarrow X$
and: $Iarrow X$ be morphisms of Set .
$m_{X}$$\eta x$$H$

Definition 3.1. The triple $(X, m_{X}, \eta_{X})$ is a monoid, iff:$m_{X^{\circ}}$$(m_{X}\otimes idx)=m_{X^{O}}(id_{X}\otimes m_{X})$$m_{X}\circ(\eta x\otimes id_{X})=l_{X}$$m_{X}\circ(id\otimes\eta_{X})=r_{X}$

;

;

.

We explain a construction of the monoid in $Set_{H}$ , which is due to Mitsuhiro Takeuchi. Let $X$ be an object of $Set_{H}$ . Suppose that$\forall\lambda,$

We will denote by$\lambda’\in H,$ $\text{ョ_{}1}x\in X$$\lambda\backslash \lambda’$such that$\lambda x=\lambda’$

.

(3.1)

the unique element $x\in X$ .

Proposition 3.2. $X$ satisfying (3.1) is a monoid, together with thephts $msm_{X}$ and:$morarrow$$\eta x$$m_{X}(\lambda)(x, y)=\lambda\backslash ((\lambda x)y);\eta_{X}(\lambda)(e)=\lambda\backslash \lambda$$(\lambda\in H, x, y\in X, I=\{e\})$.

Furthermore, this monoid structure is unique.

.
Proof. We give the proof only for the uniqueness of the morphism$X\otimes
Xarrow
X$
Suppose that:is a morphism of Set . It follows from
(2.1) and (2.2) that. By
$m_{X}$$m_{X}$$H$$\lambda m_{X}(\lambda)(x, y)=\lambda(x, y)=(\lambda x)y(\lambda\in H, x, y\in X)$

taking (3.1) into account, $m_{X}(\lambda)(x, y)$ is uniquely determined.

$\square$Example 3.3. The set $H$ with the map
is an object
of $Set_{H}$ , and obviously satisfies (3.1); hence, $H=(H, \cdot H)$ is a monoid.
$\lambda_{H}\lambda’=\lambda’(\lambda, \lambda’\in H)$

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4

Dynamical braided monoids

After introducing dynamical braided monoids in Set , we show in thissection that each dynamical braided monoid satisfying (3.1) gives birth tothe dynamical Yang-Baxter map.
Let $(X, mx, \eta_{X})$ be a monoid in the tensor category Set . Suppose thata morphism : $X\otimes Xarrow X\otimes X$ of Set satisfies:
$H$

$H$

$H$

$\sigma$

(4.1)
(4.2)
(4.3)
(4.4)

;
$(m_{X}\otimes id_{X})\circ(id_{X}\otimes\sigma)\circ(\sigma\otimes id_{X})=\sigma o(idx\otimes m_{X})$ ;;$(id_{X}\otimes m_{X})\circ(\sigma\otimes id_{X})\circ(idx\otimes\sigma)=\sigma o(m_{X}\otimes idx)$$(idx\otimes m_{X})\circ(\sigma\otimes id_{X})\circ(\eta x\otimes idx\otimes x)=l_{X\otimes X}$$(m_{X}\otimes id_{X})\circ(id_{X}\otimes\sigma)o(id_{X\otimes X}\otimes\eta_{X})=r_{X\otimes X}$

We define the morphisms$\eta x\otimes x:Iarrow X\otimes X$$m_{X\otimes X}$.

: $(X\otimes X)\otimes(X\otimes X)arrow X\otimes X$ andby:$m_{X\otimes X}=(m_{X}\otimes m_{X})\circ(id_{X}\otimes\sigma\otimes id_{X});\eta_{X\otimes X}=(\eta x\otimes\eta_{X})\circ l_{I}^{-1}$.

A straightforward computation shows
Proposition 4.1.
Definition 4.2.
satisfies $(4.1)-(4.4)$ .

$(X\otimes X, mx\otimes x, \eta_{X\otimes X})$$(X, \sigma)$

is a monoid.

