Jordan Chevalley Decomposition and Invariants for Locally Finite Actions of Commutative Hopf Algebras

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1 JOURNAL OF ALGEBRA 182, ARTICLE NO JordaChevalley ecopoitio ad Ivariat for Locally Fiite Actio of Coutative Hopf Algebra Adrzej Tyc* N. Copericu Uierity, Ititute of Matheatic, ul. Chopia 12-18, Toru, Polad Couicated by Sua Motgoery Received Augut 24, INTROUCTION Let H, : H H H, S: H H, : H k. k be a coutative Hopf algebra over a algebraically cloed field k. Bya actio of H o a vector pace V over k we ea a hooorphi of k-algebra : HEd V. Such a actio i called locally fiite rep. eiiple. if the H-odule V,. h h,. hh,v. i a uio of fiite dieioal ubodule rep. if the H-odule V,. i eiiple.. We ay that the actio i locally ilpotet if for each V there i a.. uch that H 0, where H Ker. For ay actio : H Ed V, the pace V; h h,hh 4 i called the pace of iariat of ad it i deoted by V. Give two actio, : H Ed V of H o V, the covolutio product : H Ed V i defied by h. Ý h. h, where Ýh h h Oe iply check that if ad coute that i, ht t h. for all h, t H., the i a actio of H o V. Our firt objective i thi paper i to prove the followig. THEOREM 2.6. JordaChevalley decopoitio for locally fiite actio of H o vector pace.. Let : H Ed V be a locally fiite actio of H o a ector pace V:. 1 There exit uique locally fiite actio, : H Ed VofHo V atifyig the coditio:, ad coute, i eiiple, ad i locally ilpotet. * E-ail: atyc@at.ui.toru.pl $18.00 Copyright 1996 by Acadeic Pre, Ic. All right of reproductio i ay for reerved.

2 124 ANRZEJ TYC. 2 If a actio L: H Ed V coute with, the L coute with ad.. 3 If U W are ubpace of V uch that h. W. U for all hh, the h W U ad h W U for all h H. The decopoitio i called the JordaChealley deco- poitio of. If H kx with X. 1 X X 1, S X. X, X. 0, the a locally fiite actio of H o a vector pace V i othig ele but a locally fiite edoorphi f of V, ad the decopoitio fro the theore i the well kow additive. JordaChevalley decopoitio of 1 1 f. Whe H k X, X with X X X, S X X, X. 1, the a locally fiite actio of H o V i iply a locally fiite autoorphi g of V, ad i thi cae the theore ay that there are uique locally fiite autoorphi g, g of V uch that g g g, g g g g, u u u g i diagoalizable, ad gu Id i locally ilpotet. I other word, gg g i the well-kow ultiplicative JordaChevalley decopoitio u of g. Oe ay fid aother exaple i Sectio 2. Now recall that a actio of H o a k-algebra A i a actio : HEd A of H o the uderlyig vector pace A of A uch that h 1. h1 ad h xy. Ýh xh. y. A A for h H, x, ya, where agai Ýh h h thi ea that A, together with, iah-odule algebra; ee. 8. Give uch a actio, the pace of ivariat A i a ubalgebra of A called the algebra of iariat 1 of. If Hk X rep. H k X, X. are a above, the a actio of H o a algebra A i a derivatio d rep. a autoorphi g. of A ad A Ker d rep. A a A; g a. a 4.. We ay that a actio of H o A i locally fiite, locally ilpotet, eiiple if it i locally fiite, locally ilpotet, eiiple a a actio of H o A, repectively. The ai reult of thi paper i the followig. THEOREM 3.1. If : H Ed A i a locally fiite actio of H o a k-algebra A ad i the JordaChealley decopoitio of a a actio of H o A., the i a actio of H o the algebra A. Furtherore, if H i cocoutatie, the i alo a actio of H o A. We prove alo the followig. THEOREM Suppoe that the Hopf algebra H i cocoutatie ad that i a locally fiite, eiiple actio of H o a coutatie algebra A. The the algebra of iariat A i oetheria rep. fiitely geerated., proided A i oetheria rep. fiitely geerated.. A a applicatio of above reult ad the claical Weitzebock theore oe get the followig.

