奈良教育大学学術リポジトリNEAR
Hypergroup Extensions of the Group of Order Two by the Golden Hypergroup
著者 KAWAKAMI Satoshi, MIKAMI Itsumi journal or
publication title
奈良教育大学紀要. 自然科学
volume 57
number 2
page range 11‑16
year 2008‑10‑31
URL http://hdl.handle.net/10105/711
1. Introduction
The concept of convolution of measures on a local- ly compact group has been generalized beyond the group case in the axiomatic setting of a hypergroup, due to C. F. Dunkl
(2), R.I.Jewett
(9), and R. Specter around 1975.
Let H and L be finite commutative hypergroups. A finite commutative hypergroup K is called an extension of L by H if the sequence:
1 H K L 1
is exact, i.e. if the quotient hypergroup K/H is isomor- phic to L. Here, the notions of subhypergroup, quotient hypergroup and isomorphism between hypergroups are taken from Bloom-Heyer
(1)and Wildberger
(15), a source from which all the elementary knowledge needed in the sequel will be taken.
In our previous paper (13) all extensions K of hypergroups of order two by finite abelian groups H are determined. In our paper (11) all extensions K of the Golden hypergroups L by finite abelian groups H are also determined including non-splitting case. Wildberger
(16)determined all hypergroup structures of order three in 2002. In his work, he pointed that the Golden hyper- group is in an interesting position among strong hyper- groups of order three. This is a motivation that we con- sider extension problem related with the Golden hyper- group. Hypergroup structure of order four is not yet determined. Such hypergroups arising from extensions are determined in our paper (7). Therefore it is impor- tant to make clear hypergroup structure of low orders which are greater than 3. On the other hand in our paper (4), we study the relationship between splitting extensions and extensions arising from fields.
In the present paper we study extensions K of the group L of order two by the Golden hypergroup H. We report that there are extentions K
a(t) and K
b(t) (
t √ 5 − 1) which are hermitian hypergroups of order five. By investigating the dual structures of such exten- sions, it is shown that K
a(t) [resp. K
b(t)] is strong if and only if t = 1. We should note that this is an interesting phenomenon in extension problem for the category of commutative hypergroups.
√ 5+1 4 Bull. Nara Univ. Educ., Vol. 57, No.2 (Nat. ) , 2008
Hypergroup Extensions of the Group of Order Two by the Golden Hypergroup
Satoshi KAWAKAMI and Itsumi MIKAMI *
(Department of Mathematics, Nara University of Education, Nara 630-8528, Japan) (Received May 7, 2008)
Abstract
The purpose of this paper is to investigate extension problem for the category of finite com- mutative hypergroups. In fact, we determine all strong hypergroup extensions of the group of order two by the Golden hypergroup. It is shown that one can get uncountably infinite many hypergroup extensions among which only 6 hypergroups are strong. (AMS Subject Classification : 43A62, 20N20.)
Key Words : hypergroup, extension
*completed Graduate School of Nara University of Education
2. Preliminaries
We recall some notions and facts on finite commu- tative hypergroups from Bloom-Heyer
(1)and Wildberger
(15). K : = (K, A) is called a finite commutative signed hypergroup if the following conditions (1)〜(6) are satisfied.
(1) A is a *-algebra over with the unit c
0. (2) K={ c
0, c
1, ... , c
n} is a linear basis of A.
(3) K
*=K.
(4) c
ic
j= n
i jkc
k,where n
i jk
is a real number such that c
*i=c
jn
ij0> 0,
c
*i= / c
jn
ij0=0.
(5) n
i jk=1 for any i, j.
(6) c
ic
j= c
jc
ifor any i, j.
A signed hypergroup is called a hypergroup if the coefficients n
i jk
of the structure equations are all non- negative. A hypergroup is called hermitian if c
*i= c
ifor any i. The weight of an element c
i∈ K is defined by w(c
i):= (n
ij0)
−1where c
j= c
*iand the total weight of K is given by w(K) : = w(c
i).
Let ω (K) denote the normalized Haar measure of K which is given by
ω (K) = c
k.
For a finite commutative signed hypergroup K, a com- plex valued function χ on K is called a character of K if
(1)χ (c
0) = 1, (2) χ(c
*i) = χ (c
i), (3) χ (c
i)χ (c
j) = n
i jkχ(c
k).
The set K ˆ of all characters of K also becomes a finite commutative signed hypergroup with order | K ˆ | = | K | =
n + 1, and the duality K ˆˆ K holds. A finite hypergroup K is called a strong hypergroup if the dual K ˆ of K is also a hypergroup.