is a dynamical braided monoid, iff the morphism$\sigma$

Remark 4.3. (1) By taking $(1.7)-(1.10)$ into account, the conditions $(4.1)-$(4.4) correspond to (1) in Definition 1.2, while (2) in Definition 1.2
corresponds to (2.1) for the morphism . If the monoid $X$ satisfies(3.1), then $mx(\lambda)(x, y)=\lambda\backslash ((\lambda x)y)(\lambda\in H, x, y\in X)$ because of
Proposition 3.2, and (2.1) for the morphism is equivalent to that$m_{X}o\sigma=m_{X}$ , which is similar to (2) in Definition 1.2.
$\sigma$$\sigma$(2) Let $(X, m_{X}, \eta_{X})$ and $(Y, m_{Y}, w)$ be a monoid in the tensor category$Set_{H}$ . Suppose that a morphism: $X\otimes Yarrow Y\otimes X$ of Set satisfies:$H$$\sigma$$(id_{Y}\otimes m_{X})o(\sigma\otimes id_{X})\circ(idx\otimes\sigma)=\sigma o(m_{X}\otimes id_{Y})$$(m_{Y}\otimes idx)\circ(id_{Y}\otimes\sigma)\circ(\sigma\otimes id_{Y})=\sigma\circ(id_{X}\otimes m_{Y})$$(id_{Y}\otimes m_{X})\circ(\sigma\otimes idx)\circ(\eta_{X}\otimes$$(m_{Y}\otimes idx)\circ(id_{Y}\otimes\sigma)\circ($

We define the morphisms$W\otimes x:Iarrow Y\otimes X$ by:$m_{Y\otimes X}$

idy$\otimes x)=l_{Y\otimes X}$

;

;

;

idy $\otimes X\otimes W)=r_{Y\otimes X}$ .

: $(Y\otimes X)\otimes(Y\otimes X)arrow Y\otimes X$ and$m_{Y\otimes X}=(m_{Y}\otimes m_{X})\circ(id_{Y}\otimes\sigma\otimes idx);W\otimes x=(\eta_{Y}\otimes\eta_{X})\circ l_{I}^{-1}$

.

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Then
is a monoid, which is called a matchedpair of monoids.
The following theorem is an analogue of Theorem 1.3.
$(Y\otimes X, m_{Y}\otimes x, \eta_{Y\otimes X})$

Theorem 4.4. If a dynamical bmided monoid
is a dynamical Yang-Baxter map (Definition 2.1).

$(X, \sigma)$satisfies (3.1), then$\sigma$

We give a proof of this theorem in the n’ext section.

5

Torsors (Principal homogeneous spaces)

This section is devoted to proving Theorem 4.4, in which the notion of a
torsor [11, Section 4.2] plays an essential role.
Definition 5.1. A pair $(M, \mu)$ of a nonempty set $M$ and a ternary operation: $M\cross M\cross Marrow M$ is called a torsor, iff satisfies:

$\mu$

$\mu$

$\mu(u, v, v)=u=\mu(v, v, u)$ ;(5.1)$\mu(\mu(u, v, w), x, y)=\mu(u, v, \mu(w, x, y))$$(\forall u, v, w, x, y\in M)$. $(5.2)$

Remark 5.2. (1) A Mal‘cev operation is a ternary operation satisfying(5.1) [9, Section 1]; moreover, an associative Mal‘cev operation is aternary operation satisfying (5.1) and (5.2). The torsor is also calleda herd, a Schar (in German), a flock, and a heap [17, Section 1].
(2) For a pair $(M, \mu)$ , the following conditions are equivalent (cf. [6, Section 2.1] :$)$(a) (5.1) and (5.2);(b) (5.1) and (5.3).
$\mu(\mu(u, v, w), x, y)=\mu(u, \mu(x, w, v), y)=\mu(u, v, \mu(w, x, y))$$(\forall u, v, w, x, y\in M)$

.

(5.3)

In fact, (5.1) and (5.2) induce (5.3), because$\mu(u, v, \mu(w, x, y))$$=$$\mu(u, v, \mu(w, x, \mu(\mu(x, w, v), \mu(x, w, v), y)))$$=$$\mu(u, v, \mu(\mu(w, x, \mu(x, w, v)), \mu(x, w, v), y))$$=$$\mu(u, v, \mu(v, \mu(x, w, v), y))$$=$$\mu(\mu(u, v, v), \mu(x, w, v), y)$$=$$\mu(u, \mu(x, w, v), y)$.

Thus, a pair $(M, \mu)$ satisfying (5.1) and (5.3) is exactly a torsor.

86(3) The torsor $(M, \mu)$ is a principal homogeneous space [11, Section 4.2].
Let $\mu(a, b)(a, b\in M)$ denote the map from $M$ to itself defined by$\mu(a, b)(c)=\mu(a, b, c)(c\in M)$ . The set $G=\{\mu(a,b);a, b\in M\}$ isa subgroup of Aut $(M)$ , which makes $M$ a G-principal homogeneous
space. Conversely, the principal homogeneous space gives birth to a
torsor.
on
Each group $G$ produces a torsor. Define the ternary operationby$\mu c(a, b, c)=ab^{-1}c$$(a, b, c\in G)$ .
(5.4)$\mu c$$G$The pair $(G, \mu)$ is a torsor.
Remark 5.3. Every torsor $(M, \mu)$ is isomorphic to (5.4) [17, Section 1.6]. We
first fix any element $e\in M$ . The nonempty set $M$ , together with the binary
operation$M\cross M\ni(a, b)\mapsto\mu(a, e, b)\in M$,

is a group [9, Section 1]; in fact, the unit element is , and the inverse of theelement is $\mu(e, a, e)$ . This group $M$ gives birth to the torsor (5.4), which
$e$

$a$

is isomorphic to $(M, \mu)$ .

