Actions of Finite Hypergroups and Applicationsto Extension Problem

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Actions of Finite Hypergroups and Applications to Extension Problem

著者 KAWAKAMI Satoshi, MIKAMI Itsumi, TSURII Tatsuya, YAMANAKA Satoe

journal or

publication title

奈良教育大学紀要. 自然科学

volume 60

number 2

page range 19‑28

year 2011‑11‑30

URL http://hdl.handle.net/10105/8156

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19

Key Words：action, hypergroup, extension

1．Introduction

In the present paper we investigate irreducible actions α of a fi nite hypergroup K on a fi nite set X and the invariant measure on X under the action α. Moreover we make clear the relationship between irreducible actions and extension problem. We establish the method how to determine all extension hypergroups by applying these irreducible actions of K and the invariant measure on X obtained here.

In 1998, Sunder-Wildberger [15] studied irreducible actions of fi nite commutative hypergroups and they succeeded to determine all irreducible *-actions of hypergroups Zq(2) (0 < q ≤ 1) and all irreducible *-actions of certain hypergroups of order three such as the Golden hypergroup G, the conjugacy class hypergroup K(S3) of the symmetric group S3 of order three, and the character hypergroup K( ˆ S3) of S3. Applying their results, we determine all irreducible actions of Zq(2) and all two dimensional irreducible actions of G, K(S3) and K( ˆ S3) including irreducible *-actions obtained in [15] and we characterize the *-action among irreducible actions obtained here.

On the other hand we have continued to study extension problem in the category of commutative hypergroups in [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] and [14]. Let K, H and L be finite commutative hypergroups. If H is a subhypergroup of K and there exists a homomorphism ϕ from K onto L such that Kerϕ＝H, then K is called an extension hypergroup of L by H. Extension problem is to determine all extension hypergroups of L by H when H and L are given.

This problem is important to understand the structures of hypergroups. Let L＝{0, 1,…,p} be a finite commutative hypergroup with unit 0. We denote ϕ⁻¹(j) by S(j) for j ∈ L. Then K is decomposed as K＝ ∪jp

＝0S(j). The regular action ρ^K of K on K is irreducible but the restriction ρH^K to the subhypergroup H of the action ρ^K is not an irreducible action on K. However the action ρH^K of H acts on each S(j) irreducibly which we denote by ρj . The irreducible action ρj

of H on S(j) is determined by the unique invariant probability measure v_j on S(j) which gives the parameter of the action ρj . Our developed method how to apply irreducible actions to solve extension problem is as follows.

(1) The irreducible action ρj gives the convolution product δh◦^δ^s for h ∈ H and s ∈ S(j).

(2) The invariant probability measure vj gives each weight w(s) for s ∈ S(j).

Actions of Finite Hypergroups and Applications to Extension Problem

Satoshi KAWAKAMI

(Department of Mathematics, Nara University of Education, Nara 630-8528, Japan)

Itsumi MIKAMI

(Completed Master Course, Graduate School of Education, Nara University of Education)

Tatsuya TSURII

(Graduate student of Master Course, Graduate School of Education, Nara University of Education)

Satoe YAMANAKA

(Graduate student of Doctor Course, Graduate School of Science, Osaka Prefecture University) (Received May 6, 2011)

Abstract

The purpose of the present paper is to determine all irreducible actions of hypergroups of order two and all two dimensional irreducible actions of certain hypergroups of order three. Moreover it is shown that these irreducible actions play a principal role on giving an answer to extension problem in the category of fi nite commutative hypergroups. (AMS Subject Classifi cation : 43A62, 20N20)

奈良教育大学紀要第60巻第 2 号（自然）平成23年 Bull. Nara Univ. Educ., Vol. 60, No. 2 (Nat. ), 2011

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(3) The other structure comes from the conditions of the commutativity of the regular action ρ^K of K.