3 JORANCHEVALLEY ECOMPOSITION 125 COROLLARY Suppoe that g i a degree preerig autoorphi of g the k-algebra A k X,..., X. The A a A, g a. a4 1 i a fiitely geerated ubalgebra of A. Throughout the paper k will deote a algebraically cloed field which will erve a the groud field for all vector pace, algebra, ad Hopf algebra uder coideratio. All teor product are over k. ByH we deote a fixed coutative a a rig. Hopf algebra with coultiplicatio :HHH, atipode S: H H, ad couity : H k. Ai 8 we write h. Ýh h, Idh. Id. h Ýh1. h2. h3. ad o o, for all h H. Oe kow that, S, are hooorphi of algebra, S, ad give a h H, Ýh h. h Ý h h 2., ÝSh h hýh S h., ad Sh. ÝSh Sh for baic propertie of Hopf algebra ee. 8. A eleet h H i called priitie whe h. 1 h h 1 the ecearily Shh ad h. 0.. The Hopf algebra H i aid to be cocoutatie if t, where the liear ap t: H H H H i deteried by t xy. yx. The. ideal Ker will be deoted by H. Oberve that S H H. Oe eaily check that the et of all algebra hooorphi : H k with the covolutio product defied by. h. Ý h. h i a 1 group with a the eutral eleet ad with S.. We deote thi group by GH* ad we will ue it i the equel. Notice that GH* i abelia, wheever the Hopf algebra H i cocoutative. For a vector pace V, V : H Ed V will tad for the trivial actio of H o V defied by V h. h. Id, where Id: V V i the idetity ap. Recall that a edoorphi f: V V i called locally fiite if V i a uio of fiite dieioal f-ivariat ubpace of V. Edoorphi f i aid to be locally ilpotet if for each V there exit with f. 0. Note that each locally ilpotet edoorphi i locally fiite. Alo recall that f i called eiiple if the kx-odule V,., X f., i eiiple. If f i locally fiite, the clearly k beig algebraically cloed. f i eiiple if ad oly if f i diagoalizable. I what follow we eed the followig kow propoitio. PROPOSITION 1.1 Additive JordaChevalley decopoitio for locally fiite edoorphi of vector pace.. Let f be a locally fiite edoorphi of a ector pace V:. 1 There exit uique locally fiite f, f Ed V atifyig the codi- tio: f f f, f i eiiple, f i locally ilpotet, ad f, f co- ute.. 2 Edoorphi f ad f coute with ay g Ed V cout- ig with f.

4 126 ANRZEJ TYC. 3 If U W are ubpace i V ad f W. U, the f W. U ad f W. U. The decopoitio f f f i called the additie. JordaChealley decopoitio of f. For fiite dieioal V the propoitio i proved i 2, 4.2. The proof for arbitrary V i a iple applicatio of the fiite dieioal cae. COROLLARY 1.2. pace V: Let f be a locally fiite edoorphi of a ector. 1 If g Ed V i locally fiite ad fg gf, the f g ad fg are locally fiite with f g. f g, f g. f g, fg. fg,ad fg. fgfg fg. I particular, arbitrary fiite u ad copoi- tio of pairwie coutig, locally fiite, ad eiiple rep. locally ilpotet. edoorphi of V are locally fiite ad eiiple rep. locally ilpotet... 2 For each k, V, f. 4 V, f Id For fiite dieioal V the corollary i kow ad eay to prove, uig the above propoitio ad the defiitio of f ; ee 2, The geeral cae reduce eaily to the fiite dieioal cae. Fially recall that a higher derivatio of a algebra A i a equece of liear ap d : A A, i 0, uch that d Id ad d xy. i 0 Ý d x. d y. for all 0 ad x, y A. ij i j 2. JORANCHEVALLEY ECOMPOSITION FOR LOCALLY FINITE ACTIONS OF H ON VECTOR SPACES Let V be a vector pace. By a actio of H o V we ea a hooorphi of algebra : H Ed V. Such a actio i called locally fiite rep. locally ilpotet. if the H-odule V,. h h,hh,v.. i a uio of fiite dieioal ubodule rep. if for each V there exit a uch that H. 0.i. aid to be eiiple if the H-odule V,. i a u of iple ubodule. Whe i locally fiite, the it i eiiple if ad oly if V,. i a u of fiite dieioal iple ubodule. Equivaletly k beig algebraically cloed ad H beig coutative., a locally fiite i eiiple if ad oly if i diagoalizable, i.e., if there exit a bai e, j J4 j of V uch that h e. j kej for all j J. It i obviou that for each vector pace V, the trivial actio V : H Ed V i locally fiite, locally ilpotet, ad eiiple.

