Finite Commutative Hypergroups Associated with Actions of Finite Abelian Groups

奈良教育大学学術リポジトリNEAR

Finite Commutative Hypergroups Associated with Actions of Finite Abelian Groups

著者 HEYER Herbert, JIMBO Toshiya, KAWAKAMI Satoshi, KAWASAKI Ken‑ichiroh

journal or

publication title

奈良教育大学紀要. 自然科学

volume 54

number 2

page range 23‑29

year 2005‑10‑31

URL http://hdl.handle.net/10105/115

1  Introduction

For a finite group L, one defines canonically two commutative finite hypergroups K(L)and K(Lˆ ), called the class hypergroup and the character hypergroup of L respectively. The character hypergroup K(Lˆ )is obtained by the fusion rule on representations of G. In the previous paper [K-I], where L is given as a semidirect product group N αGunder an action αof a finite abelian group G on a finite abelian group N, we constructed directly the character hypergroup K(N _αG)in the ＊-algebra A(Nˆ ) A(Gˆ )with the help of an action αˆ of Gon Nˆ.

For an action αof a finite abelian group Gon a finite commutative hypergroup H, in the previous paper [K-I], we defined the finite commutative hypergroup K(H,G,α) as the dual of K(Hˆ, Gˆ, αˆ ). In the present paper we are going to construct the hypergroup K(H,G,α)associated with the action αof Gon Hand to show that K(Hˆ, Gˆ, αˆ )is isomorphic to the dual of K(H,G,α)as hypergroups. The construction of K(H,G,α)related to the action αof Gon H=(H, A(H)), where A(H)is the associated ＊-algebra over of the hypergroup H, runs as follows. Let E be the canonical conditional expectation from the crossed prod- uct A(H) _αGonto the center Z(A(H) _αG)of A(H) _αG.

Then K(H, G, α)is defined as the image of the canonical base {h_jλg; hj∈Hand g∈G} of A(H) _αGunder E, under the assumption that αis regular. Here, λ denotes the canonical representation of Gimplementing the action α, namely, αg=Adλg.

This work was completed when the first named author visited the Department of Mathematics at Nara University of Education in the spring 2004.

2  Preliminaries

We recall some notions and facts about finite com- mutative signed hypergroup according to Wildberger’s paper [W]. K= (K, A)is called a finite commutative signed hypergroups if the following conditions (1)〜(6) are satisfied.

(1)Ais a ＊-algebra over with the unit c0. (2)K= {c0, c1, ... ,cn} is a linear basis of A.

(3)K^＊= K.

(4)cicj= n^kijck, where n^kijis a real number such that c^＊i= cj n⁰ij> 0and c^＊i ≠cj n⁰ij= 0.

(5) n^k_ij= 1for any i, j.

Σ

Σ

Bull. Nara Univ. Educ., Vol. 54, No.2 (Nat.),2005

Finite Commutative Hypergroups Associated with Actions of Finite Abelian Groups

Herbert HEYER^*, Toshiya JIMBO, Satoshi KAWAKAMI and Ken-ichiroh KAWASAKI

(Department of Mathematics, Nara University of Education, Nara 630-8528, Japan) (Received May 2, 2005)

Abstract

This paper is devoted to constructing finite commutative hypergroups associated with actions of finite abelian groups, which are given as modifications of crossed products in operator algebras. This construction relies on the application of some notions of operator algebras, crossed products and con- ditional expectations.

Key Words：finite commutative hypergroup, group action, crossed product, conditional expectation

* Tübingen University

(6)c_ic_j=c_ic_jfor any i,j.

In the case that n^kij>_ 0for any i, j, k, K= (K, A) = (K, A(K))is called a finite commutative hypergroup.

The weight of an element ci∈Kis defined by w(ci) = (n⁰_ij)⁻¹where cj=c^＊_i. The total weight of Kis defined by w(K) =Σ^ni=0w(ci).

For a finite commutative signed hypergroup K, a complex-valued function χon Kis called a character of K if

χ(c_i)χ(c_j) = n^k_ijχ(c_k)whenever cic_j= n^k_ijc_k. The set Kˆ of all characters of Kalso becomes a finite commutative signed hypergroup, and the duality Kˆˆ〜=K holds in the sense of signed hypergroups.

For a finite not necessary abelian group G the two hypergroups K(G)and K(Gˆ )are canonically defined. K(G)is introduced as the class hypergroup for which by definition each element of K(G)corresponds to a conjugacy class of Gunder the adjoint action of G. K(Gˆ )is defined as the set of normalized characters χof G, i.e. of all functions g χ(g) =tr(ρ(g)), where ρis an irreducible representation of Gand trdenotes the normalized trace. K(Gˆ )also becomes a finite commutative hypergroup. In this case we have K(Gˆ )〜=K(G).