Let K
1and K
2be two extensions of L by H and ϕ
1[resp. ϕ
2] be a canonical quotient mapping from K
1[resp. K
2] onto L. Then K
1is called to be equivalent to K
2as an extension if there exists a hypergroup isomor- phism ψ from K
1to K
2such that ψ (h)= h for all h∈ H and ϕ
2○ψ = ϕ
1.
3. Extensions of
2by the Golden hypergroup Let H = {h
0, h
1, h
2} be the Golden hypergroup where
h
0is the unit of H. The hypergroup structure of H is given by
h
21= h
0+ h
2, h
*1= h
1, h
22= h
0+ h
1, h
*2= h
2,
h
1h
2= h
1+ h
2Let L = { r
0, r
1} be the group of order two such that r
0is the unit of L and r
12
= r
0.
We investigate structures of extensions K of L by H. Let ϕ be a homomorphism from K onto L such that Ker ϕ = H, where H is assumed to be a subhypergroup of K. Then K is written as the disjoint union of H and S where H= ϕ
−1( r
0) and S= ϕ
−1( r
1). Let H( r
1) denote the stability subhypergroup of H at s
0∈ S, i.e.
H( r
1) = {h∈ H: hs
0= s
0}.
First of all, we consider the case that K = {h
0, h
1, h
2, s
0, s
1}, s
*0= s
0. In this case it is easy to see that H( r
1) = {h
0}. Set the structure equations by
h
1s
0= α s
0+ (1 − α )s
1, h
1s
1= γ s
0+ (1 −γ )s
1, h
2s
0=β s
0+ (1 − β )s
1, h
2s
1= δ s
0+ (1 − δ )s
1, (0 α ,β ,γ ,δ 1),
s
0s
0= a
0h
0+ a
1h
1+ a
2h
2,
(a
0+ a
1+ a
2= 1, a
0> 0, a
10, a
20), s
0s
1= b
1h
1+ b
2h
2, (b
1+ b
2= 1, b
10, b
20), s
1s
1= c
0h
0+ c
1h
1+ c
2h
2,
(c
0+ c
1+ c
2=1, c
0> 0, c
10, c
20).
We get the following equalities by associativity.
+ β= α
2+ (1− α )γ by (h
1h
1)s
0= h
1(h
1s
0).
α + β= α β+ (1 − α )δ= α β+ (1 − β )γ by (h
2h
1)s
0= h
2(h
1s
0) = h
1(h
2s
0).
+ α =β
2+ (1 − β)δ by (h
2h
2)s
0= h
2(h
2s
0).
δ= α γ+ (1 − γ)γ by (h
1h
1)s
1= h
1(h
1s
1).
γ+ δ=βγ+ (1 − γ)δ, by (h
2h
1)s
1= h
2(h
1s
1).
γ=βδ+ (1 − δ)δ by (h
2h
2)s
1= h
2(h
2s
1).
By these equalities we have
α
2+ β
2− 3 α β + α + β − 1 = 0, γ = ( + β − α
2),
2 1 2 1 1 − α
1 2
1 2 1 2 1 2
1 2 1 2
1 2 1 2
1 2 1 2
1 2 1 2 1 2 1 2
1 2 1 2
n
Σ
k=0w(c
k) w(K)
n
Σ
k=0n
Σ
i=0 nΣ
k=0n
Σ
k=0Satoshi Ka w a k a m i
・Itsumi Mikami
12
δ = γ .
Next, we calculate a
i, c
j(i, j = 0 , 1 , 2) and b
1, b
2in a simi- lar way.
a
1= 2 α a
0by h
1(s
0s
0) = (h
1s
0)s
0. a
2=2 β a
0by h
2(s
0s
0) = (h
2s
0)s
0. b
1=2 γ a
0by h
1(s
1s
0) = (h
1s
1)s
0. b
2= 2 δ a
0by h
2(s
1s
0) = (h
2s
1)s
0. c
1=2(1 − γ )c
0by h
1(s
1s
1) = (h
1s
1)s
1. c
2= 2( 1 − δ)c
0by h
2(s
1s
1) = (h
2s
1)s
1. Hence we see that
(1 + 2 α + 2 β )a
0= 1, (2 γ+ 2 δ )a
0= 1, {1 + 2(1 − γ) + 2(1 − δ)}c
0= 1.