According to this new method, we give some typical examples. Example 4.1 (the case that H＝Zq(2), L＝Z2) agrees with the paper [8]. Example 4.2 (the case that H＝Zq(2), L＝Zp(2)) agrees with the paper [6]. Example 4.3 (the case that H ＝G, L＝Z2) agrees with the paper [11]. However we would like to stress that Example 4.4 (the case that H＝K(S3), L＝

Z2), 4.5 (the case that H＝K( ˆ S3), L＝Z2) and 4.6 (the case that H＝Z₋¹

2 (2), L＝Z₋¹

2 (2) and │K│＝5) are new results.

2．Preliminaries

In this section we recapitulate some fundamental notions and facts of a finite hypergroup and an action of a hypergroup, referring to Bloom-Heyerʼs book [1] and Wildbergerʼs paper [16].

For a finite set X＝{x1, x2,…, xn }, we denote by M^b(X) and M¹(X) the set of all measures on X and the set of non- negative probability measures on X respectively, namely

M^b(X) :＝{a1δx1＋a2δx2＋ … ＋anδxn : a1, a2, … , an ∈ C}, M¹(X) :＝{a¹δx1＋a²δx2＋ … ＋anδxn : aj ≥ 0 (j＝1, 2, … , n),

Σ

n

j＝1

aj＝1},

where the symbol δx stands for the Dirac measure at x ∈ X. For μ＝a1δx1+a2δx2+…+anδxn∈ M^b(X), the support of μ is given by

supp(μ) :＝ {xj ∈ X : aj ≠ 0}.

Defi nition. A fi nite hypergroup K consists of a fi nite set K＝{c⁰, c¹, … , cm} together with an associative product (called convolution) ◦ and an involution * in M^b(K) satisfying the following conditions.

(1) The space M^b(K) admits a convolution ◦ and an involution * such that (M^b(K),◦, *) is an associative *-algebra with unit δc0 .

(2) For ci, cj ∈ K, the convolution δci◦^δ^cj belongs to M¹(K).

(3) There exists an involutive bijection ci c^*i on K such that δ^c*_i＝δ^c^*_i and cj＝c^*i if and only if c⁰ ∈ supp(δci◦^δ^cj) for all ci, cj ∈ K.

A fi nite hypergroup K is said to be commutative if the convolution ◦^{in M}^b(K) is commutative, and hermitian if the involution * is the identity mapping.

The weight w(ci) of an element ci ∈ K is defi ned by w(ci) :＝(n⁰_i)⁻¹ where δ^*c_i◦^δ^ci＝n⁰_iδc0 + n¹_iδc1＋…＋ n^m_iδcm. Here we note that n⁰_i > 0 by the above conditions (2) and (3). The total weight w(K) of K is defi ned by w(K) :＝Σ^m_i＝0w(ci). The normalized Haar measure eK of K is given by

eK＝

Σ

m

i＝0

w(ci) w(K)δc_i .

A complex-valued function χ on K is called a character if χ is a function on K which is extendable linearly on M^b(K) by χ(δci)＝χ(ci) and satisfying χ(δc0)＝1, χ(δc◦i ^δ^cj)＝χ(δci)χ(δcj) and χ(δ^c*_i )＝ χ(δci) for all ci, cj ∈ K. The set of all

characters of K is denoted by ˆ K . We note that ˆ K is not necessarily a hypergroup in general even if K is commutative.

3．Actions of hypergroups

In this section we study irreducible actions of fi nite hypergroups. The notion of actions of hypergroups is investigated in Sunder-Wildbergerʼs paper [15]. We give a reformulated version of the defi nition of actions of hypergroups in [15].

For a fi nite set X＝{x1, x2, … , xn}, B(M^b(X)) denotes the algebra of all linear transformations on the linear space M^b(X) over C.

Defi nition. We call an action α of a fi nite hypergroup K on a set X if α satisfi es the following conditions.

(1) α is homomorphism from M^b(K) to B(M^b(X)) as algebra such that α(δc0) is the identity mapping on M^b(X).

(2) For ci ∈ K andμ∈ M¹(X), α(δc_i)μ∈ M¹(X).

We often denote α(δci) by α(ci). A subset S of X is called invariant under the action α of K if supp(α(ci)δx) ⊂ S for any

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ci ∈ K and x ∈ S.