5 JORANCHEVALLEY ECOMPOSITION 127 I geeral, both locally ilpotet ad eiiple actio eed ot be locally fiite. For exaple, if H kx, X, with all X i priitive, the the atural actio of H o V H H. 3 a a vector pace. i locally ilpotet but it i ot locally fiite. If k i coutable, the for the ae H there exit a urjective hooorphi of algebra g: H kt, where kt i the quotiet field of the algebra of polyoial kt e.g., if kt 4 1,t 2,..., et g Xi Ttj whe i 2 j ad g Xi Ttj 1 whe i 2 j 1.. Hece the atural actio of H o HKer g a a vector pace. i iple but it i ot locally fiite. Oe ca eaily how that if H i fiitely geerated a a algebra, the both locally ilpotet ad eiiple actio are locally fiite. Oberve alo that if H i of fiite dieio a a vector pace., the all actio of H o vector pace are locally fiite. Whe a actio : H Ed V i locally fiite, the clearly all h are locally fiite edoorphi of the vector pace V. LEMMA 2.1. Let : H Ed V be a locally fiite actio of H o a ector pace V. The i eiiple rep. locally ilpotet. if ad oly if the liear ap h : V V i eiiple rep. locally ilpotet for each h H. Proof. Iplicatio i obviou. To prove iplicatio we ay aue that V i of fiite dieio. The H. ² h,...,h : 1 r. liear pa of h i. for oe hi H, i 1,...,r. Moreover, all h. i coute, becaue H i a coutative algebra. Now uppoe that all h,hh, are eiiple. The, a oe kow, there exit a bai,..., of V with h 1 i j kj for all i, j. It follow that all k j are ubodule of V,., which ake it clear that i eiiple. If all h:vv,hh, are locally ilpotet, the h i i 0 for oe, i 1,...,r, becaue V i fiite dieioal. Hece H. q. i 0 for q 1 r r 1, which how that the actio i locally ilpotet. The lea i proved. COROLLARY 2.2. Let : H Ed V be a locally fiite actio which i both eiiple ad locally ilpotet. The V. Proof. I view of the auptio, h,hh, are both diagoalizable ad locally ilpotet edoorphi of the vector pace V. It follow that h0for h H, whece hhh1. h1. H H h. 1. h. Id, h H, a wa to be how. H EFINITION 2.3. Give actio, : H Ed V, we defie the ap 1 ad : H Ed V a 1 S, Ý h h h.

6 128 ANRZEJ TYC Reark. i the covolutio. product of ad i the covolutio algebra Ho H, Ed V. ee 8, pp k. Sice the alge- bra Ho H, Ed V. k i aociative, operatio o actio i aocia- tive. We ay that the actio ad a above. coute if hg g. h. for all h, g H i geeral, it doe ot ea that ad coute i Ho H, Ed V.. k. Note that the trivial actio V: H Ed V coute with ay other actio T: H Ed V. The followig lea i a iple coequece of the correpodig defiitio ad it proof i left to the reader. LEMMA 2.4. ector pace V: Let, : H Ed V be coutig actio of H o a. 1 1 i a actio.. 2 i a actio V ad V V.. 4 If H i cocoutatie, the. LEMMA 2.5. Let, T: H Ed V be locally fiite, coutig actio of Hoaector pace V:. 1 1 The actio ad T are locally fiite If ad T are eiiple rep. locally ilpotet, the ad T are alo eiiple rep. locally ilpotet.. Proof. Local fiitee of 1 S i obviou. To prove local fiitee of T take a V ad chooe 1,...,r V i uch a way that T H. ²,..., : 1 r. Sice i alo locally fiite, the vector pace W H. H. 1 r i a fiite dieioal ubodule of the H-odule V,.. Hece, for ay h H, T. h. Ýh Th W, which iplie that T. H i of fiite dieio. Thi prove. 1. For. 2, firt aue that ad T are eiiple ad oberve that, i view of Corollary 1.2.1, T. h. Ýh Th i a eiiple edoorphi of the vector pace V for each h H. Fro Lea 2.1 it follow that the actio T i eiiple. The ae Lea 2.1 give eiiplicity of 1. If ad T are locally ilpotet, the iilar arguet how that 1 ad T are locally ilpotet. Thi coplete the proof of the lea. THEOREM 2.6 JordaChevalley decopoitio for locally fiite actio of H o vector pace.. Let : H Ed V be a locally fiite actio of H o