Therefore the weights of s
0and s
1are given by w(s
0) = = 1 + 2 α + 2 β = 2 γ + 2 δ ,
w(s
1) = = 1 + 2 (1 − γ) + 2(1 − δ)= 5 − (2 γ + 2 δ).
We note that w ( s
0) + w ( s
1) = 5.
One can rewrite β, γ,and δ by α in the following two cases (a) and (b).
(a) K = K
a(t).
β= + α , γ= + α , δ= + α ,
w(s
0) = √ 5 + ( 5 −√ 5 ) α , w(s
1) = (5 −√ 5 )(1 − α ).
We take a parameter t by
t = = , 0 α .
By these equalities we have
α = − + ,
β= + ,
γ= , δ= ,
a
0= , a
1= − t, a
2= + t, b
1= , b
2= ,
c
0= (1 + ), c
1= − , c
2= +
．
(b) K = K
b( t ).
β= + α , γ= + α , δ= + α ,
w(s
0) = −√ 5 + ( 5 + √ 5 ) α , w(s
1) = (5 + √ 5 )(1 − α ).
We also take a parameter t by
t = = , α .
By these equalities we have
α = + ,
β= + ,
γ= , δ= ,
a
0= , a
1= ＋ t, a
2= − t, b
1= , b
2= ,
c
0= (1+ ), c
1= + , c
2= −
．
Hence we have the following theorem.
Theorem 1. Let K be an extension of the group L of order two by the Golden hypergroup H. Then all her- mitian hypergroup extensions K = {h
0, h
1, h
2, s
0, s
1} of order five are determined by K
a(t) and K
b(t) where t = w(s
1)/w(s
0) and t √ 5 − 1.
The structure equations of K
a(t) are given by
h
1s
0= ( − + ) s
0+ ( − )s
1, h
2s
0= ( + ) s
0+ ( − )s
1, h
1s
1= ( )s
0+ (1 − )s
1,
h
2s
1= ( )s
0+ (1 − )s
1,
s
0s
0= h
0+ ( − t )h
1+ ( + t)h
2, s
0s
1= h
1+ h
2,
s
1s
1= (1+ )h
0+ ( − ) h
1+ ( + 1 )h
2. t
− 1 + √ 5 10 2 5 1
t 1+ √ 5
10 2 5 1 t 1 5
5 −√ 5 10 5+ √ 5
10
− 1+ √ 5 10 2 5 1+ √ 5
10 2 5 t + 1
5
1 t + 1 5 −√ 5
4 1
t + 1 5 −√ 5
4
1 t + 1 5+ √ 5
4 1
t + 1 5+ √ 5
4
1 t + 1 5 −√ 5
4 5 −√ 5
4 1
t + 1 5 −√ 5
4
− 1+ √ 5 4
1 t + 1 5+ √ 5
4 5 + √ 5
4 1
t + 1 5 + √ 5
4
√ 5 + 1 4
√ 5+1 4
1 t 1+ √ 5
10 2 5
1 t
− 1 + √ 5 10 2 5 1
t 1 5
5 + √ 5 10 5 −√ 5
10
1 + √ 5 10 2 5
− 1+ √ 5 10 2 5 t + 1
5
1 t + 1 5 + √ 5
4 1
t + 1 5 −√ 5
4
1 t + 1 5 + √ 5
4
− 1 −√ 5 4
1 t + 1 5 −√ 5
4
− 1+ √ 5 4
5 −√ 5 4
√ 5 −1 2 ( √ 5 + 1)(1 − α )
− 1 + (√ 5＋1) α
w(s
1) w(s
0)
3 ＋√ 5 2
− 1 −√ 5 4
1 −√ 5 4 3 ＋√ 5
2
− 1 −√ 5 2
1 t
− 1 + √ 5 10 2 5
1 t 1 + √ 5
10 2 5 t
5 1 5
5 −√ 5 10 5 + √ 5
10
− 1 + √ 5 10 2 5 1 + √ 5
10 2 5 t + 1
5
1 t + 1 5 −√ 5
4 1
t + 1 5 + √ 5
4
1 t + 1 5 −√ 5
4
− 1 + √ 5 4
1 t + 1 5 + √ 5
4
√ 5 + 1 4
3 −√ 5 4 (√ 5 − 1)(1 − α )
1 + ( √ 5 − 1) α
w(s
1) w(s
0)
3 −√ 5 2
− 1 + √ 5 4
1+ √ 5 4 3 −√ 5
2
− 1 + √ 5 2 1 c
01 a
01 − β
1 − α
The structure equations of K
b(t) are given by
h
1s
0= ( + ) s
0+ ( − )s
1, h
2s
0= ( + ) s
0+ ( − )s
1, h
1s
1= ( )s
0+ (1 − )s
1,
h
2s
1= ( )s
0+(1 − )s
1,
s
0s
0= h
0+ ( + t )h
1+ ( − t)h
2, s
0s
1= h
1+ h
2,
s
1s
1= (1 + )h
0+ ( + )h
1+ ( − )h
2. Proof. These structure equations are already cal- culated. So we omit the details.
[Q.E.D.]