Definition. An action α of a finite hypergroup K on X is called irreducible if a non-empty subset S of X which is invariant under the action α must be X.

When we take X＝K and ρ^K(ci)δcj＝δci◦^δ^c^j^{for c}ⁱ^{, c}^j ∈ K, we get an action ρ^K. We call this action ρ^K the (left) regular action of K. It is easy to check that the regular action ρ^K of K is irreducible.

Proposition 3.1. An irreducible action α of a fi nite hypergroup K on X has the unique invariant probability measure on X.

Proof. Since supp(μ) of an invariant measureμis invariant in X, we see that supp(μ)＝X ifμ≠ 0 by irreducibility of the action α of K on X. For the normalized Haar measure eK of K and x ∈ X, it is easy to see that α(eK)δx is an invariant probability measure on X which means that there exists an invariant probability measure on X. Let μ¹ and μ² be two invariant measures on X and written by

μ1＝t1δx1＋t2δx2＋…＋ tnδxn , μ2＝s1δx1＋s2δx2＋…＋ snδxn. Put μ＝s1μ1−t1μ2. Then the measureμis also invariant under the action α and

μ＝0・δx1＋(s1t2−t1s2)δx2＋…＋ (s1tn−t1sn)δxn.

This equality means that supp(μ) ∈ ／ x1, namely, supp(μ) ≠ X. Hence we see that supp(μ)＝φ which implies thatμ＝ 0.

Therefore we obtain

μ1＝cμ2, where c＝ , s1 ≠ 0.

Then we get the desired conclusion. [Q.E.D.]

Corollary 3.2. ([15]) Let α be an irreducible action of K on X and eK the normalized Haar measure of K. Then α(eK) is a rank one projection.

Proof. Since α(eK)²＝α(eK)α(eK)＝α(eKeK)＝α(eK), α(eK) is a projection. Take μ ∈ Im(α(eK)). Then there exists v ∈ M^b(X) such thatμ＝ α(eK)v which implies thatμis an invariant measure on X. By Proposition 3.1,μ＝cμ⁰ for the invariant probability measure μ0 on X. Then we see that the rank of α(eK)＝the dimension of Im(α(eK)) is one. [Q.E.D.]

Remark 3.3. When the invariant probability measureμon X is written by μ＝ t1δx1＋t2δx2＋…＋tnδxn , the representing matrix of α(eK) associated with the basis δx1, δx2 , …, δxn is

α(eK)＝

by the fact that α(eK)δx1＝μ, α(eK)δx2＝μ,…, α(eK)δxn＝μ.

An action α of a fi nite commutative hypergroup K on X is called a *-action by Sunder-Wildberger [15] if α(c^*i)＝α(ci)^* as a matrix associated with the linear basis δx1 , δx2 ,…, δxn in M^b(X). They studied irreducible *-actions of such hypergroup K that K is a hypergroup Zq(2) (0 < q ≤ 1) of order two, the Golden hypergroup G, the conjugacy class hypergroup K(S3) of the symmetric group S3 of order three and the character hypergroup K( ˆ S3) of S3.

In the present paper we report to succeed to determine all irreducible actions of hypergroups Zq(2) (0 < q ≤ 1) and all two dimensional irreducible actions of hypergroups K＝G, K(S3) and K( ˆ S3).

Defi nition. An action α of a fi nite hypergroup K on X is called to be equivalent to an action β of K on Y if there exists a bijection ψ from X to Y such that

β(δcj)＝ψ_*◦^α(^δ^cj)◦^ψ−*1

for all cj ∈ K where ψ_* is a linear isomorphism from M^b(X) to M^b(Y) given by ψ_*(δx)＝δψ(x) for x ∈ X.

t1

s1

Actions of Finite Hypergroups and Applications to Extension Problem

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Example 3.1. Let K＝{c0, c1} be a hypergroup of order two with unit c0 where the structure is characterized by a parameter q (0 < q ≤ 1) as follows.

δc1◦^δ^c¹^＝q^δ^c⁰^＋(1−q)^δ^c¹^.