3  Group actions and fixed point hypergroups Let αbe an action of a finite group Gon a finite commutative signed hypergroup K= (K, A(K)), where, for each g∈G, αgoperates on the set K = {c0, c1, …,cn} as ＊-preserving permutations satisfying

αg(c_i)αg(c_j) = n^k_ijαg(c_k)

if cic_j= n^k_ijc_k, and αgαh=αghfor g, h∈G. The action α of Gon Kinduces the action of Gon the associated ＊- algebra A(K) of K, which will also be denoted by α. In this case, the canonical conditional expectation Efrom A(K) onto the fixed point algebra A(K)^α is canonically defined by

E(x) = αg(x)for x∈A(K).

Let K^αdenote the image of K={c0, c1, …,cn} under the conditional expectation E. Then K^αbecomes a finite com- mutative signed hypergroup, and A(K^α) = A(K)^α. K^α is called the fixed point hypergroup of Kunder the action α.

The action αof Gon Kalso induces an action αˆ of G on the dual Kˆ ={χ0, χ1, . . . , χn} of Kby

αˆ_g(χj)(c_i) =χj(α−1g (c_i))for χj∈Kˆand ci∈K.

In this way we obtain a finite commutative signed hyper- group Kˆ^αˆ which satisfies K^α〜= Kˆ^αˆ.

Let Gbe a finite group and λbe the regular repre- sentation of G. Then the set K ={λg; g∈G} is a finite hypergroup, and the associated algebra A(K) is the ＊- algebra λ(G)''generated by λg(g∈G)on the Hilbert space l²(G). Let αbe the adjoint action of Gwhich induces the action αof Gon A(K) = λ(G)' '. Then we also obtain the fixed point hypergroup K^α= (K^α, A(K)^α). It is easy to check that A(K)^αis the center Z(A(K))of A(K)and K^αis the class hypergroup K(G)of G. This interpretation gives rise to the construction of hypergroups associated with actions of G.

This construction will be described in the next section.

4  Construction of hypergroups associated with actions

Let H= {c0, c1, …, cn} ⊂A(H)be a finite commuta- tive hypergroup and Hˆ = {χ0, χ1, …, χn} be the dual of H.

Then, there exist mutually orthogonal minimal projec- tions e0, e1, …, en∈A(H)satisfying

c_ie_j=χj(c_i)e_j.

The relationships between {c0, c1, …,cn} and {e0, e1,

…, en} in A(H)are given by

e_j= w(c_i)χj(c_i)c_i, c_i= χj(c_i)e_j. In particular,

e0= w(ci)ci

is called the Haar measure of Hwhich satisfies cie0= e0

for any ci∈H. The characters χ0, χ1, …, χnare known to be orthogonal with respect to the inner product

〈f, g〉= w(c_i)f(c_i)g(c_i).

Let Gbe a finite abelian group and αbe an action of Gon the finite commutative hypergroup H = {c0, c1, …, c_n} ⊂A(H). Then the action αof Gon the associated ＊- algebra A(H)and the crossed product

A(H) _αG=

{

}

is defined in the canonical way. For each element x∈ A(H) _αG, put

F(x) = w(c_i)c_iλgxλ^＊gc^＊_i

Then we have the following lemma.

Lemma 1. The mapping Fis an operator-valued

Σ

Σ

1

|G|

1 w(H)

Σ

Σ

0

1 w(H)

Σ

1 w(H)

n

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w(χj) w(H)

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1

|G|

n

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n

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n

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n

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Herbert HEYER ・Toshiya JIMBO ・Satoshi KAWAKAMI ・Ken-ichiroh KAWASAKI 24

weight from A(H) _αGonto the center Z(A(H) _αG)of A(H) _αGsatisfying

F(c_iλg) =_ c_if(g)λg, where

_ci= αg(ci) and f(g) = 〈χj, αg(χj)〉ej. In particular,

F(c0) = 〈χj,χj〉e_j

is a positive and invertible operator in Z(A(H) _αG).

Proof.

F(c_iλg) = w(c_j)c_jλk(c_iλg)λ^＊kc^＊_j

= w(c_j)c_jαk(c_i)αg(c^＊_j )λg

=

(

) (

)

=_ cif(g)λg

where

_ci= αk(ci) and f(g) = w(cj)cjαg(c^＊j).