7 JORANCHEVALLEY ECOMPOSITION 129 a ector pace V:. 1 There exit uique locally fiite actio, : H Ed V atify- ig the coditio:, ad coute, i eiiple, ad i locally ilpotet.. 2 If a actio T: H Ed V coute with, the T coute with ad.. 3 If U W are ubpace of V uch that h. W. U for all hh, the h W U ad h W U for all h H. The decopoitio i called the JordaChealley deco- poitio of the actio. The actio ad are called the eiiple copoet of ad the locally ilpotet copoet of, repectively. Proof. Let u defie the ap : H Ed V by h. h for hh. The accordig to Propoitio 1.1 ad Corollary 1.2.1, i a locally fiite actio of H o V. Furtherore, fro Lea 2.1 it reult that i eiiple. So, if we et. 1, that i, h. Ý Sh. h , the Lea 2.5 ay that i a locally fiite actio of H o V. We how that i locally ilpotet. For GH*, let V V; h. h. 4 h H. The V GH*. V, becaue the actio i diagoalizable. Further, a ad coute, V i a ubodule of the H-odule V,. for each GH*, which i tur iplie that V i a ubodule of V,.. Therefore, i order to prove that i locally ilpotet it uffice to verify that the actio : HEd V with h. h. i locally ilpotet for every GH*. To thi ed fix a GH* ad defie the ap S : H H by S h. Ý Sh. 1. h 2.. The oe iply check that S i a hooor-. phi of algebra uch that S H H. Moreover, for h H ad V we have Ý Ý h.. Sh h Sh h Ý Sh h h Ý Ý Sh h. Sh h Ý Ý Ý. Ý Sh h Sh h Sh h Sh h S h.., becaue ÝSh h h. 0 ad Ý Sh. h. Ý Sh. h S h.., by Corollary Hece h. S h.., which, to-

8 130 ANRZEJ TYC gether with Lea 2.1, yield that the actio : H Ed V i locally ilpotet. Thu it ha bee proved that the actio i locally ilpotet. Fro Propoitio we kow that ad coute, ad fro Lea it follow that Part 2. V of the theore i a coequece of Propoitio For part. 3, aue that U W are ubpace of V uch that h W Ufor all h H. The h W U for h H, by Propoitio Hece h U. U ad U. U for all all h H, becaue H H k1 H. Uig thee icluio we verify that. give a h H, h W U.AH HHHH, oe ca write h Ýi1 ti h i, where h 1,...,h r,t r1,...,th for oe r,1r. Now applyig the coutativity of with, Propoitio 1.1.3, ad the above icluio, we get r Ý i i Ý i i i1 ir1 r Ý i. Ý i. i1 ir1 h. W. St. h. W. h. St. W. St U h U UUU. It reai to etablih the uiquee aertio i. 1. Suppoe that. i aother uch a decopoitio, o oe ha 1 1. By part. 2, all actio i ight coute, o that fro Lea we derive that rep.. 1. i a eiiple rep. locally ilpotet. actio of H o V. But, i view of Corollary 2.2, V i the uique locally fiite, eiiple, ad locally ilpotet actio of H o the vector pace V. Coequetly,. 1 V 1, which ea that ad, by Lea The theore i proved. COROLLARY 2.7. If H i cocoutatie ad T, F: H Ed V are locally fiite, coutig actio of H o V, the T F i a locally fiite actio of H o V with T F. T F ad T F. T F. Proof. The corollary i a coequece of part. 1 of the theore, Lea 2.4.4, ad Lea 2.5. Give a actio T: H Ed W of H o a vector pace W, the pace w W, h H T h. w. h. w4 i called the pace of iariat of T ad it i deoted by W T.IfTi locally fiite, the for each GH* we defie the vector pace Oberve that W W T. W w W ; h H T h. w. h. w 4.