We often denote this hypergroup K by Zq(2). The total weight w(K) of K is w(K)＝ and the normalized Haar measure eK of K is given by

Let α be an irreducible action of K on X＝{x1, x2,…, xn} andμthe unique invariant probability measure on X which is written by

μ＝ t1δx1＋t2δx2＋…＋tnδxn, where tj > 0 (j＝1, 2,…, n) and t1＋t2＋…＋tn＝1. For t＝(t1, t2,…, tn) we get

α^t(c¹)＝(1＋q)α^t(eK) −qα^t(c⁰), where

α^t

1 1 1

2 2 2

1 0 0

0

0 1 α^t 0

as the representing matrices associated with the basis δx1 , δx2 ,…, δx_n in M^b(X). Hence we obtain

α^t 1

1 1 1

1 1

1

1 1

2 2

1 2

*

with a parameter t＝(t1, t2,…,tn). Let Sn be the symmetric group of order n. Forσ∈ Sn and t＝(t1, t2,…,tn), we denote (tσ(1), tσ(2),…, tσ(n)) by σ(t). Then we have the following proposition for irreducible actions of Zq(2).

Proposition 3.4. (1) When 2 ≤ n ≤ , ≤ tj ≤ (j＝1, 2,…, n) and t1＋t2＋…＋tn＝1, α given by (*) with the ^t parameter t＝(t1, t2,…, tn) is an irreducible action of Zq(2).

(2) All irreducible actions of Zq(2) are obtained in this way.

(3) For two irreducible actions α^t and α^t′ of Zq(2), α^t is equivalent to α^t′ if and only if there exists σ ∈ Sn such that t＝

σ(t^′).

(4) An irreducible action α^t of Zq(2) with t＝(t1, t2,…,tn) is a *-action if and only if t1＝t2＝…＝tn＝ ─n ¹.

Example 3.2. Let K＝{c0, c1, c2} be a hypergroup of order three with unit c0 and ˆ K＝{χ0, χ1, χ2} where χ0(c)＝1 for c ∈ K. Let α be an irreducible two dimensional action of K on X＝{x1, x2} andμ＝ tδx1＋(1−t)δx2 (0 < t < 1) be the unique invariant probability measure on X.

Lemma. Under the above situation there exists a measure v on X such that α(c)v＝χ(c)v for someχ∈ ˆ K whereχ ≠ χ0. Moreover, α(eK)v＝0 for the normalized Haar measure eK of K.

Proof. We may assume that there exists an eigen vector v ∈ M^b(X) with an eigen valueλ(c1) ≠ 1 such that α(c1)v ＝ λ(c1)v for c1 ∈ K by irreducibility of the action α of K on X. Then we see that

α(eK)v＝α(eK)α(c1)v＝λ(c1)α(eK)v.

The fact λ(c1) ≠ 1 implies that α(eK)v＝0. Since α(eK) is a linear combination of α(c0), α(c1) and α(c2), we obtain α(c2)v＝ λ(c²)v for someλ(c²) ∈ C.

By the fact that α(cicj)＝α(ci)α(cj), we see that λ(cicj)＝ λ(ci)λ(cj). Hence λ(c)＝χ(c) for someχ∈ ˆ K such that χ ≠ χ0. [Q.E.D.]

The representing matrices of α(eK), α(c0), α(c1) and α(c2) associated with eigen vectorsμand v on M^b(X) are

1＋q q

eK = q

1＋qδc0 + 1 1＋qδc1.

1＋q

q q

1＋q 1

1＋q

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23

where λ1＝χ(c1) and λ2＝χ(c2).

The representing matrix E(t) of α(eK) associated with δx1 and δx2 is

Take a matrix T(t) which satisfi es that

For example, we take and put

1 ₁

0 0 Then we have

We note that A(t,λ) does not depend on the choice of T(t). Hence we obtain irreducible actions α^t1 and α₂^r of K on X＝ {x¹, x²} whose representing matrices associated with δx1 and δx2 are respectively

(1) α1^t (c1)＝A(t,χ1(c1)) and α^t1 (c2)＝A(t,χ1(c2)), (2) α2^r (c1)＝A(r,χ2(c1)) and α^r2 (c2)＝A(r,χ2(c2)),

where all entries of matrices are non-negative real numbers. Moreover it is easy to check that α^t₁ and α₂^r is never mutually equivalent and that α^t₁ (resp. α₂^r ) is equivalent to α₁^t′ (resp. α^r′₂) if and only if t′＝ t or t′＝ 1−t (resp. r′＝ r or r′

＝ 1−r).