Applying the formulae

c_j= χi(c_j)e_i, αg(c^＊_j) = αg(χi)(c_j)e_i, we obtain

f(g) =

(

)

= 〈χi, αg(χi)〉e_i

= 〈χi,χi〉ei

where

J(g) ={i; αg(χi) =χi} ={ i; αg(e_i) =e_i}.

Since we assumed Gto be an abelian group, J(g) ={ei; αg(e_i) =e_i} is α-invariant, namely, αk(J(g)) =J(g)for any k

∈G. This fact implies that λkf(g)λ^＊k=αk(f(g)) =f(g)for any k∈G. Moreover, f(g)αg(ci) =cif(g)holds for any c_i∈H. It is clear that _

c_i∈A(H)^α. Then, we can see that F(c_iλg) =c_if(g)λgis an element of the center Z(A(H) _αG) of A(H) _αG. Since {ciλg; g∈G, ci∈H} is a basis of A(H)

αG, F(x)is in Z(A(H) _αG)for any x∈A(H) _αG. The bimodule property of Ffollows directly from the definition of F.

[Q.E.D.]

For each x∈A(H) _αG, set E(x) =F(a⁻¹x)

where

a=F(c0) = 〈χj,χj〉e_j is a positive invertible operator in Z(A(H) _αG).

Then Ebecomes a conditional expectation from A(H)

αGonto Z(A(H) _αG)satisfying E(ciλg) =cie(g)λg

where

_c_i= αg(c_i), e(g) = e_j, J(g) ={j; αg(e_j) =e_j}.

We note that _

c_i∈H^αand that e(g)is in A(H)^α. For each g

∈G, set

X(g) ={χj∈Hˆ ; αg(χj) =χj} and

Y(g) ={ci∈H; χj(c_i) = 1for χj∈X(g)}.

Here, we note that J(g) ={j; χj∈X(g)}.

Proposition 2. If X(g)is a subhypergroup of Hˆ for each g∈G, we have the following:

(1)e(g)is the Haar measure of Y(g)and also of Y(g)^α. (2){die(g)λg; di∈H^α, g∈G} is a linear basis of Z(A(H)

αG).

Proof.(1)For χj∈Hˆ and ei∈A(H), χj(ei) =δij

holds. Then e(g) = e_j satisfies χj(e(g)) = 1if χj∈X(g)

0if χj∈

/

The Haar measure η(g)of Y(g)is given by

η(g) = w(ci)ci

where w(Y(g)) = w(c_i). For each χj∈X(g)we have χj(η(g)) = 1since χj(c_i) = 1for ci∈Y(g). For each χj∈

/

X(g)there exists ci∈Y(g)such that χj(c_i) ≠ 1because X(g)is characterized by

X(g) ={χ∈Hˆ ; χ(c_i) = 1for any ci∈Y(g)}

if X(g)is a subhypergroup of Hˆ. The Haar measure η(g) satisfies ciη(g) =η(g), so that χj(ci)χj(η(g)) =χj(η(g))holds.

Applying the fact that χj(c_i) ≠ 1, χj(η(g))must be 0. Since each element of A(H)is characterized by the values of χj∈ Hˆ, we obtain that e(g) =η(g). Let O0, O1, …, Okbe the orbits of X(g)under the action αˆ of Gon X(g). Then the

c

Σ

ci

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1 w(Y(g))

i∈J(g)

Σ

i∈J(g)

Σ

g∈G

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1

|G|

n

Σ

i∈J(g)

Σ

n

Σ

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1 w(H)

Σ

n

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1

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k∈G

1

|G|

n

Σ

1

Σ

k∈G

1

|G|

Σ

Σ

1

|G|

1 w(H)

k∈G

Σ

n

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1

|G|

1 w(H)

n

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Σ

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1

|G|

canonical minimal projections p0, p1, …, pkin Y(g)^αare given by pj= el(j= 0, 1, …, k)so that p0+p1+ … + pk=e(g). By similar arguments as above it is easy to see that e(g)is also the Haar measure of Y(g)^α.

(2) It is known that D(g) ={die(g) ; d_i∈H^α} is a hypergroup isomorphic to the quotient hypergroup Q(g) = H^α/Y(g)^α. Then the set of D(g)is a basis in A(H)^αe(g). Hence we see that the elements of K={die(g)λg; di∈H^α, g∈G}

are mutually independent in Z(A(H) _αG). It is clear that the linear hull of the set K={die(g)λg; di∈H^α, g∈G} is Z(A(H) _α G) because Ewas a conditional expectation from A(H) _αGonto Z(A(H) _αG). Hence we see that the set Kis a linear basis of Z(A(H) _αG).

