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REPRESENTATIONS OF FINITE DIMENSIONAL HOPF ALGEBRAS Martin Lorenz

Department of Mathematics Temple University Philadelphia, PA 19122-6094 e-mail: [email protected]

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Abstract. Let denote a nite dimensional Hopf algebra with antipode over a eld . We give a new proof of the fact, due to Oberst and Schneider [OS], that is a symmetric algebra if and only if is unimodular and 2 is inner. If is involutory and not semisimple, then the dimensions of all projective -modules are shown to be divisible by char . In the case where is a splitting eld for , we give a formula for the rank of the Cartan matrix of , reduced mod char , in terms of an integral for . Explicit computations of the Cartan matrix, the ring structure of 0 ( ), and the structure of the principal indecomposable modules are carried out for certain speci c Hopf algebras, in particular for the restricted enveloping algebras of completely solvable -Lie algebras and of (2 ). H

S

S

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Introduction

This article is a study of representations of nite dimensional Hopf algebras H in the spirit of Larson's \Characters of Hopf Algebras" [L], but with the emphasis on the non-semisimple case. Thus particular attention is given to the properties of projective modules. To a large extent, we work inside the Grothendieck groups G0(H) and K0 (H) of the categories of nitely generated H-modules and nitely generated projective H-modules, respectively, and the various connections between G0(H) and K0 (H) are among our main focal points: The comultiplication of H causes G0(H) to be a ring and K0 (H) to be a module over G0(H); there is a canonical duality between K0 (H) and G0(H) which has a very natural interpretation in terms of Hattori-Stallings ranks (for K0 (H)) and ordinary characters (for G0(H)); and, of course, there is the Cartan map c : K0 (H) ! G0 (H). Here are the main results of the article. Throughout, H denotes a nite dimensional Hopf algebra over a eld |of characteristic p  0, and S is the antipode of H. Theorem 1. If H is involutory (that is, S 2 = Id) and not semisimple, then p divides the dimension of every projective H-module. The next result determines the rank of the map ec = Id c : | K0 (H) ! | G0(H). This is a lower bound for the rank of c, and the two ranks are identical for p = 0. We let C denote the Cartan matrix, that is, the matrix of the Cartan map with respect to the canonical Z-bases of G0(H) and K0 (H) that are aorded by the irreducible H-modules and their projective covers, respectively.

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1991 Mathematics Subject Classi cation. 16W30, 16G99. Key words and phrases. Finite dimensional Hopf algebra, character, Hattori-Stallings rank, Grothendieck group, Cartan matrix, projective module, principal indecomposable module, duality, trace function, orthogonality, symmetric algebra, group algebra, restricted enveloping algebra, p-Lie algebra. The author was supported in part by NSF Grant DMS{9400643. Typeset by AMS-TEX 1