Example 3.2.1. The case that K is the Golden hypergroup G＝{c0, c1, c2} with unit c0. The structure equations are given by

1 2

1 1 0 2 2 2 0 1 1 2 1 2

and ˆ K＝{χ0,χ1,χ2}. Since χ1(c1)＝a＝ , χ1(c2) ＝b＝ , χ2(c1) ＝b and χ2(c2)＝a, we have

1 1

2

α

α1

1 1 1

1

1 1 1

1

1 and

1 1

1 1 1

1 1 1 1

1 1 1

1

2

α2

where ≤ t ≤ and ≤ r ≤ .

Remark. α₁^t (resp. α^r₂ ) is a *-action if and only if t＝ (resp. r＝ ). Moreover, α^t1 (resp. α^r₂ ) is equivalent to α₁^t′ (resp. α^r′₂ ) if and only if t′＝ t or t′＝1−t (resp. r′＝r or r′＝1−r).

Example 3.2.2. The case that K is the conjugacy class hypergroup K(S3)＝{c0,c1,c2} of S3 with unit c0. The structure α

α

0

2 1

1 0 1 0

0 1 0 0

1 1

0 0 0 0

1 1

1 0 1

0 0 1

1 1

1

1 1

1 1 1

−1＋√5

4 −1−√5

4

−b

1−b 1

1−b −b

1−b 1

1−b

1 2

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equations are given by

1 2 2

0 2 2 2 0 1

1 1

1 3

2 3

1 2

and ˆ K＝{χ0,χ1,χ2}. Since χ1 (c1)＝−1, χ1 (c2)＝1, χ2 (c1)＝0 and χ2 (c2)＝− , we have 0 1

1 0

1 0 0 1

2 1 1

1 α

α and

1 1

α2 α2 2

1 2

3 2

3 2 3

2 3

2 3 2

where ≤ r ≤ .

Remark. α1 is a *-action. α2^r is a *-action if and only if r＝ . Moreover, α^r2 is equivalent to α^r′2 if and only if r′＝r or r′＝1−r.

Example 3.2.3. The case that K is the character hypergroup K( ˆ S3) = {c0,c1,c2} of S3 with unit c0. The structure equations are given by

and ˆ K = {χ0, χ1, χ2}. Since χ1(c1)＝−1, χ1(c2)＝0, χ2(c1)＝1 and χ2(c2)＝− , we have

and

1 1

α2 α2 2

1 2

3 2

3 2 3

2 3

2 3 2

1 0 0 1 where ≤ r ≤ .

Remark. α₁ is a *-action. α₂^r is a *-action if and only if r＝ . Moreover, α^r₂ is equivalent to α^r′₂ if and only if r′＝r or r′＝1−r.

4．Applications to extension problem

Let K, H and L be fi nite commutative hypergroups. If H is imbedded in K as a subhypergroup of K and there exists a homomorphism ϕ from K onto L such that Kerϕ＝H, then K is called an extension hypergroup of L by H.

Extension problem：When two hypergroups H and L are given, determine all extension hypergroups K of L by H.

We have solved extension problem for certain concrete cases as in [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]

and [14] in the category of commutative hypergroups by using associativity of convolutions. Our strategy to solve the extension problem in this section is to apply irreducible actions which we already determined. Next proposition plays an essential role on our strategy.

Let L＝{⁰, ¹,…,p} be a fi nite commutative hypergroup with unit ⁰ and set S(j)＝ϕ⁻¹(j) for j∈ L. Then K is decomposed as K＝∪jp

＝0 S(j) where S(0)＝H.