Remark 1. We note that K
a(t
1) and K
b(t
2) are not equivalent as extensions for any t
1and t
2. However it is easy to observe that K
a(t) is isomorphic to K
b(t) as hyper- groups. In order to see this fact we remark the case of t = 1. The structure equations of K
a(1) are given by
h
1s
0= s
0+ s
1, h
1s
1= s
0+ s
1, h
2s
0= s
0+ s
1, h
2s
1= s
0+ s
1, s
0s
0= s
1s
1= h
0+ h
1+ h
2,
s
0s
1= h
1+ h
2.
The structure equations of K
b(1) are given by
h
1s
0= s
0+ s
1, h
1s
1= s
0+ s
1, h
2s
0= s
0+ s
1, h
2s
1= s
0+ s
1, s
0s
0= s
1s
1= h
0+ h
1+ h
2,
s
0s
1= h
1+ h
2.
Next we consider the case that extensions K of L by H are non-hermitian hypergroups of order five. We obtain the following theorem.
Theorem 2. Let K be an extension of the group L of order two by the Golden hypergroup H. Then all non-
hermitian hypergroup extensions K = {h
0, h
1, h
2, s
0, s
1} of order five are only two hypergroups K
cand K
d. The structure equations of K
care given by
h
1s
0= s
0+ s
1, h
1s
1= s
0+ s
1, h
2s
0= s
0+ s
1, h
2s
1= s
0+ s
1, s
0s
1= h
0+ h
1+ h
2,
s
0s
0= s
1s
1= h
1+ h
2.
The structure equations of K
dare given by
h
1s
0= s
0+ s
1, h
1s
1= s
0+ s
1, h
2s
0= s
0+ s
1, h
2s
1= s
0+ s
1, s
0s
1= h
0+ h
1+ h
2,
s
0s
0= s
1s
1= h
1+ h
2.
Proof. These statements are obtained by associa- tivity of products of hypergroups in a similar way to the above arguments. So we omit the details.
[Q.E.D.]
Here, we investigate the structure of the dual K ˆ of extensions K described in Theorem 1 and Theorem 2.
(1) The character table of K
a(t) is given by K ˆ
a
(t) = { χ
0, χ
1, χ
2, χ
3, χ
4}.
The structure equations of K ˆ
a
(t) are given by χ
1χ
1= χ
0, χ
1χ
2= χ
2, χ
1χ
3= χ
4, χ
1χ
4= χ
3, χ
2χ
2
= χ
0
+ χ
1
+ χ
3
+ χ
4
, χ
2χ
3= χ
2χ
4= χ
2+ χ
3+ 1 χ
4,
4 1 4 1 2
1 4 1 4 1 4 1 4
5+ √ 5 10 5 −√ 5
10
3 −√ 5 10 3+ √ 5
10 2
5
3 −√ 5 8 5 + √ 5
8 5+ √ 5
8 3−√ 5
8
3 ＋√ 5 8 5 −√ 5
8 5 −√ 5
8 3+ √ 5
8
5 −√ 5 10 5 + √ 5
10
3 + √ 5 10 3 −√ 5
10 2
5
3 + √ 5 8 5 −√ 5
8 5 −√ 5
8 3+ √ 5
8
3 −√ 5 8 5 + √ 5
8 5+ √ 5
8 3 −√ 5
8
5 + √ 5 10 5 −√ 5
10
3 −√ 5 10 3+ √ 5
10 2
5
3 −√ 5 8 5 + √ 5
8 5 + √ 5
8 3 −√ 5
8
3 + √ 5 8 5 −√ 5
8 5 −√ 5
8 3 + √ 5
8
5 −√ 5 10 5 + √ 5
10
3 + √ 5 10 3 −√ 5
10 2
5
3 + √ 5 8 5 −√ 5
8 5 −√ 5
8 3 ＋√ 5
8
3 −√ 5 8 5 + √ 5
8 5+ √ 5
8 3 −√ 5
8
1 t 1 + √ 5
10 2 5 1
t
− 1+ √ 5 10 2 5 1
t 1 5
5+ √ 5 10 5 −√ 5
10
1+ √ 5 10 2 5
− 1+ √ 5 10 2 5 t + 1
5
1 t + 1 5+ √ 5
4 1
t + 1 5+ √ 5
4
1 t + 1 5 −√ 5
4 1
t + 1 5 −√ 5
4
1 t + 1 5+ √ 5
4 5+ √ 5
4 1
t + 1 5 + √ 5
4
− 1 −√ 5 4
1 t + 1 5 −√ 5
4 5 −√ 5
4 1
t + 1 5 −√ 5
4
− 1+ √ 5 4
Satoshi Ka w a k a m i