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MARTIN LORENZ

Theorem 2. Suppose that k is a splitting eld for H. Then

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rank ec = dim (H / t) : Here t is any nonzero left integral of H and / denotes the right adjoint action of H on itself. Moreover, if S 2 is inner, then H is semisimple i C = Id i p does not divide det C. The article also contains explicit computations of the Cartan matrix C and of the ring structure of G0(H) for a number of speci c Hopf algebras H that are of interest, in particular for the restricted enveloping algebras of completely solvable p-Lie algebras and of sl(2; |). These algebras display features that contrast sharply with the classical case of nite group algebras: C is always singular, and the ring G0(H) is not semiprime for the restricted enveloping algebra of sl(2; |). A brief summary of the contents of the individual sections is as follows. x1 reviews the basic pertinent material on Grothendieck groups in the more general setting of nite dimensional associative algebras. This section is, to a large degree, a summary of parts of [Ba2], specialized to nite dimensional algebras. x2, on symmetry and dimensions, is independent of Grothendieck groups and is entirely based on a few simple observations about duality and traces. Theorem 1 is proved in x2.3 and, in x2.5, unimodularity of H is shown to be equivalent with self-duality of the projective cover of the \trivial" H-module. As a consequence, one obtains that H is a symmetric algebra if and only if H is unimodular and S 2 is inner. x3 takes up the material of x1 in the context of Hopf algebras and contains the proof of Theorem 2 (in x3.4). The main emphasis is on the ring and module structures that are now carried by the various objects of x1. This section also contains an analysis of the special case where the Jacobson radical of H is a Hopf ideal. x4 is devoted to a detailed discussion of some explicit examples: the Sweedler algebra H4, a class of algebras that were constructed by Radford [R3] (based on earlier examples due to Taft [T]), and the restricted enveloping algebras of completely solvable p-Lie algebras and of sl(2; |). In each case, the ring G0(H), its module K0 (H), the principal indecomposable modules, and the Cartan matrix are determined. The author would like to thank Susan Montgomery for her thorough reading of the rst version of this article which lead to a number of improvements, clari cations, and proper accreditations. In particular, she pointed out to us that the aforementioned characterization of symmetry was rst observed by Oberst and Schneider [OS]. The present proof is a simpli cation of our original argument which incorporates a suggestion of hers and H.-J. Schneider. After the rst version of the article was circulated, the author learned from Jim Humphreys that most of the features of the sl(2; |)example exhibited in x4.4 have previously been discovered by Pollack [Po] and have been rederived by Humphreys in [H]. There is also previous work of Humphreys on symmetry [H2]. Related material on representations of nite dimensional cocommutative Hopf algebras, phrased in the language of ( nite) algebraic groups, can be found in [V]. Notations and conventions. All algebras, Hopf and otherwise, considered in this article are nite dimensional over a commutative base eld |of characteristic p  0, and all modules are left modules and are assumed to be nite dimensional over |. Finally, stands for and :  = Hom ( : ; |) denotes the linear dual. Further assumptions will be explicitly stated at the beginning of each section.

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1. Background from the theory of finite dimensional algebras

Throughout this section, A will denote a ( nite dimensional) algebra over |and J = rad A is the Jacobson radical of A. We x a full set of nonisomorphic irreducible A-modules V1 ; : : : ; Vt and we let Pi = P(Vi) denote their projective covers, the principal indecomposable A-modules (cf. [CR], p. 131).

REPRESENTATIONS OF HOPF ALGEBRAS

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1.1 Grothendieck groups. Let G0(A) denote the Grothendieck group of the category of ( nite dimensional left) A-modules. This is the abelian group that is generated by the isomorphism classes [V ] of A-modules V modulo the relations [V ] = [U] + [W] for each short exact sequence of A-modules 0 ! U ! V ! W ! 0. It is a classical fact (cf. [Ba1], p. 404) that G0(A) is a free abelian group with basis given by the classes [Vi] (i = 1; : : : ; t). Similarly, K0 (A) denotes the Grothendieck group of the category of projective ( nite dimensional left) A-modules, that is, the abelian group that is generated by the isomorphismclasses [P] of projective A-modules P modulo the relations [P  Q] = [P] + [Q]. Again, K0 (A) is free abelian, with basis f[Pi]ji = 1; : : : ; tg. The two Grothendieck groups are related via the Cartan map c = cA : K0(A) ! G0 (A) ; [P] 7! [P] : The matrix C 2 Mt (Z) which represents c with respect to the above bases is called the Cartan matrix : C = (ci;j )tt with ci;j = multiplicity of Vj as composition factor of Pi . 1.2 Orthogonality. Putting h : ; : i : K0 (A)  G0(A) ! Z; hP; V i = dim HomA (P; V ) ; one obtains a well-de ned bilinear map (cf. [Ba2], x4.3; for simplicity, the parentheses [ : ] are omitted). Since Pi =JPi  = Vi , we have HomA (Pi ; Vj )  = HomA (Vi ; Vj ) which yields the orthogonality relations  0 if i 6= j hPi; Vj i = d if i = j ; where di = dim EndA (Vi ) : i If |is a splitting eld for A then all di = 1. De ning f : ; : g : K0(A)  K0 (A) ! Z; fa; bg = ha; c(b)i one obtains a bilinear form on K0 (A) whose matrix with respect to the given basis of K0 (A) is C 0 = (c0i;j )tt where c0i;j = di cj;i : P P Indeed, c([Pj ]) = ` cj;`[V` ] implies fPi ; Pj g = ` cj;`hPi ; V` i = c0i;j , by the orthogonality relations. 1.3 Trace spaces. Following [Ba2], we write T = TA : A ! T(A) = A=[A; A] for the natural projection to the quotient of A by the |-linear span [A; A] of the Lie commutators [a; b] = ab ? ba in A. We let [ : ] : A ! T(A)reg = A= ([A; A] + J)  = T(A=J) denote the canonical map. The space of |-valued trace functions on A is the |-subspace of A = Hom (A; |) that is de ned by C(A) = ff 2 A : f(ab) = f(ba) for all a; b 2 Ag  = T(A) : We de ne the space of regular |-valued trace functions by C(A)reg = ff 2 C(A) : f vanishes on J g  = C(A=J)  = T(A)reg : If |is a splitting eld for A and p = char |> 0, then [A; A]+J = fa 2 A : apn 2 [A; A] for some n  0g (e.g., [P], p. 56), and hence C(A)reg = ff 2 A : f(a) = 0 for all a 2 A with apn 2 [A; A] for some n  0g :

