alexa Dynamical Yang-Baxter Maps Associated with Homogeneous Pre-Systems | Open Access Journals
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
Make the best use of Scientific Research and information from our 700+ peer reviewed, Open Access Journals that operates with the help of 50,000+ Editorial Board Members and esteemed reviewers and 1000+ Scientific associations in Medical, Clinical, Pharmaceutical, Engineering, Technology and Management Fields.
Meet Inspiring Speakers and Experts at our 3000+ Global Conferenceseries Events with over 600+ Conferences, 1200+ Symposiums and 1200+ Workshops on
Medical, Pharma, Engineering, Science, Technology and Business

Dynamical Yang-Baxter Maps Associated with Homogeneous Pre-Systems

Noriaki Kamiya1 and Youichi Shibukawa2

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

2Department 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

Abstract

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.

MSC 2010

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

Introduction

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

Equation

Here Equation, and others are the maps from X × X × X to itself defined as follows: for u, v,wX,

Equation

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 : LM 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 kK.

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

Equation

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 × GG, 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 Equation concerning the ternary systems, which is equivalent to a category Equation 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 Equation, isomorphic to the category Equation, by means of the homogeneous pre-systems with (4.1). Because the category Equation is equivalent to the category Equation, each object of Equation 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 Equation concerning the left quasigroups satisfying (5.1) and an essentially surjective functor Equation 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,wL, there uniquely exists vL such that u·v = w (resp. v ·u = w). For the simplicity, one uses the notation uv instead of u·v (u, vL).

(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 ×MM.

By this definition, the left quasigroup L = (L, ·) has another binary operation \L called a left division [38, Section I.2.2]. For u,wL, we denote by u\Lw the unique element vL satisfying uv = w,

u\Lw = vuv = w. (2.1)

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.

Table

Each ternary system M = (M,μ) satisfying

Equation (2.2)

Equation (2.3)

for any a, b, c, dM, 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 : LM a (set-theoretic) bijection. For λ, uL, we define the maps Equation and Equation as follows: for vL,

Equation (2.4)

Equation (2.5)

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

Equation (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 Equation and Equation 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 Equation (cf. the category Equation2 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, bM, (2.7)

Equation (2.8)

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

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

Equation (2.9)

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

Equation

the composition gf is the usual one of the maps Equation

Proposition 4. Equation is a category.

The next task is to describe the category Equation, which is exactly the category Equation2 in [32, Section 6]. The object of this category Equation is a pair (L,R) of a left quasigroup L = (L, ·) and a dynamical Yang-Baxter map R(λ) : L×LL×L (λ ∈ L) satisfying

Equation

Here, (ηλ(v)(u), ξλ(u)(v)) := R(λ)(u, v) (λ, u, vL).

The morphism Equation of Equation is a homomorphismEquation of left quasigroups satisfying

Equation

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

We can construct functors Equation which establish an equivalence of the categories Equation and Equation (cf. [32, Proposition 6.15]): for (L,M, π) ∈ Equation, 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 Equation, write Equation denotes the triple (L, (M, μ), idL), where M = L as sets and Equation for a morphism f of Equation, set T(f) = f.

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

Theorem 6. Equation is an equivalence of categories.

Homogeneous pre-systems

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, yG, (3.1)

Equation (3.2)

for all x, y, u, v,wG.

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

Equation (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

Equation (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

Equation (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

Equation (4.1)

In fact, we present a category Equation concerning the homogeneous pre-systems with (4.1); this Equation is isomorphic to the category Equation in Section 2, and, on account of Theorem 6, every object of Equation 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 : LG a (set-theoretic) bijection. The object of Equation is a triple (L, G, π).

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

Equation

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

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

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

Equation (4.2)

Proposition 10.Equation

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),

Equation

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 Equation, the following proposition holds.

Proposition 11. Equation is a functor.

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

Equation (4.3)

Proposition 12.Equation

Proof. It suffices to prove that MG 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

Equation

From (3.1) and (3.2),

Equation

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

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

Proposition 13. Equation is a functor.

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

Theorem 14. The categories Equation and Equation are isomorphic.