・Itsumi Mikami 14
χ
0χ
1χ
2
χ
3χ
4h
01 1 1
1
1
h
11 1
h
21 1
s
01
− 1 0
s
11
− 1
− 1+ √ 5 0 4
− 1 −√ 5 4
− 1 −√ 5 4
− 1 −√ 5 4
− 1+ √ 5 4
− 1+ √ 5 4
√ t
√ 2
√ t
√ 2
−
1 2t 1 2t
−
χ
3χ
3= χ
4χ
4= χ
0+ χ
2+ ( √ t − ) χ
3− ( √ t − ) χ
4, χ
3χ
4
= χ
1
+ χ
2
− (√ t− ) χ
3
+ (√ t− ) χ
4
. When t = 1, the structure equations of K ˆ
a
(1) are χ
1χ
1= χ
0, χ
1χ
2= χ
2, χ
1χ
3= χ
4, χ
1χ
4= χ
3, χ
2χ
2= χ
0+ χ
1+ χ
3+ χ
4,
χ
2χ
3
= χ
2
χ
4
= χ
2
+ χ
3
+ χ
4
, χ
3χ
3
= χ
4
χ
4
= χ
0
+ χ
2
, χ
3χ
4= χ
1+ χ
2. (2) The structure of the dual K ˆ
b
(t) of K
b(t) is similar to the above.
(3) The character table of K
care given by K ˆ
c
= { χ
0, χ
1, χ
2, χ
3, χ
4}.
The structure equations of K ˆ
c
are
χ
1χ
1
= χ
0
, χ
1
χ
2
= χ
2
, χ
1
χ
3
= χ
4
, χ
1
χ
4
= χ
3
, χ
2χ
2
= χ
0
+ χ
1
+ χ
3
+ χ
4
, χ
2χ
3
= χ
2
χ
4
= χ
2
+ χ
3
+ χ
4
, χ
3χ
3= χ
4χ
4= χ
1+ χ
2, χ
3χ
4
= χ
0
+ χ
2
.
(4) The structure of the dual K ˆ
d
of K
dis similar to the above.
Theorem 3. K
a(t) and K
b(t) of hermitian type with order five are strong hypergroups if and only if t = 1, i.e.
w(s
0)= w(s
1). Extensions K
cand K
dof non-hermitian type with order five are strong.
Proof. By the structure equations of the dual K ˆ
a
(t) of K
a(t), it is easy to see that K ˆ
a
(t) is a hypergroup if and
only if t = 1, i.e. w(s
0) = w(s
1) because t = w(s
1)/w(s
0). In a similar way one can see that K ˆ
b
(t) is a hypergroup if and only if t = 1. It is clear that K
cand K
dare strong by the above structure equations.
[Q.E.D.]
Remark 2. By the formula | L | + | H | − 1 | K |
| L || H | on order | K | of extensions K of L by H, the possi- bility of orders | K | of K are | K | = 4, 5, 6 when | L | = 2 and
| H | = 3 as in our situation. In the case of | K | = 5 we
determined all extensions K of L =
2by the Golden hypergroup H and characterlized when they are strong.
We note that all these extensions K of order five are not splitting
(4). In the case of | K | = 6, it is easy to see that a strong hypergroup extension K is only one which is H×
L, owing to model 1 in our paper (11). In the case of | K |
= 4, the extension K must be the join H ∨ L of H by L in general. We note that both H×L and H ∨L are strong and splitting. Hence we conclude that strong hyper- group extensions K of
2by the Golden hypergroup H are H × L, K
a(1), K
b(1), K
c, K
d, and H ∨ L. Splitting extensions K of L by H are H × L and H ∨ L.
References
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(2) Dunkl, C.F. and Ramirez , D.E. : A family of compact P
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