9 JORANCHEVALLEY ECOMPOSITION 131 COROLLARY 2.8. Let : H Ed V be a locally fiite actio of H o a ector pace V. The. 1 VGH*. V.. 2 V V, hhid.. 04 h H for all GH*,. 3 if F: H Ed U i a arbitrary locally fiite actio of H o a ector pace U ad p: V,. U, F. i a hooorphi of H-odule, the p: V,. U, F. i alo a hooorphi of H-odule. Moreoer,. F if p i urjectie, the p V U. Proof. Part. 1 i true, becaue the actio i diagoalizable. Part. 2 i a coequece of Corollary For part. 3, part. 2 iplie that pv U,GH*, whece, i virtue of 1., p h. F h. p for ay hh. Thi ea that p: V,. U, F. i a hooorphi of H- F odule. If p i urjective, the the fact that V V, U U, part. 1, ad the above icluio pv U give together the equality pv. F U ad eve pv U for all GH*.. COROLLARY 2.9. Let : H Ed V be a locally fiite actio of H o a ector pace V. The V i a ubodule of the H-odule V,. ad. V V, where : H Ed V. i the retrictio of. Moreoer, the actio i eiiple. Siilarly, V i a ubodule of the H-odule. V, ad V V, where : H Ed V i the retrictio of. Moreoer, the actio i locally ilpotet.. Proof. For each h H, h V V, becaue ad co-. ute. The equality V V i a coequece of part. 3 of the theore applied to W V ad U 0. Seiiplicity of i obviou. The proof of the ecod part of the corollary i aalogou. EXAMPLE If H kx with X priitive, the each actio of H o a vector pace V i of the for f., where f Ed V ad. f. f a X a f. It i clear that V Ker f ad that f. i locally fiite rep. eiiple, locally ilpotet. if ad oly if the edoorphi f i locally fiite rep. eiiple, locally ilpotet.. For coutig f, ged V, f. g. fg., f. 1 f., ad the above theore aout to Propoitio EXAMPLE If H k X, X with X X X, S X X, X. 1, the every actio of H o a vector pace V i of the for G f., where f Aut V ad G f a X, X a f, f.. It i obviou that G f. V V, f 4 ad that G f. i locally fiite rep. eiiple. if ad oly if f i locally fiite rep. eiiple.. Moreover, the actio G f. i locally ilpotet if ad oly if f i a uipotet autoor-

10 132 ANRZEJ TYC phi of V, that i, if f IdV i a locally ilpotet edoorphi of V. Give coutig f, g Aut V, G f. G g. G fg. ad G f. 1 Gf 1.. I thi cae the theore ay that a locally fiite f Aut V poee a uique decopoitio f f f, where f i a eiiple autoorphi of V, f i a uipotet autoorphi of V, ad ffff. I particular, if V k, the thi i the well-kow ultiplicative Jorda decopoitio of f ee. 3. EXAMPLE Let H kx, X,..., X 1, with X. o 1 o Ý X X, X. Kroecker delta. ij i j, o for 0, ad with S X. uiquely deteried by the equalitie Ý S X. i ij i Xj0, 0. The every actio of H o a vector pace V i of the for. f, where f i a equece f. 0of coutig edoorphi of V with fo Id ad f:. HEd V i deteried by. f X. i f i, i 0. It i clear that f. V V, f 0, i 0. 4 If h ad g. coute i i.e., if hggh for all i, j., the h. g. b. i j j i, where b Ýijhg. i j Theore 2.6 tate i thi cae that each locally fiite actio. f ha a uique decopoitio. f b. g., where b. i j k j for a certai bai of V, i 0, 1,..., all g are locally ilpotet, ad b., g. j i coute. Reark Uig 9, 9.1, oe ca iediately prove that Theore 2.6 hold, ot oly for algebraically cloed field, but alo for all perfect field. 3. JORANCHEVALLEY ECOMPOSITION FOR LOCALLY FINITE ACTIONS OF H ON ALGEBRAS Let u recall that a actio of H o a algebra A i a actio : H Ed A of H o the uderlyig vector pace A of A uch that h 1. h1 ad h xy. Ýh xh. y. A A for all h H ad x, y A i.e., A, together with, ia H-odule algebra; ee. 8. Oberve that for H fro Exaple 2.10 each actio of H o A i of the for d, where d i a derivatio of A. If Hi a i Exaple 2.11, the each actio of H o A i of the for G g., where g i a autoorphi of the algebra A. Whe the Hopf algebra H coe fro Exaple 2.12, every actio of H o A i of the for d., where d i a higher derivatio of A. If i a actio of H o a algebra A, the oe eaily check that the vector pace of ivariat A a A; h aha. 4 h H i a ubalgebra of A which i called the algebra if iariat of i particular, o it i for actio of H fro Exaple A actio of H o a algebra A i called locally fiite rep. eiiple, locally ilpotet. if,