[Q.E.D.]

We note that the dimension of Z(A(H) _αG)is equal to

|K| = |D(g)| = |X(g)^αˆ|

by the fact that the dual of Q(g)is isomorphic to X(g)^αˆ. In the previous paper [K-I] we constructed the finite commutative signed hypergroup K(Hˆ, Gˆ, αˆ ) associated with an action αof a finite abelian group Gon a finite commutative signed hypergroup H. Here, we recall the construction of K(Hˆ, Gˆ, αˆ ). The action αof Gon Hinduces the action αˆ of G on the dual Hˆ ={χ0, χ1, …, χn} of H. Let O0, O1, …,Ombe all orbits of Hˆ under the action αˆ of G and Gibe the stabilizer of Gcorresponding to each Oi(i= 0, 1, …, m), namely,

G_i={g∈G ; αg(χ) =χ} for χ∈O_i

which does not depend on the choice of χ∈O_i, because Gis an abelian group. For each g∈G, X(g)was defined by X(g) ={χ∈Hˆ;αg(χ) =χ}.

Let Hˆαˆ

={ρ0, ρ1, …, ρm} be the fixed point hyper- group of Hˆunder the action αˆ of Gon Hˆ. Here, we note that each ρiis defined by

ρi= χ (i= 0, 1, …,m).

Let τibe the Haar measure of the annihilator Zi={τ∈Gˆ ; τ(g) = 1for all g∈Gi} of Gi, namely,

τi= τ (i= 0, 1, …,m).

Then K(Hˆ, Gˆ, αˆ )was defined by

K(Hˆ, Gˆ, αˆ ) ={ρi τ τi; i= 0, 1, …,m, τ∈Gˆ}

={ρi τ τi; i= 0, 1, …,m, τ∈Gˆ/Zi〜=Gˆ

i}.

In [K-I] we said that the action α(or the action αˆ )is regular if the following condition (＊)is satisfied for any g

∈G.

(＊)If αˆ_g(χi) =χiand αˆ_g(χj) =χjfor χi, χj∈Hˆ, then αˆ_g(χk) =χkholds for every ksuch that m^kij≠ 0, where χiχj= m^k_ijχk.

We will characterize the above regularity condition in the following lemma.

Lemma 3. In the above situation the following statements are equivalent:

(1)The action αis regular.

(2)Gi∩Gj⊂Gkfor every ksuch that m^kij≠ 0.

(3)X(g)is a subhypergroup ofHˆ. (4)X(g)^αˆ is a subhypergroup of Hˆ^αˆ.

We obtained the subsequent theorem in 1999.

Theorem([K-I]). If the action αis regular, then K(Hˆ, Gˆ, αˆ )is a finite commutative signed hypergroup.

Lemma 4. If the action αis regular, then for ρi

∈Hˆ^αˆ and g∈Gwe get the following:

(1)ρi∈X(g)^αˆ if and only if g∈G_i. (2)ρi(e(g)) =τi(g).

Proof.

Statement (1)follows immediately from the defini- tions. Since e(g)is the Haar measure of Y(g)^α(the annihi- lator of X(g)^αˆ)and τiis the Haar measure of Zi(the anni- hilator of Gi), one can see that

ρj(e(g)) = 1if ρj∈X(g)^αˆ 0if ρj∈

/

,

τj(g) = 1if g∈Gi

0if g∈

/

.

The proof goes similar to the proof of (1) in Proposition 2. Hence we arrive at the equality (2).

[Q.E.D.]

Under these preparations we obtain our main theo- rem as follows.

Theorem 5. Let αbe an action of a finite abelian group G on a finite commutative hypergroup H={c0, c1,

…, cn} ⊂A(H). If the action αis regular, then (1)K(H,G, α)is a finite commutative hypergroup.

(2)A(K(H,G, α)) =Z(A(H) _αG).

(3)Kˆ (H,G, α)〜=K(Hˆ, Gˆ, αˆ ).

m

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|Zi|

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|Oi|

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Herbert HEYER ・Toshiya JIMBO ・Satoshi KAWAKAMI ・Ken-ichiroh KAWASAKI 26

(4)Kˆ (Hˆ, Gˆ, αˆ )〜=K(H,G, α).

Proof. (1)In Proposition 2we saw that {die(g)λg; d_i∈H^α, g∈G} was a linear base of A(K(H,G, α)) =Z(A(H)

αG). We note that e(g1)e(g2) ^<_e(g1g2) for g1, g2∈G, since X(g1) ∩X(g2)⊂X(g1g2). We examine products of elements of K(H,G, α).