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1.4 Hattori-Stallings ranks. Let P be a projective A-module. The trace map is de ned by 

P T(A); = Hom (P; A) P ?! trP = trP=A : EndA (P) ?! A A

P (f v) = T(f(v))

P

(cf. [Ba2]). If f(fi ; vi)gn1  HomA (P; A)  P is a dual basis for P, that is, v = i fi (v)vi holds for all v 2 P, then X trP () = fi ((vi )) + [A; A] ( 2 EndA (P)) : i

The Hattori-Stallings rank map is de ned by (cf. [Ba2], x2.4) r : K0(A) ! T(A); [P] 7! rP = trP (IdP ) :

P

Explicitly, writing P = An
e for some idempotent matrix e = (ei;j ) 2 Mn (A), we have rP = T(ei;i ). In particular, if P is free of rank n over A then rP = T(n). Since each Pi has the form Pi = Aei for some primitive idempotent ei 2 A, the image of the Hattori-Stallings rank map is exactly the additive subgroup of A=[A; A] that is generated by the (primitive) idempotents of A. Finally, we de ne r T(A)  T(A) ;  : K0 (A) ?! reg

where the last map is the canonical epimorphism A=[A; A]  A= ([A; A] + J), and we put i = ([Pi ]) = [ei ] 2 T(A)reg

(i = 1; : : : ; t):

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1.5 Characters. Let V be an A-module and denote its structure map A ! End (V ) by a 7! aV (a 2 A). Then the character V of V is de ned by

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V (a) = trV= (aV ) 2 |

(a 2 A) :

The characters i = Vi are called the irreducible characters of A. Each character V is an element of C(A) and if 0 ! U ! V ! W ! 0 is a short exact sequence of A-modules then V = U + W . In particular, since the irreducible characters clearly belong to C(A)reg , so do all characters V , and the character map

 : G0(A) ! C(A)reg ; [V ] 7! V is a well-de ned group homomorphism which satis es V (1) = dim (V ). 1.6 Theorem. (a) The following diagrams commute.

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K0 (A)  G0(A)

? ? y

h:;:i ???? !

T(A)reg  C(A)reg ??????! evaluation

Z

cA ! G (A) K0 (A) ???? 0

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T(A)reg ????t! C(A)reg

?? ycan. and

?

?? y :

? y

(:)

Here, the map  is as in x1.4 and ( : )t : T(A)reg ! C(A)reg = T(A)reg is de ned by

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([a])t([b]) = trA= (Lb  Ra)

(a; b 2 A);

where Lb ; Ra 2 End (A) are left multiplication with b and right multiplication with a, respectively.

