ON POINTED HOPF ALGEBRAS OF DIMENSION 2n

Transcription

1 ON POINTED HOPF ALGEBRAS OF DIMENSION 2n STEFAAN CAENEPEEL AND SORIN DA SCA LESCU ABSTRACT We give a structure theorem for pointed Hopf algebras of dimension 2n having coradical kc where k is an algebraically closed field of characteristic zero 1 Introduction Let k be an algebraically closed field of characteristic zero A classical problem is to classify all finite-dimensional Hopf algebras of a given dimension over k Basically apart from some small dimensions there is only one case where this classification has been completed that is the situation where the Hopf algebra is of prime dimension (see [10]) Recently some attempts to classify pointed Hopf algebras have been made For dimension p the classification is well-known and in fact goes back to [6]: a pdimensional pointed Hopf algebra is either a group algebra or a Taft algebra Pointed Hopf algebras of dimension pn with coradical the group algebra of an abelian group of order pn were classified in [2] and the proof was essentially based on the Taft Wilson Theorem In [3] the classification of pointed Hopf algebras of dimension p was completed by giving a description of all pointed Hopf algebras of dimension p with coradical kc p In this paper we give a classification of all pointed Hopf algebras of dimension 2n with coradical kc It turns out that there is only one such Hopf algebra which we denote by E(n1) generated by a grouplike element c and n1 (c 1)-primitives all anticommuting This Hopf algebra can be obtained by starting with kc adding indeterminates by repeated Ore extensions and then factoring out a certain Hopf ideal (see [1]) Note that E(1) is the non-commutative non-cocommutative fourdimensional Hopf algebra of Sweedler and E(2) is a unimodular ribbon Hopf algebra that was defined in [9] and then used in [4] for computing Kauffman s invariant for knots and Henning s invariant for 3-manifolds As in the previous publications [2] and [3] our major tool is the Taft Wilson Theorem For further investigations of properties of pointed Hopf algebras of dimensions 2n we refer to the forthcoming papers [7] and [8] Throughout k is an algebraically closed field of characteristic 0 We use the notation of [5] If H is a Hopf algebra then G(H) will denote the group of grouplike elements of H and H H will denote the coradical filtration of H We call H Received 14 November Mathematics Subject Classification 16W30 Research supported by the FWO research network WO01196N and the project Hopf algebras and co-galois theory of the Flemish Community Bull London Math Soc 31 (1999) 17 24

2 STEFAAN CAENEPEEL AND SORIN DA SCA LESCU pointed if H kg(h) If g h G(H) then P gh x H (x) xghx is the set of (g h)-primitive elements If H is finite-dimensional then there are no non-zero (1 1)-primitive elements For g h G(H) let P gh be a complement of k(gh) inp gh We shall need the version of the Taft Wilson Theorem proved in [5 Theorem 541] THEOREM 1 Let H be a pointed Hopf algebra Then (1) H H ( gh G(H) P gh ); (2) for any n 1 and c H n there exist (c gh ) gh G(H) in H and w H n H n such that c gh G(H) c gh and (c gh ) c gh ghc gh w The first part of the theorem shows that G(H) is not trivial if H is pointed and finite-dimensional 2 Main result Let P i i i s be a non-empty ordered set of integers such that i i i s and let F i j i jr be a subset of P Then we define S(F P) 2 3 ( j j r )r(r1)2 if F i j i jr 0 if F We also set S( ) 0 LEMMA 1 Let F be a totally ordered finite set such that 2 F and let i and j i j be elements of F Then (1)S(F ij F i )(1)S(F i F) (1)S(F ij F j )(1)S(F j F) (21) Proof Write F i i l i i k j i n It is easy to compute that S(Fi F) (1n)l(1(n1)) nl S(F j F) nk S(Fi j Fi ) (n1)(k1) S(Fi j F j ) (n1)l and the integer S(Fi j Fi )S(Fi F)S(Fi j F j )SF j F) 1 is odd so (21) follows Let E(n) be the Hopf algebra with generators c x x n subject to the relations c 1 x i 0 c c x j x j (c) cc ( ) c 1