Proposition 4.1. Let ρ^K be the regular action of K and ρ^KH the action of H which is the restriction of ρ^K to H. Then ρ^KH

is decomposed as actions of H by

where ρj is an irreducible action of H on S(j) and ρ0 is the regular action ρ^H of H.

Remark. Let vj be the invariant probability measure on S(j)＝{s0, s1,…,sk}, written by v_j＝t0δs0＋t1δs1＋ … ＋tkδsk. Then we note that the weight w(si) is given by w(si)＝tiw(S(j))＝tiw(j)w(H).

1 2

1 3

2 3

1 2

1 1 0 2 2 0 1 2 1 2 2

1 4

1 2

0 1

1 0 ¹ ²

1

1 α

α

1 2

1 2 1 2

1 2

1 3

2 3

1 2

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25

Example 4.1. ([8]) The case that H＝Zq(2)＝{c0, c1}, L＝Z2＝{0, 1}, K＝H ∪ S(1)＝{c0, c1, s0, s1}.

Case1. The case that K is hermitian, namely s^*⁰＝s⁰, s^*¹＝s¹ and the irreducible action α^t of Zq(2) on S(¹) is given by

α^t 1

1 1

where ≤ t ≤ and the invariant probability measure v on S(1) under the action α^t is v＝tδs0＋(1−t)δs1.

Since w(S(¹))＝ , we have w(s⁰)＝ and w(s¹)＝ . Therefore we obtain the structure equations:

0 0

1 1

0 1

1

0

1

1 1

1 1 1

Case2. The case that K is not hermitian, namely s0^*＝s1, s^*1＝s0. In this case it is easy to see that w(s0)＝w(s1). Hence t ＝ . Therefore we obtain the structure equations:

0 0 1 1 1 0 1

2 1

1

0 1 1

Example 4.2. ([6]) The case that H＝Zq(2)＝{c0, c1}, L＝Zp(2)＝{0, 1}, K＝H∪S(1)＝{c0, c1, s0, s1} is hermitian. By similar arguments in Example 4.1, we have the irreducible action α^t of Zq(2) on S(1) with ≤ t ≤ , w(s0)＝ and w(s1)＝ .We put

0 0

1 1

0 1

0

1

0

1

1 1

1

1 1

1

0 1

0

0 1

1

Then the regular action ρ^K(s0) and ρ^K(s1) are given by

0

1

0 0

0

0 0

0

1 1 1

1 1

1

1 0 0 1 1

0

1

0

1 1

1 0

1 1

1

0

1 1 1

0 0 0 0 1

One can determine the structure by applying the commutativity condition ρ^K(s⁰)ρ^K(s¹)＝ρ^K(s¹)ρ^K(s⁰) as follows.

0 0

1 1

0 1

1 1

1 1 1

1

1 1

1 1 1

1 1

1 ₀ ₁

0

1

0 1

0

1

1 0

where ( ≤ t ≤ , 0 ≤ r ≤ ) or ( ≤ t ≤ , ≤ r ≤ 1 ).

Remark. The case that K is not hermitian, namely s^*0＝s1, s1^*＝s0. In this case it is easy to see that w(s0)＝w(s1) and t＝ . Therefore we obtain the structure equations:

q

1＋q 1

1＋q

q (1＋q) t

q (1＋q) (1−t)

q

1 2

q

1＋q 1

1＋q (1＋q) t

pq (1＋q)(1−t)

pq

q

1＋q 1

2 t

1−t 1

2 q

1＋q 2t−1 t

1 2

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0

1

0 1

1

0 0 1

1 0

0 1

1 2

1 2 1 1

2 2

1 1

2

1 2

1

0 1

Example 4.3. ([11]) The case that H＝G＝{c⁰, c¹, c²}, L＝Z²＝{⁰, ¹}, K＝H∪ S(¹)＝{c⁰, c¹, c², s⁰, s¹}.

Case1. By Example 3.2.1 an irreducible action α^t1 is given by 1

1 1

1 1 1

1 1

α

1 2

α where ≤ t ≤ and a＝ , b＝ .