REPRESENTATIONS OF HOPF ALGEBRAS

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(b) The nonzero irreducible characters i 2 C(A)reg are linearly independent over |. If |is a splitting eld for A then the i (i = 1; : : : ; t) form a |-basis of C(A)reg and  induces an isomorphism of |-vector spaces = e = Id  : G^ 0 (A) = | G0(A) ?! C(A)reg : (c) If |is a splitting eld for A then the i (i = 1; : : : ; t) form a |-basis of T(A)reg and  induces an isomorphism of |-vector spaces

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| Z Proof. The diagrams in (a) are minor modi cations of [Ba2], 4.3 and 4.7, respectively. To see that ( : ) is a well-de ned map, note that tr | (L  R ) = 0 if a 2 J or a 2 [A; A], because L  R = R  L is a nilpotent endomorphism in the former case and a commutator of endomorphisms in the latter. Similarly for b in place of a. (b) Assume rst that is a splitting eld for A. Then all d = 1 inPthe orthogonality relations in x1.2, . Thus, if k  = 0 for k 2 then P and the rst diagram in (a) yields  ( ) =  1| 0 = k  ( ) = k for all j, thereby proving linear independence of the  's. Furthermore, the number, t, of non-isomorphic irreducible A-modules, equals the number of matrix components of A=J which in turn is equal to the dimension of T(A=J)  C(A) , and so the  = C(A) . Hence t = dim| span C(A) . 

= e = Id  : K^ 0(A) = | K0 (A) ?! T(A)reg :

A=

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For independence of the nonzero irreducible characters i in the general case, choose a eld extension e = K C(A) denote the character K=|so that K is a splitting eld for Ae = K A and let  ei 2 C(A) e e e of the A-module Vi = K Vi . So ei = 1 i. For i 6= j, the A-modules Vei and Vej have no common composition factor ([CR], p. 170, Exercise 7.9). Therefore, writing each ei in terms of irreducible e there is no overlap between the various i's. In view of the foregoing, this implies that characters of A, the nonzero ei 's are independent over K, and hence the nonzero i 's are independent over |. (c) The proof of (c) is dual to the rst part of the proof of (b).  1.7 Symmetric algebras. The |-algebra A is called symmetric if there exits a nondegenerate |bilinear form :AA !| which is associative and symmetric, that is, (ab; c) = (a; bc) and (a; b) = (b; a) holds for all a; b; c 2 A. We recall some well-known facts about symmetric algebras: (1) Group algebras of nite groups and all nite dimensional semisimple algebras are symmetric. (2) If A is symmetric then, for any irreducible A-module V , the socle of P(V ) is isomorphic to V . In particular, it follows that P(V  )  = P(V ) as (right) A-modules. (In case A is a Hopf algebra, the switch from left to right modules is unnecessary, because the side of the action is preserved for duals by means of the antipode. See x2.1 below.) (3) If A is symmetric then the matrix C 0 = (c0i;j ) of x1.2 is symmetric and, consequently, the form f : ; : g is symmetric. Facts (1) and (2) can be found in [CR], p. 198, and (3) is [La], Theorem 9.8 (at least for |large enough; the general case follows along the same lines). Part (b) of the following Lemma, apparently a well known fact, has been pointed out to us by S. Montgomery and H.-J. Schneider who also provided the proof given below. It is included here for lack of a suitable reference. Lemma. Let A is a symmetric algebra with form . Then: (a) ([a; b]; c) = (a; [b; c]) holds for all a; b; c 2 A. In particular, C(A)  = Z(A), the center of A, and C(A)reg  = annZ (A) (J) as |-vector spaces.