3 ON POINTED HOPF ALGEBRAS OF DIMENSION 2n 19 E(n) is a pointed Hopf algebra with coradical kc and it has dimension 2n+ We note that E(1) is Sweedler s four-dimensional non-commutative non-cocommutative Hopf algebra For P i i i s 1 n such that i i i s we shall write x P s IfP then x 1 The set cjx P P 1 n j 0 1 is a basis of E(n) The integers S(F P) are very useful for finding an elegant formula for (x P ) LEMMA 2 For any subset P of 1 n we hae (x P ) (1)S(FP)c F x P F x F (22) F P Proof We proceed by induction on m P For m 0 the statement is obvious Assume that the formula is valid for P m and take a set P with m1 elements Let i be the greatest element of P Then (x P ) (x P i ) ( ) 0 F P i (1)S(FP i )c F x P i F x F1 (c 1) (1)S(FP i )+ P i F c F +x P i F x F i F P i (1)S(FP i ) c F x P F x F F P i It is easy to see that for F Pi we have S(F Pi)PiF S(Fi P) and S(F Pi) S(F P) proving the required formula for P From now on H will be a pointed Hopf algebra of dimension 2n for some integer n 1 with Corad(H) kc Let us first show that H contains a copy of Sweedler s four-dimensional Hopf algebra LEMMA 3 H contains a Hopf subalgebra that is isomorphic to E(1) Proof Using the Taft Wilson Theorem we see that P k(c1) Clearly c P is invariant under the conjugation by c So we have a linear map φ: P P c c c φ(y) cyc for any y P Since φ Id P has a basis of eigenvectors for φ Thus c c φ has an eigenvector x P kc If the corresponding eigenvalue of s 1 then the c Hopf subalgebra of H generated by c and x has dimension 4 and is commutative thus it is a group algebra and this contradicts the fact that Corad(H) kc The only remaining possibility is that the corresponding eigenvalue of s 1 and thus cx x c Now (x ) cx x 1 and this implies that x is (1 1)-primitive and equal to 0 The Hopf subalgebra generated by c and x is then isomorphic to E(1)

4 20 STEFAAN CAENEPEEL AND SORIN DA SCA LESCU LEMMA 4 Suppose that the Hopf algebra H contains E(d) as a Hopf subalgebra for some d Let a H g G(H) and let m n be an integer such that for some ip H Then (a) gaa1 ip cix P i= P m (a ) g(a )(a )1 x P P (23) P m and for any non-empty P we hae that ( ) g c P (1)S(P P) c P x P P P P P P (24) P P P m equals Proof We have that ( I) (a) ggaga1a11 ip cix P 1 ( ip )cix P i= P m i= P m (I ) (a) ggaga1a11 g ip cix P (1)S(FP) ci+ F ip x P F cix F i= P m i= P m F P Looking at the terms with 1 on the third tensor position we see that ip cix P ( i= ) g x P m P P (25) P m Then (23) follows after we replace ip cix P i= P m by (a)gaa1 in (25) Fixing a non-empty P and looking at the terms with x P on the third tensor position in (25) (24) follows LEMMA 5 Suppose that H contains a Hopf subalgebra E isomorphic to E(d) for some d n Then H E Proof Suppose that H E Then there exists 1 m d such that H m E m and H m+ E m+

5 ON POINTED HOPF ALGEBRAS OF DIMENSION 2n 21 Case 1 Assume that m 1 Pick some h H m+ E m+ and write h uv G(H) h uv as in the Taft Wilson Theorem Pick some h uv H m+ E m+ Denoting g u we have that a u h uv H m+ E m+ and (a) gaa1 ip cix P i= P m for some ip E m Using Lemma 4 we see that b a H m+ E m+ and changing the notation from P to P we have (b) gbb1 P x P (26) P m with P E m Fix a subset P of 1 d containing m elements Lemma 4 shows that ( P ) g P P cm If g cm then P is a (cm cm)-primitive and is equal to zero In this case (b) H HHH m which shows that b H m a contradiction The only possibility that remains is that g cm+ In this case P is a (cm cm+)- primitive and since H E this means that P β P (cm+cm) d α Pi cm i= for some scalars β P α Pi We claim that α Pi 0 for any i P Indeed let P Pi a set containing m1 elements Using Lemma 4 we see that ( P ) cm+ P P cm (1)S(P P) P cm x P P P P P =m cm+ P P cm (1)S(P P j ) P cm x j j (27) j P If α Pi 0 then in the right-hand side of (27) the term cm cm appears with a non-zero coefficient But P H E so P is a linear combination of ct ctx u ctx u x v for t 0 1 and u Since (ctx u x v ) ctctx u x v ctx u x v ctct+x v ctx u ct+x u ctx v (28) we see that no term of the form cm cm can appear in ( P ) providing a contradiction We conclude that for P m we can write P β P (cm+cm) α Pi cm (29) i P If m d (29) shows that P H and (26) implies that b H m which is a contradiction Let m d and let F 1 d be a set containing m1 elements Pick two elements i j F and write (24) for P Fi j :