11 JORANCHEVALLEY ECOMPOSITION 133 a a actio of H o A, i locally fiite rep. eiiple, locally ilpotet.. Now uppoe that : H Ed A i a locally fiite actio of H o a algebra A, ad let be the JordaChevalley decopoitio of a a locally fiite actio of H o A.. I thi ituatio of iportace i the followig. Quetio. Are ad alo actio of H o the algebra A? THEOREM 3.1. Let : H Ed A be a locally fiite actio of H o a algebra A ad let be the JordaChealley decopoitio of. The i a actio of H o the algebra A. Furtherore, if H i cocouta- tie, the i alo a actio of H o A. For the proof of thi theore we eed everal lea. I what follow A will deote a algebra. If T: H Ed A i a fixed actio of H o A ad h H, the h: A A will tad for the liear ap T h.. LEMMA 3.2. If the Hopf algebra H i cocoutatie ad R, T: H 1 Ed A are coutig actio of H o the algebra A, the R ad RT are actio of H o A. Proof. Fro Lea 2.4 we kow that R 1 ad RT are actio of H o A. Sice H i cocoutative, S hýsh. 1. Sh2. for each h H, 1 whece give x, y A, R h. xy. RSh xy. ÝRSh x RSh y ÝR h x R h y. Furtherore, R h A 1 RSh 1A Sh 1A h 1 A, becaue S. Thi how that R i a actio of H o A. A for RT, agai akig ue of the cocoutativity of H we have Ý RT. h. xy. R h T h xy.. Ý R h T h x. T h y. Ý R h T h x. R h T h y. Ý R h T h x. R h T h y. Ý RT. h x. RT. h y.. Moreover, RT. h. 1. ÝRh Th 1. h1 A A A, becaue 1A A T A R. Hece RT i alo a actio of H o A, ad the lea i proved.

12 134 ANRZEJ TYC LEMMA 3.3. Let T: H Ed A be a actio of H o A, ad let for a gie GH*, ½ h H. 5 A aa; h h Id a 0. The A A A for all, GH*,ad 1 A. A Proof. Fix, GH*,hH with h Ýi1hi t i, ad x, y A. Firt, by iductio o, we prove that for all 1, Ý ž /. h. h. Id xy. L.. hold, where L Ý1i,...,i i...i fi fi x g i... gi t i t y ad h, f h h Id, g t t Id a uual. i i i i i i i i i!.!!.. Sice T i a actio, h. h. Id. xy. Ý h x. i1 i t y. h. t. xy. Ý h x. xt. y. xt y. t. y. i i i i1 i i i i i i i Ý f x t y xg y L L. Thi prove forula i1 i i i i o 1 for 1. Suppoe that it hold for oe 1. The h. 1 h. Id. xy. h. h. Id. Ý L. Ý 0 0 h.. h Id L.. Furtherore, fro forula for 1 ad the fact that all f i, g j, ad t coute, oe get r h h Id. L h h. Id Ý i1 i 1i,...,i 1 ž i1 i i1 i i1 i / f... f x. g... g t...t y. Ý i1 i i1 i j i1 i i1 i j 1i 1,...,i, j... f... f f x. g... g t...tt y. Ý i1 i j i1 i i1 i j i1 i 1i 1,...,i, j... f... f x. g... g g t...t y. Ý i2 i1 i1 i1 i2 i1 i1 i1 1i 1,...,i1... f... f x. g... g t...t y. Ý i1 i1 i1 i i1 i1 i1 i 1i 1,...,i1... f... f x. g... g t...t y..

13 JORANCHEVALLEY ECOMPOSITION 135 Hece, 1 h. h. Id xy. ž / 2 Ý... f... f x Ý i i i i i 1,...,i1 g... g t...t y. i2 i1 i1 i1 ž / 1 Ý... f... f x Ý i i i i i 1,...,i1 g... g t...t y. 1 i1 i1 i1 i ž / r ž / r1 Ý Ý i...i f i... fi x. r1 1 1 r r0 1i 1,...,i1 g... g t...t y. 1 ir1 i1 i1 ir ž / 1 1. Ý L r. r r0 Thu forula i how. Now, to prove the aertio of the lea, take x A, y A for oe, GH* ad h H with h. Ý h t. The there exit a q uch that f q x. 0g q y. i1 i i i i, i 1,...,, where, a above, f h h. Id ad g t t. i i i j j j Id. Apply- ig forula to 2 q 1. 1 we obtai h. h. Id xy. ž / 1 Ý... f... f x. g... g t...t y Ý i i i i i i i i ž / 1 1i,...,i 1 Ý Ý i...i f i... fi x. t i...t i 0 1i,...,i 1 g... g y. 0, i1 i 1 1 becaue each product i thi u cotai a factor of the for f q x. i or q gj y.. Coequetly, xy A, which prove that A AA. Sice h h for all h H, 1 A, a wa to be how. A A Reark. Forula i a geeralizatio of forula fro Lea B i 2, 4.2.