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

Corollary 15. Each object of Equation 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

Equation (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

Equation

and if G is a homogeneous system (3.5), then

Equation

A relation between homogeneous pre-systems and left quasigroups

Because of the work in [32, Section 6] and the fact that the categories Equation and Equation are isomorphic, every homogeneous pre-system G (Definition 7(1)) in the object Equation 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 Equation concerning the left quasigroups with (5.1) and an essentially surjective functor Equation (see [32, Proposition 6.17]). The functors Equation, Equation in Section 2, and Equation 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 Equation. L1, L2 = (L2, ∗) be left quasigroups. We assume that the left quasigroup L2 satisfies

Equation (5.1)

Here the symbol Equation is the left division (2.1) of L2. Let π : L1L2 be a (set-theoretic) bijection. An object of Equation is such a triple (L1, L2, π).

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

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

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

Equation (5.2)

Proposition 18. Equation

Proof. It suffices to prove that Equation 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,wG(= L2). From (5.2) we have

Equation (5.3)

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

Equation

which is exactly Equation. This is the desired conclusion.

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

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

Proposition 19. Equation is a functor.

This functor J is essentially surjective. In fact, for any Equation, we can construct a left quasigroup L2 such that Equation and J(L,L2, π) = (L, G, π). We fix any element λ0G. Set L2 = G as sets and

Equation (5.4)

Due to (3.1) and (4.1), L2 is a left quasigroup; its left division is defined by

Equation (5.5)

Proposition 20. Equation.

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

Equation (5.6)

From (3.2) and (4.1),

Equation

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

Equation

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,L2, π) = (L, G, π), and consequently, the following holds.

Proposition 21. The functor Equation is essentially surjective.

Corollary 22. The functor Equation is essentially surjective.

The final task of this section is to construct dynamical Yang-Baxter maps by means of the functor Equation 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

Equation (5.7)

Equation (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.Equation.

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

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 Equation is

Equation (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,

Equation

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.

References

Select your language of interest to view the total content in your interested language
Post your comment

Share This Article

Relevant Topics

Recommended Conferences

Article Usage

  • Total views: 11898
  • [From(publication date):
    December-2011 - Sep 25, 2017]
  • Breakdown by view type
  • HTML page views : 8116
  • PDF downloads :3782
 

Post your comment

captcha   Reload  Can't read the image? click here to refresh

Peer Reviewed Journals
 
Make the best use of Scientific Research and information from our 700 + peer reviewed, Open Access Journals
International Conferences 2017-18
 
Meet Inspiring Speakers and Experts at our 3000+ Global Annual Meetings

Contact Us

Agri, Food, Aqua and Veterinary Science Journals

Dr. Krish

[email protected]

1-702-714-7001 Extn: 9040

Clinical and Biochemistry Journals

Datta A

[email protected]

1-702-714-7001Extn: 9037

Business & Management Journals

Ronald

[email protected]

1-702-714-7001Extn: 9042

Chemical Engineering and Chemistry Journals

Gabriel Shaw

[email protected]

1-702-714-7001 Extn: 9040

Earth & Environmental Sciences

Katie Wilson

[email protected]

1-702-714-7001Extn: 9042

Engineering Journals

James Franklin

[email protected]

1-702-714-7001Extn: 9042

General Science and Health care Journals

Andrea Jason

[email protected]

1-702-714-7001Extn: 9043

Genetics and Molecular Biology Journals

Anna Melissa

[email protected]

1-702-714-7001 Extn: 9006

Immunology & Microbiology Journals

David Gorantl

[email protected]

1-702-714-7001Extn: 9014

Informatics Journals

Stephanie Skinner

[email protected]

1-702-714-7001Extn: 9039

Material Sciences Journals

Rachle Green

[email protected]

1-702-714-7001Extn: 9039

Mathematics and Physics Journals

Jim Willison

[email protected]

1-702-714-7001 Extn: 9042

Medical Journals

Nimmi Anna

[email protected]

1-702-714-7001 Extn: 9038

Neuroscience & Psychology Journals

Nathan T

[email protected]

1-702-714-7001Extn: 9041

Pharmaceutical Sciences Journals

John Behannon

[email protected]

1-702-714-7001Extn: 9007

Social & Political Science Journals

Steve Harry

[email protected]

1-702-714-7001 Extn: 9042

 
© 2008-2017 OMICS International - Open Access Publisher. Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version
adwords