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(b) If : A  A ! |is any nondegenerate bilinear form which is associative then there exists a unit u 2 A such that (a; b) = (u?1 bu; a) holds for all a; b 2 A. Proof. (a) First, (ba; c) = (c; ba) = (cb; a) = (a; cb) and so ([a; b]; c) = (ab; c) ? (ba; c) = (a; bc) ? (a; cb) = (a; [b; c]). Thus, a 2 Z(A) if and only if (a; : ) vanishes on [A; A], whence the rst isomorphism follows. For the second isomorphism, observe that (a; J) = 0 is equivalent with (aJ; A) = 0, and hence with aJ = 0. (b) Viewing A as left A-module via (af)(b) = f(ba) for f 2 A and a; b 2 A, the form determines a left A-isomorphism G : A ! A , G(a) = ( : ; a). Letting B : A ! A denote the analogous isomorphism corresponding to , we obtain a left A-isomorphism  = B ?1  G : A ! A. Thus (a) = au, where u = (1) is a unit of A, and so G(1) = B(u). Now, for a; b 2 A,

(a; b) = (ab; 1) = G(1)(ab) = B(u)(ab) = (ab; u) and, consequently, (u?1 bu; a) = (u?1 bua; u). Finally, using associativity and symmetry of , one computes (u?1 bua; u) = (u; u?1bua) = (bu; a) = (a; bu) = (ab; u) which entails the claimed identity (a; b) = (u?1 bu; a).  2. Dimensions and Symmetry

Throughout this section, H denotes a nite dimensional Hopf algebra P over |, with counit ", antipode S, and comultiplication
. The latter will be written
(h) = h1 h2 for h 2 H. Recall that all H-modules are left modules and are assumed to be nite dimensional over |. 2.1 Homomorphisms. Let V and W be H-modules. Then Hom (V; W) can be made into an Hmodule by de ning

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| In the special case where W = = is the trivial H-module, so H acts on via the counit ", this simpli es to the following formula describing the H-action on V = Hom| (V; ): (hf)(v) =

X

h1f(S(h2 )v)

| |"

(hf)(v) = f(S(h)v)

(h 2 H; v 2 V; f 2 Hom (V; W)): 

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(h 2 H; v 2 V; f 2 V  ):

Viewing tensor products of H-modules as H-modules by means of the diagonal map
, the canonical isomorphisms (cf. [Bou], p. II.77 and II.80)

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 = Hom (V; W) ; W V  ?!  = (W V ) ; V  W  ?!

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w f 7! (v 7! f(v)w) f g 7! (w v 7! g(w)f(v)) :

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are in fact an H-module isomorphisms. Finally, the space of H-invariants Hom (V; W)H = ff 2 Hom (V; W) : hf = "(h)f for all h 2 H g coincides with the space of H-module maps HomH (V; W) ([L], Proposition 2.3 or [Zhu], Lemma 1). 2.2 Duality. Let V be a H-module. Viewing V  as H-module as in x2.1, the canonical isomorphism : V ! V  ; (v)(f) = f(v) satis es (hv) = S ?2 (h) (v). Therefore, as H-modules, ?2 V   = V (S ) ; the S ?2 -twist of V . Since twists by inner automorphisms do not aect the isomorphism type, we see in particular: If S 2 is inner then V   = V holds for all H-modules V .

REPRESENTATIONS OF HOPF ALGEBRAS

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Using ( : ) to denote transpose maps, we have an isomorphism of |-vector spaces 

= Hom (W  ; V  ) : ( : ) : HomH (V; W) ?! H

Thus ( : ) becomes an exact contravariant automorphism of the category of nite dimensional Hmodules. Clearly, ( : ) respects direct sums, and some power of ( : ) is equivalent to the identity, because S has nite order ([R3]). In particular, V  is irreducible (resp., indecomposable) if and only if V is. The H-module V is called self-dual if V   = V . The trivial H-module |= |" and the regular left H-module H are self-dual, the former as a consequence of the identity "  S = ", the latter by [Sw], Theorem 5.1.3. Consequently, projectivity also transfers between V and V  . 2.3 Theorem. Assume that H is involutory (that is, S 2 = Id). (a) ([L], Theorem 2.8) If H is semisimple then p does not divide the dimensions of any absolutely irreducible H-module. (b) If H is not semisimple, then p divides the dimension of every projective H-module. Recall that an H-module V is absolutely irreducible if for every eld extension K=|, K V is an irreducible K H-module or, equivalently, if End (V ) = |(cf. [CR], Theorem 3.43). Proof. Both parts follws from the observation that, for any H-module V , the trace map trV= : End (V ) ! |= |" of x1.4 is an H-module map. Indeed, identifying End (V ) with V V  as in xP2.1, it suces to check H-linearity of the map  : V V  ! |; v f 7! f(v). Using the identity S(h2 )h1 = "(h)1H for h 2 H (from S 2 = Id) we compute