Let $H=(H, \cdot H)$ denote the object of the category $Set_{H}$ in Example 3.3.
Here,
. Suppose that an object $X$ of Set satisfies(3.1). We define the map $i:Harrow$ Map $(H, X)$ by$\lambda\cdot H\lambda’=\lambda’(\lambda, \lambda’\in H)$$H$$i(\lambda)(u)=\lambda\backslash u$

Proposition 5.4. The map$i$$(\lambda, u\in H)$.

is an isomo rphism of Set

$H$

from

$H$

to $X$ .

In fact, its inverse is as follows.
$(\lambda\in H, x\in X)$$i^{-1}(\lambda)(x)=\lambda x$.

: $X\otimes Xarrow X\otimes X$ be a morphism of Set . By virtue of (2.1) for: $H\otimes Harrow H\otimes H$ ,the morphism
Let

$H$

$\sigma$$i^{-1}\otimes i^{-1}o\sigma oi\otimes i$

Proposition 5.5. The second component of $(i^{-1}\otimes i^{-1}o\sigma oi\otimes i)(\lambda)(u, v)$$(\lambda,u, v\in H)$ is.
$v$

We define the ternary operation on the set $H$ by the first component; that is,
of
$\mu$

$(i^{-1}\otimes i^{-1}o\sigma oi\otimes i)(\lambda)(u, v)$$(i^{-1}\otimes i^{-1}o\sigma oi\otimes i)(\lambda)(u, v)=(\mu(\lambda, u, v), v)$$(\lambda, u, v\in H)$.

87
Proposition 5.6. $(H, \mu)$ is a torsor, if and only if
bmided monoid.

$(X, \sigma)$

is a dynamical

Proof. We first observe (4.1) is equivalent to that$\mu(u, v, \mu(v, w, x))=\mu(u, w, x)$.

$(\forall u, v, w, x\in H)$

On account of Proposition 5.4, the morphism$\sigma$(5.5)

satisfies (4.1), if and only if$(id_{H}\otimes(i^{-1}om_{X}oi\otimes i))\circ((i^{-1}\otimes i^{-1}o\sigma oi\otimes i)\otimes id_{H})$$\circ(id_{H}\otimes(i^{-1}\otimes i^{-1}o\sigma oi\otimes i))$$=(i^{-1}\otimes i^{-1}o\sigma oi\otimes i)\circ((i^{-1}om_{X}oi\otimes i)\otimes id_{H})$

.

(5.6)

Because $(i^{-1}om_{X}oi\otimes i)(\lambda)(u, v)=v(\lambda, u, v\in H),$ $(5.5)$ and (5.6) areequivalent.

Similar argument implies to: (4.2) is equivalent to that$\mu(\mu(u, v, w), w, x)=\mu(u, v, x)$$(\forall u, v, w, x\in H)$

;

(5.7)

(4.3) is equivalent to that $\mu(v, v, u)=u(\forall u, v\in H)$ ; and (4.4) is equivalent
to that $\mu(u, v, v)=u(\forall u, v\in H)$ .
An easy computation shows that (5.2) is equivalent to (5.5) and (5.7), if
satisfies (5.1); in fact, (5.5) and (5.7) induce (5.2), because$\mu$$\mu(\mu(u, v, w), x, y)=\mu(\mu(u, v, w), w, \mu(w, x, y))=\mu(u, v, \mu(w, x, y))$Hence, $(H, \mu)$ is a torsor, if and only if
monoid.

Proof of Theorem 4.4. Let$(X, \sigma)$.

is a dynamical braided$\square$be a dynamical braided monoid satisfying(3.1). From (3.1) and Proposition 5.6, $(H, \mu)$ is a torsor. If $(H, \mu)$ is a torsor,
then the morphism $(i^{-1}\otimes i^{-1})\circ\sigma o(i\otimes i):H\otimes Harrow H\otimes H$ satisfies thebraid relation (2.3), and so does the morphism . Thus, is a dynamical
Yang-Baxter map (Definition 2.1).
$(X, \sigma)$$\sigma$$\sigma$$\square$Acknowledgment$s$

The author wishes to express his thanks to the organizers of the Conference on Quantum Groups and Quantum Topology for the invitation andhospitality.

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