Case1-1. The case that K is hermitian, namely s0^* ＝s0, s^*1＝s1. We obtain the structure equations:

1 1 1 5 2 5 1 5

2

5 1 1 2

5 1 1

1 2 5 1 5 1

2 5 2 1

1

0 0

1 1

0 1

0

1

2

2 2

Case1-2. The case that K is not hermitian, namely s^*0＝s1, s1^*＝s0. In this case it is easy to see that w(s0)＝w(s1) and t＝ . Therefore we obtain the structure equations:

0

1

2

2 5

2 1 5 2 1 5 2 1

5 2 5

0 1

0 0

1

1 1

Case2. An irreducible action α^r2 is also given by

1

1 1

1 1 1

2 1

α

2 2

α

where ≤ r ≤ and a＝ , b＝ .

Case2-1. The case that K is hermitian, namely s0^*＝s0, s^*1＝s1. We obtain the structure equations:

1

5 1 1 1

1 1 1

1

1 1

2 5

2 5 5

2 5 2

5

2 1 5

1 1

2 5

0 0

1 1

0 1

0

1

2

Case2-2. The case that K is not hermitian. In this case it is easy to see that w(s0)＝w(s1) and r＝ . Therefore we obtain the structure equations:

2 1 5 2 5 1 2

5 2

5 1

2 5 1

0 0

0 1

1 1

0 1

1 2

2

Example 4.4. The case that H＝K(S3)＝{c0, c1, c2}, L＝Z2, K＝H∪S＝{c0, c1, c2, s0, s1}.

Case1. By Example 3.2.2 an irreducible action α1 is given by

−b

1−b 1

1−b

−1＋√5 4

−1−√5 4

1 2

−b

1−b 1

1−b −1＋√5

4 −1−√5

4

1 2

(10)

27

1 1

α 0 1 α1 2

0 1

1 0 Case1-1. The case that K is hermitian. We obtain the structure equations:

1 3 2 3 2

3 1 3

0 0 1 1 0 2 0 1 1 2

Case1-2. The case that K is not hermitian. We obtain the structure equations:

2 3 1 3 1

3 2

0 0 1 1 3 1 2 0 1 0 2

Case2. An irreducible action α^r2 is also given by

1

1 1 2

1

1 2

3 2

3 2 3

2 3

2 3 2

α2 α₂

where ≤ r ≤ .

Case2-1. The case that K is hermitian. We obtain the structure equations:

1 6

1 6 1

1 6 1 3

2 3

1 3

2 1

3 2 3 1

3

6 1

0 0

1 1

0 1

0

1 2

1

2

Case2-2. The case that K is not hermitian. In this case it is easy to see that w(s0)＝w(s1) and r＝ . Therefore we obtain the structure equations:

1 3

1 3 2

3 ² ⁰ ¹ ⁰ ¹ ²

1 1 1 0 0

Example 4.5. The case that H＝K( ˆ S3)＝{c0, c1, c2}, L＝Z2＝{0, 1}, K＝H∪ S＝{c0, c1, c2, s0, s1}.

Case1. By Example 3.2.3 an irreducible action α1 is given by 1 0

0

α 1 1 1 α 1 2

1 2 1 2

Case1-1. The case that K is hermitian. We obtain the structure equations:

0 0 1 1 0 2 0 1 1 2

1 3

2 3

1 3 Case1-2. The case that K is not hermitian. We obtain the structure equations:

2 3

2 3 1

3

1

3 ⁰ ²

2 1

0 0 1 1 0 1

Case2. An irreducible action α^r2 is also given by

1 2

3 2

3 2 3

2 3

2 1

1 0

0

α2 1 1 α₂ 2

where ≤ r ≤ .

Case2-1. The case that K is hermitian. We obtain the structure equations:

0 0

1 1

0 1

1

1 1 1

6

6 1 3 1 2 1

1 1

1 3

1

0 2

2

0 1

1 2

2

Case2-2. The case that K is not hermitian. In this case it is easy to see that w(s⁰)＝w(s¹) and r＝ . Therefore we obtain the structure equations:

1 3

2 3

0 0 1 1 2 0 1 0 1

1 3

2 3

1 2

1 3

2 3

1 2

(11)

Example 4.6. The case that H＝Z₋¹

2 (2)＝ {c0, c1}, L＝Z₋¹

2 (2), K＝H∪ S＝{c0, c1, s0, s1, s2}.