14 136 ANRZEJ TYC LEMMA 3.4. Let : H Ed A be a locally fiite actio of H o the algebra A ad, a i Sectio 2, gie a GH*,let A a A: h H h. a. ha.the. 4 we hae. 1 A A for all GH*.. 2 A A ad A A A for, GH*. GH*.. 3 AA ad 1 A. A Proof. Part. 1 i a coequece of Lea Part. 2 we derive fro Lea 2.8.1, part. 1, ad Lea 3.3. A for part. 3, the equality A A i obviou, ad the relatio 1 A A i a coequece of the equality A A A ad Lea 3.3. The lea follow. Havig the above lea we ca prove the theore. Proof of Theore 3.1. Oberve that h. 1. h. 1. A A, becaue fro Lea we kow that 1 A A.If xa, ya for oe, GH*, the i view of Lea 3.4.2, xy A, whece h. xy. Ý h. x. h. y. Ý h. x. h. y for all h H. Sice A A, agai by Lea it follow that i a actio of H o the algebra A. Now aue that H i cocoutative. The, i virtue of Lea 3.2,. 1 i a actio of H o A. Moreover, 1 coute with. 1 Agai akig ue of Lea 3.2, we ee that i a actio of H o A. Thi coplete the proof of the theore. Reark 3.5. Theore 3.1 applied to the Hopf algebra H fro Exaple 2.10 tate that for each locally fiite derivatio d: A A with the additive. JordaChevalley decopoitio d d d a a locally fiite edoorphi of A., d ad d are derivatio of A. I the cae where the field k i of characteritic 0, thi ha bee proved i 10. If H i a i Exaple 2.11, the fro Theore 3.1 we derive that for each locally fiite autoorphi g of the algebra A with the ultiplicative Jorda Chevalley decopoitio g g g a a locally fiite autoorphi of A., g, g are autoorphi of A. Whe H coe fro Exaple 2.12, Theore 3.1 ay that if d d. i i a locally fiite higher derivatio of A with coutig d ad d. d. d. i i the JordaChevalley decopoitio of d. a a actio of H o A., the d, d are higher derivatio of A. Fro ow o, it i aued that all rig uder coideratio are coutative. Below we hall eed the followig which are kow ad eay to prove.. LEMMA 3.6. Suppoe that B i a ubrig of a oetheria rig Q ad that there exit a B-liear ap R: Q B uch that R b. b for all b B. The B i a oetheria rig, too.

15 JORANCHEVALLEY ECOMPOSITION 137 THEOREM 3.7. If the algebra A i oetheria ad : H Ed A i a locally fiite, eiiple actio of H o A, the A i a oetheria algebra. Proof. Fro Lea 3.4 we kow that A GH*. A, ad A A A for, GH*, where A a A; h aha.. 4 h H Moreover, A A. Let A A. Hece, A A A ad A A A. Thi ea that if we defie R: A A to be the atural projectio, the R i a A -liear ap with Raafor all a A. The theore ow follow fro Lea 3.6. COROLLARY 3.8. Suppoe that the group G H*. i abelia ad that the algebra A i fiitely geerated. The for ay locally fiite ad eiiple actio : H Ed A of H o A, the algebra of iariat A i alo fiitely geerated. Proof. Sice A G H *. A, there are a 1,...,a i A uch that Aka,...,a ad a A for oe GH*, 1 i i i1,...,. Let i PkX,..., X 1 ad let be the actio of H o P uiquely deteried by the coditio h. X. h. X, i1,..., i i i uch a actio exit, becaue the group GH* i abelia.. The i clearly eiiple ad preerve the atural gradig give by degree. i P. Furtherore, if p: PA i the hooorphi of algebra defied by p X. i a i, i 1,...,, the p i a urjective hooorphi of the correpodig H-odule. Now fro Corollary oe get that A pp.. Therefore, it reai to prove that the algebra P i fiitely geerated. Accordig to the above theore, P i a oetheria algebra. O the other had, P i a graded ubalgebra of P. The cocluio i that P i fiitely geerated, a wa to be how. For H kx, X-priitive, ad k, the corollary i cotaied i 10. If : H Ed A i a arbitrary actio of H o A, the a iple calculatio how that A a A: h a0a. 4 o h H i a ubalgebra of A. Moreover, if i locally fiite, the A A o A A, by Corollary 3.4. Coequetly, Theore 3.7 ad Corollary 3.8 yield the followig. COROLLARY 3.9. Let A be oetheria ad let be a locally fiite actio of H o the algebra A. The A A o i a oetheria algebra. Furtherore, if the group G H*. i abelia, the Ao i fiitely geerated, wheeer A i alo. Aother coequece of Theore 3.7 ad Corollary 3.8 i the followig. THEOREM Suppoe that H i cocoutatie ad i a locally fiite actio of H o the algebra A. The A i a ubalgebra i A ad the algebra

