Article信州大学教養部紀要. 第二部, 自然科学 9: 7-10(1975)departmental bulletin pape

NISHIKAWA,  Akira

English

NISHIKAWA  Akira

Shinshu University Institutional Repository

UNIQUENESS OF THE ALGEBRA OF POLYNOMIALFUNCTIONS ON A FINITE  UPAkira NISHIKAWA   § 1. Introduction and notation    Let S be a set, F a field. Denote by MF (5) the  F  -  algebra of all F-valued functions on S.    Let G be a group, x an element in G, and f in MF  (G). We define  x-  f and  f-  x as the elements in MF (G) by  •.f)  (y) =  f  (yx),  (f-x)  (y) =  f  (xY) (y  E  C). Then, MF (G) is a two sided module on the group ring FG.    The group multiplication  GxG---,-G induces an  F-algebra-homomorphism  MF  (G)  MF  (G  x  G).       On the other hand, there is a canonical injection  7C:  MF  (G)  OF  MF  MF  (G x  G), and we identify MF  (G)  OF MF  (G) with  Ima-.    Now it is easily seen that for f in MF(G),  6(f)  Ima if and only if  FG  f is finite dimensional. A function with this property is called a representative function on G.    We denote by RF  (G) the set of all representative functions on G. Then the following results are known:     (1) RF (G) is a F-subalgebra of MF (G).      (2) Denote by  7- the restriction of  o- on  RF(G), then  Imr c RF (G)  OF  RF (G).     (3) RF  (G) is a Hopf algebra, where r is the comultiplication, c:RF (G)  — F such that  c  (f) =  f  (1) is the counit, and  3:  RF  (G) RF (G) such that  5(f) (x) = f  (x-1) (x  E G) is the antipode.    The structure of an affine algebraic group over the field F consists of a pair (G, A), where G is a group, and A is a sub Hopf algebra of RF (G) satisfying the following conditions:     (1) A is a finitely generated F-algebra.      (2) A separates the elements of  G,  i.e.  , for x, y in G such that x  1 y, there8exists　a　in　A　such　that　a（．T）　－a（y）．　　　　　（3）　Every　F－algebra－morphism・A　一’　．　F　is　the，　ev41uation　a’．1？．．scO（a）’・＝＝a（x）　atan　element　x　of　G．　　　　The　second　and　the　third　conditions　rnean　that　the　canonical　map　x　1．xO　of　G　intoAlgF　（z〈t，　F）　is　bijective，　and　we　shall　identify　G　with　AlgF　（A，　F）．　　　　When　（G，　A）　is　the　structure　of　an　afllne　algebraic　group　over　F，　A　is　called　thealgebra　of　polynomial　functions　on　G．　　　　Although　all　groups　are　not　a缶ne　algebraic　gro如s（亡3］），　any丘nite　group　is　anaffine　algebraic　group．　　　　If　G　is　a　finite　group，　then　the　natural　map　　　　　　　　　r，　：　MF　（G）　opF　MF　（G）　一”　MF　（G　×　G）is　bijective，　hence　　　　　　　　　M．　（G）　＝＝　RF　（G）．　　　　In　g　2，　we　shall　g．ive　a　proof　that　for　一a　finite　．qroup　G，　（G，　MF　〈G））　is　the　structureof　an　athne　aigebraic　group．　　　　In§3，　we　shall　prove　that　for　a丘nite　group　G，　if（G．A’）is　the　structure　of　a且affine　al．aebraic　group，　then　A　is　identical　with　L14F　（G）．　　　　g2．　A　pregf　that　fipmgte　grgups　are　affgne　aggebraie　groups　　　　If　S　is　a　non－empty　set，　then　we　have　canonical　identification　　　　　　　　　（FS）＊　＝＝　MF　（S），where　（FS）’　is　the　dual　of　FS，　F－space　with　the　basis　S．　　　　Especially　if　G　is　a　finite　group，　then・　　　　　　　　　（．FG）＊　＝＝　RF　（G）．　　　　亘£灘澱．1．　∬G　is　aプinite　9フ・oup，　then　（G，（FG）＊）is．the　stl’zacture　O！α7Zαがπ6algebraic　group．　　　　ProoL　Let　G　be　a　finite　group．　（FG）’“　is　then　finite　dimensiona｝，　especially　a丘nitely　generated　F－a工gebra．As　（FG）’　is　vhe　set　of　all　F－valued　functions，　it　naturally　separatep　the　elemepts　ofG．　　Hence，　　　　　　　　　G　c　AlgF　（（FG）’i‘，　F）．　　　　　　　　　　Uniqueness　of　the　algebra　of　polynomial　functions　on　a　finite　group　　On　the　other　hand，　for　any　Hopf　algebra　if，　　　　　　　　　A1gF　（H，　F）　＝　G　（Ho），where　G（HO）　is　the　set　of　all　grouplike　elements　of　the　dual　Hopf　algebra　　Hence　here，　　　　　　　　　AlgF　（（FG）”’　，　F）　＝：　G　（（FG）““）　＝＝　G　（17G）　D　G　　By　linear　independence　of　grouplike　elements，　we　have　　　　　　　　　lG（FG）　i　；：E！　dimFFG　＝　］G1．　　Therefore　　　　　　　　　AlgF　（（FG）”，　F）　＝　G，which　completes　the　proof．9NO　（［2］）．　　　§3．VitiqveePteSS　of鰍e　aggebra　of　geo亙yEeom量ag　f腿麟且。醜S　　　　Tkeorem　2．びGis　a　finite　9プ・妙，　and（G，　A）is　the　structure・！an　aflinealgeろraic　9プ。妙，　then　A　is　equal　to（FG）＊．　　　Proo　f．　Write　n　as　the　order　of　G，　then　　　　　　　　　dimF（FG）＊＝　［Gi＝n．　　By　the　definit圭On　of　the　af丑ne　algebraic　group，　　　　　　　　　　　　　A⊂RF（G）・・（FG）＊，　　　　　　　　　　　　　G窒A19F値，　F）．　　Using　the　result　of　Lemma　1，　we　think　the　following　composition　of　maps＝　　　　　　　　　　　　　　　　　　　　　　　　　　　プヨお　　　　　　　　　G　＝＝　AlgF（（FG）＊，　F）一→AlgF（A，　F）窪G．　　　　　　　　　（＊）　　Observing　the　process　of　construction，　this　compos三tion　is　the　identity　map　on　C．　　On　the　other　hand，　　　　　　　　　A19F（A．F’）⊂HomF（A，　F）＝A＊，　　and　all　the　elements　of　AlgF（A．F’）are　linearly　independent，　since　　　　　　　　　A19F（A，　F）＝G（五。）＝G（．A＊）．　　So　by　the　equality（＊），　AlgF（A，　F）consists　of　n　elements．　　Therefore　　　　　　　　　dimFA　・dim〆A＊＝n＝dimF（FG）＊，10and　soA　＝＝　（FG）＊，which　completes　the　proof．葺式e歪ere亙亙ces1．　HocHsCHILD，　G．：Introduction　to　Affnne　Algebraic　Groups，（1971）Ho工den・Day，　Inc．2．　SwEEDLER，　M．　E．　：　Hopf　Algebras，　（1969）　Benjamin3．　TAFT，　E．　J．　：　Refiextvility　of　algebras　and　coalgebras，　American　Journal　of　Mathematics　（1972）　1111－1130．

UNIQUNESS OF THE ALGEBRE OF POLYNOMIAL FUNCTIONS ON A FINITE GROUP

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UNIQUNESS OF THE ALGEBRE OF POLYNOMIAL FUNCTIONS ON A FINITE GROUP

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