6 22 STEFAAN CAENEPEEL AND SORIN DA SCA LESCU ( P ) cm+ P P cm t P (1)S(P P t ) P t cm x t cm+ P P cm t P (1)S(P P t )β P t (cm+cm)cm x t (1)S(P P t )α P t s c mx s cm x t t P s P t But P E so because of (28) the coefficient of cmx j cm in the last sum of the above equation must be the opposite of the coefficient of cm cm x j We obtain that (1)S(F ij F i )α F i i (1) S(F ij F j )α F 0 (210) j j Dividing (210) by (21) we find that where γ F does not depend on i F Now we have that α F i i (1)S(F i F) α F j j (1)S(F j F) γ F γ F (x F ) (1)S(GF)γ F c G x F G x G F =m+ F =m+ G F (1)S(F i F)γ F cm x F w i F =m+ i F α F i F =m+ i c m x F w i i F where w H HHH m On the other hand it follows from (29) that P x P α Pi cm x P w P =m P =m i P α F i F =m+ i c m x F w i i F for some w H HHH m Therefore we have that P =m P x P F =m+ γ F (x F ) H HHH m Looking at (26) we see that and this shows that 0 b γ F x F 1 H HHH m F =m+ b F =m+ γ F x F H m E m and we find that b E which is a contradiction

7 ON POINTED HOPF ALGEBRAS OF DIMENSION 2n 23 Case 2 If m 1 then proceeding as in Case 1 we can find some b H E such that (b) 1bb1 i i where i k(cc) j kcx j Since we can assume that ( )1 1 (c1) (b) 1bb1 α ij c x j (211) ij for some scalars α ij (subtract a linear combination of the from b) Then clearly bccb is a (c c)-primitive and bccb 0 Also for a fixed t we have that and (bx t ) cbx t bcx t α ij x j x t ij x t bbx t 1 α ij c x t x j ij (x t b) cx t bcbx t α ij x t x j ij x t bx t b1 α ij cx t x j ij showing that bx t x t b is (1 c)-primitive thus it is in H E Obviously bx t x t b E+ EKer(ε) Let K be the subalgebra of H generated by E and b Then (211) shows that K is a Hopf subalgebra of H Moreover the commutation rules between b c and the x t show that E+K KE+ which means that E is a normal Hopf subalgebra of K producing an extension of Hopf algebras E K KE+K Since in KE+K the classes of the x t are zero the class of c is 1 and the class of b is a (1 1)- primitive thus zero we see that KE+K k This implies that dim(e) dim(k) a contradiction If d 1 then Case 1 does not occur in the proof of Lemma 5 COROLLARY 1 With notation as in Lemma 5 there exists x d+ H E such that cx d+ x d+ c (x d+ ) cx d+ x d+ 1 and x d+ x d+ for any 1 i d Proof P has a basis of eigenvectors of φ We already know that c1 x x c d are eigenvectors for φ Lemma 5 shows us that dim(p ) dim(e ) d1 This c c means that we can find an eigenvector x d+ H E for φ such that (x d+ ) cx d+ x d+ 1 If the corresponding eigenvalue of x d+ is 1 then as in Lemma 3 the Hopf subalgebra of H generated by c and x d+ has dimension 4 and is commutative and this is not possible Therefore the corresponding eigenvalue of x d+ is 1 and thus cx d+ x d+ c

8 24 STEFAAN CAENEPEEL AND SORIN DA SCA LESCU Now we have ( x d+ x d+ ) ( x d+ x d+ )11( x d+ x d+ ) for any 1 i d therefore x d+ x d+ THEOREM 2 Let H be a pointed Hopf algebra of dimension 2n n 1 such that Corad(H) kc Then H E(n1) Proof We prove by induction on 1 d n1 that H contains a Hopf subalgebra isomorphic to E(d) If d 1 then this follows by Lemma 3 For the induction step if H contains some E(d) with d n1 then Corollary 1 shows that H also contains E(d1) and the proof is finished References 1 M BEATTIE SDA SCA LESCU and L GRU NENFELDER Constructing pointed Hopf algebras by Ore extensions preprint M BEATTIE SDA SCA LESCU and L GRU NENFELDER On pointed Hopf algebras of dimension pn Proc Amer Math Soc to appear 3 S CAENEPEEL and S DA SCA LESCU Pointed Hopf algebras of dimension p J Algebra to appear 4 S GELAKI On pointed ribbon Hopf algebras J Algebra 181 (1996) S MONTGOMERY Hopf algebras and their actions on rings (American Mathematical Society Providence RI 1993) 6 W D NICHOLS Bialgebras of type one Comm Algebra 6 (1978) F PANAITE and F VAN OYSTAEYEN Quasitriangular structures for some pointed Hopf algebras of dimension 2n preprint F PANAITE and F VAN OYSTAEYEN Clifford type algebras as cleft extensions for some Hopf algebras preprint D E RADFORD On Kauffman s knot invariants arising from finite-dimensional Hopf algebras Adances in Hopf algebras Lecture Notes in Pure and Appl Math 158 (ed J Bergen and S Montgomery Dekker New York 1994) Y ZHU Hopf algebras of prime dimension Internat Math Res Notices 1 (1994) Faculty of Applied Sciences Faculty of Mathematics University of Brussels VUB University of Bucharest Pleinlaan 2 Str Academiei 14 B-1050 Brussels RO Bucharest 1 Belgium Romania