16 138 ANRZEJ TYC A i a oetheria rep. fiitely geerated, proided A i oetheria rep. fiitely geerated.. Proof. By the ecod part of Theore 3.1, i a actio of H o the algebra A. Hece A i a ubalgebra i A. I view of Corollary 2.9 ad. the firt part of Theore 3.1, A A, where i a locally fiite, eiiple actio of H o the algebra A. The theore ow follow fro Theore 3.7 ad Corollary 3.8. I the corollarie below A deote the algebra of polyoial kx,..., X for oe 0. 1, ad M deote the et of all t atrice over k. If t tji M, the d : A A will deote the deriva- t j tio of A deteried by d X Ý t X, i1,...,. Whe t t. i j1 ji j ji t GL, k ivertible atrice i M, the g : A A will deote the t j autoorphi of A deteried by g X Ý t X, i1,...,. i j1 ji j t COROLLARY For each atrix t M ad d d, the algebra of cotat A d Ker d i fiitely geerated. Proof. Whe char k p 0, the the corollary i obviou. So, let char k 0 ad let t t t, t, t M, be the JordaChevalley de- copoitio of t a a edoorphi of V k i the tadard bai.. The oe eaily how that d d t d t d t i the JordaChevalley t. t decopoitio of d: A A, which ea that d d d. i the JordaChevalley decopoitio of the actio d:hed A, where H i the Hopf algebra fro Exaple I view of the above theore ad the equality A d. A d, it follow that for the proof of the t corollary we ca aue that d d or, equivaletly, that the atrix t i.. ilpotet. But i thi cae, a Ýi0 d a!, k, aa,ia liear algebraic actio of the algebraic group k o A uch that A d k A a A, a a, k 4. The cocluio ow follow, uig the claical Witzebock theore 4, 6. t COROLLARY For each atrix t GL, k ad g g : A A, g the algebra A a A, g a. a4 i fiitely geerated. Proof. Proceedig iilarly a above with H fro Exaple 2.10 replaced by H fro Exaple 2.11., oe ca aue that the atrix t i p uipotet. If char k p 0, the t I idetity atrix. for oe, p tt t t I whece g Id, becaue g g g for t, t GL, k ad g Id. Thi ea that cyclic ubgroup G of Aut A geerated by g i fiite. g G Moreover, it i clear that A A a A; g a. a, g G 4 i i. Uig, g for exaple, 7, Propoitio ad Theore 2.4.9, it follow that A i fiitely geerated.

17 JORANCHEVALLEY ECOMPOSITION 139 Now uppoe that char k 0, ad defie the ap d: A A by forula j dlog g Ý gid. j j j1 a g Id i locally ilpotet, d i well defied.. It i ot difficult to prove ee, e.g., the proof of Theore 4 i. 5 that d i a derivatio of A, ad that A g Ker d. Moreover, d d t for oe ilpotet. atrix t GL, k.. Makig ue of Corollary 3.11, we agai coclude that A g Ker d i a fiitely geerated algebra. Thi coplete the proof of the corollary. Reark Applyig 1, Theore 1.5 itead of Propoitio 1.1, oe ca prove couterpart of Theore 2.6, 3.1, ad 3.7 for actio of H o profiite locally copact. vector pace ad for correpodig actio of H o profiite algebra. REFERENCES 1. R. Gerard ad A. H. M. Levelt, Sur le Coexio a Sigularite ` Reguliere ` da le ca de Pluieur Variable, Fukcjal. Ekac , J. E. Huphrey, Itroductio to Lie Algebra ad Repreetatio Theory, Graduate Text i Math., Vol. 9, Spriger-Verlag, New YorkBerli, J. E. Huphrey, Liear Algebraic Group, Spriger-Verlag, New YorkBerli, M. Nagata, Lecture o the Fourteeth Proble of Hilbert, Bobay, Tata Ititute of Fudaetal Reearch, C. Praaga, Iteratio ad logarith of foral autoorphi, Aequatioe Math , C. S. Sehadri, O a theore of Weitzebock i ivariat theory, J. Math. Kyoto Ui , T. A. Spriger, Ivariat Theory, I Lecture Note i Math., Vol. 585, Spriger- Verlag, New YorkBerli, M. E. Sweedler, Hopf Algebra, Bejai, New York, W. C. Waterhoue, Itroductio to Affie Group Schee, Graduate Text i Math., Vol. 66, Spriger-Verlag, New YorkBerli, V.. Zurkowki, Locally fiite derivatio, preprit, 1993.