By Example 3.1 an irreducible action α of Z−¹2 (2) on S(¹) is known to be unique which is given by

α 1

0 ¹₂ ¹₂

1 2 1

2 1

2 1 2

0 0 and the invariant probability measure v on S(1) under the action α is given by

1 3

1

0 1 3 2

Then we get only two extension hypergroups up to equivalence as extensions which are si ＝s^* (i＝0, 1, 2),i

0 0

1 0

0 1

0

2 2

1 1 0

1 1

2 2 2

2 0

1 0

1 2 1 2

1 2 1 2 or

0 0

0 1

0

1 1

0

0 2

1 1 0

1 2

2 1

2 2 0

2 0

1 1

1 2 1 2

5．References

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[2] Heyer, H., Jimbo, T., Kawakami, S. and Kawasaki, K. : Finite commutative hypergroups associated with actions of fi nite abelian groups, Bull. Nara Univ. Educ., Vol. 54 (2005), No. 2, pp.23-29.

[3] Heyer, H. and Kawakami, S. : Extensions of Pontryagin hypergroups, PMS(Probability and Mathematical Statstics), Vol. 26, Fasc. 2 (2006), pp.245-260.

[4] Heyer, H. and Kawakami, S. : A cohomology approach to the extension problem for commutative hypergroups, to appear in Semigroup Forum.

[5] Heyer, H., Katayama, Y., Kawakami, S. and Kawasaki, K. : Extensions of fi nite commutative hypergroups, Scientiae Mathematicae Japonicae, 65, No. 3 (2007), pp.373-385.

[6] Ichihara, R. and Kawakami, S. : Strong hypergroups of order four arising from extensions, Scientiae Mathematicae Japonicae, 68, No.

3(2008), pp.371-381.

[7] Ichihara, R. and Kawakami, S. : Estimation of the orders of hypergroup extensions, Scientiae Mathematicae Japonicae, 72, No. 1 (2010), pp.307-317.

[8] Ichihara, R., Kawakami, S. and Sakao, M. : Hypergroup extensions of fi nite abelian groups by hypergroups of order two, Nihonkai Mathematical Journal,Vol. 21 (2010), No. 1, pp.47-71.

[9] Kawakami, S. : Extensions of commutative hypergroups, Infi nite Dimensional Harmonic Analysis, World Scientifi c, 2008, pp.135-148.

[10] Kawakami,S., Kawasaki, K. and Yamanaka, S. : Extensions of the Golden hypergroup by fi nite abelian groups, Bull. Nara Univ. Educ., Vol. 57(2008), No. 2, pp.1-10.

[11] Kawakami, S. and Mikami, I. : Hypergroup extensions of the group of order two by the Golden hypergroup, Bull. Nara Univ. Educ., Vol. 57(2008), No. 2, pp.11-16.

[12] Kawakami, S., Sakao, M. and Takeuchi, T. : Non-splitting extensions of hypergroups of order two, Bull. Nara Univ. Educ., Vol.

56(2007), No. 2, pp.7-13.

[13] Kawakami, S., Sakao, M., and Yamanaka, S. : Extensions of hypergroups of order two by locally compact abelian groups, Bull. Nara Univ. Educ., Vol. 59, No.2.(2010), pp.1-6.

[14] Kawakami, S. and Yamanaka, S. : Extensions of the Golden hypergroup by locally compact abelian groups, to appear in Acta Sci.

Math.

[15] Sunder, V.S. and Wildberger, N.J : Action of hypergroups and examples, (International Conference on Harmonic Analysis, 1995, University of Delhi), Birkhauser, 1998, pp.145-163.

[16] Wildberger, N.J. : Finite commutative hypergroups and applications from group theory to conformal fi eld theory, Applications of Hypergroups and Related Measure Algebras, Amer. Math. Soc., Providence, 1994, pp.413-434.