(d_ie(g1)λg₁)(d_je(g2)λg₂) =d_id_je(g1)e(g2)λg₁λg₂

=d_id_je(g1)e(g2)e(g1g2)λg₁g₂, where

d_id_je(g1)e(g2) = n^k_ij(g1, g2)d_k. Then

(die(g1)λg₁)(dje(g2)λg₂) = n^kij(g1, g2)dke(g1g2)λg₁g₂, where the structure constants n^kij(g1, g2) are non-nega- tive.

Next, we examine the ＊-operation (d_ie(g)λg)^＊=d^＊_ie(g)^＊λ^＊g = d^＊_ie(g)λg−1, (d_ie(g)λg)(d_ie(g)λg)^＊=d_ie(g)λgd_i^＊e(g)λg−1=d_id^＊_ie(g).

Since did＊i = n^kidk,

d_id_i^＊e(g) = n^k_id_ke(g) = n^k_id_ke(g) + n^k_id_ke(g)

=

(

)

The Haar measure e(g)of Y(g)was defined by e(g) = w(d)d= d0+ (others).

Hence we see that

(d_ie(g)λg)(d_ie(g)λg)^＊=

(

)

The structure constant at d0is positive, and clearly the weight of die(g)λgis

w(d_ie(g)λg) =w(Y(g))

(

)

Next we examine the case that dje(g1)λg₁≠ (d_ie(g)λg)^＊. (¡)In the case that g1≠g⁻¹, λg₁λg=λgg₁and gg1≠e.

Hence the structure constant at d0 in the product (d_ie(g)λg)(d_je(g1)λg₁)is zero.

(™)In the case that g1=g⁻¹, but dje(g1) ≠ (d_ie(g))^＊, (d_ie(g)λg)(d_je(g⁻¹)λg⁻¹) =d_id_je(g)λe=d_id_je(g).

If diand djis in Y(g), d_ie(g) =d_je(g) =e(g)holds, conse- quently (d_je(g))^＊=e(g)^＊=e(g) =d_ie(g). Therefore the assumption dje(g⁻¹) ≠ (d_ie(g))^＊implies that did_j∈

/

this case the structure constant at d0in didje(g)must be

zero.

From (¡)and (™)we conclude that K(H, G, α)satisfies the axiom of a finite commutative hypergroup.

Assertion (2)follows from Proposition 2.

(3)Kˆ (H,G, α) ⊃K(Hˆ, Gˆ, αˆ )is clear. We show that Kˆ (H,G, α)⊂K(Hˆ, Gˆ, αˆ ). Let χbe a character of K(H, G, α).

Then there exists ρj∈Hˆαˆ

such that χ(d_ie(e)λe) =χ(d_i) = ρj(di), since Hˆ^αˆ 〜=H^α. On the other hand, there exists τ∈ Gˆ such that χ(λg) =τ(g).

Hence we see that

χ(d_ie(g)λg) =ρj(d_i)ρj(e(g))τ(g)

=ρj(d_i)τj(g)τ(g). (by (2)of Lemma 4.) Therefore

χ=ρj τjτ∈K(Hˆ, Gˆ, αˆ ).

Statement (4)follows immediately from (3)by duality.

[Q.E.D.]

Remark 1. When H=K(N)for some finite abelian group N, we see that

K(H,G, α) =K(N _αG),

where K(N _αG)is the class hypergroup of the semidirect product group N _αG.

Remark 2. The sequence

1→H^α→K(H,G, α)→K(G)→1

is exact, namely, K(H,G, α)/H^α〜=K(G). On the other hand, we also note that

1→K(Gˆ )→K(Hˆ, Gˆ, αˆ )→Hˆ^αˆ →1 is exact.

Example 1. H=K( 3), G= 2, α: 2→Aut ( 3), namely, G={e, g} (g²=e), H={c0, c1, c2} (c²1=c2, c²2=c1, c1c2=c0), and αg(c0) =c0, αg(c1) =c2, αg(c2) =c1. In this case we get

H^α={d0, d1}, d²1= d0+ d1, where

d0=c0, d1= (c1+c2).

The Haar measure e(g)of Y(g)^α=H^αis given by e(g) = d0+ d1= (c0+c1+c2).

Then we see that

K(H,G, α) ={d0, d1, d2=e(g)λg}, 1

3 2 3 1 3

1 2

1 2 1 2

k∈Y(g)

Σ

k∈Y(g)

Σ

1 w(Y(g))

1 w(Y(g))

d∈Y(g)

Σ

1 w(Y(g))

k∈Y(g)

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k∈

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m

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m

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