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(h
v f) = (

X

h1 v h2f) =

X

(h2f)(h1 v) =

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X

f(S(h2 )h1 v) = "(h)f(v) = h
(v f) ;

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as required. (a) Since H is semisimple, there is a left integral t in H with "(t) 6= 0 in |. Note that t
End (V ) consists of H-invariants and hence is contained in EndH (V ), by x2.1. Thus, if f 2 End (V ) and V is absolutely irreducible, then tf = cf IdV for some cf 2 |, and hence

| | | | Choosing f 2 End| (V ) with tr | (f) = 1, we deduce that dim| V 6= 0 in . (b) Now let P be a projective H-module and assume, by way of contradiction, that p does not divide dim| P. Then the trace map tr | splits via  : ! End| (P); k 7! (dim| P) k
Id . Therefore, is isomorphic to a direct summand of End| (P). Inasmuch as End| (P)  = P P is projective, by the cf dim V = trV= (tf) = t trV= (f) = "(t) trV= (f) : |

V=

P=

|"

?1 

P

|"

Fundamental Theorem of Hopf Modules ([Mo], Theorem 1.9.4.), we conclude that |" is projective as well. This forces H to be semisimple, contrary to our assumption.  Remarks. The semisimplicity hypothesis in (a) is de nitely necessary. Indeed, the restricted enveloping algebra H = u(g) of the restricted p-Lie algebra g = sl(2; |) over a eld |of characteristic p > 2 has an irreducible module of dimension p. See x4.4 below for a detailed discussion of this example. Part (b) implies in particular the known result ([L], Theorem 4.3) that any involutory Hopf algebra H with dim H not divisible by p is semisimple. Since this is false in general if H is not involutory (cf. x4.1), this hypothesis is necessary for (b) to hold. In (a), semisimplicity of H may very well entail that H is involutory (Kaplansky's conjecture [K]; known to be true in characteristic 0 [LR2]. For an alternative proof, see [PQ2]). 2.4 Local Hopf subalgebras. Stronger estimates than the one provided by Theorem 2.3(b) are sometimes possible by using local Hopf subalgebras. These are exactly those Hopf subalgebras L of H such that the augmentation ideal L+ = Ker "L is nilpotent.

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MARTIN LORENZ

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Lemma. Let L be a local Hopf subalgebra of H. Then dim L divides the dimension of every projective

H-module. Proof. Since H is free as (left and right) L-module, by the Nichols-Zoeller Theorem ([Mo], Theorem 3.1.5), every projective H-module is projective, and hence free, over L.  Remarks and Examples. If H = |G is the group algebra of a nite group G then the local Hopf subalgebras of H are the group subalgebras |P, where P is a p-subgroup of G (0-subgroup means h1i). The local Hopf subalgebras of the restricted enveloping algebra H = u(g) of a nite dimensional restricted p-Lie algebra (g; [p]) over a eld |of characteristic p > 0 are the enveloping subalgebras u(p), where p is a p-nilpotent p-Lie subalgebra of g. Here, p is p-nilpotent if some power of the p-map [p] is 0 on p. If H is a semisimple Hopf algebra then all Hopf subalgebras are semisimple as well ([Mo], Corollary 3.2.3). Therefore, |is the only local Hopf subalgebra in this case. Even if H is not semisimple, local Hopf subalgebras 6= |need not exist; the Hopf algebra H4 in characteristic p 6= 2 is an example. (Note, however, that H4 is free over the local non-Hopf subalgebra L = |[x], in the notation of x4.1, and so the conclusion of the Lemma applies to L as well.) 2.5 Projective covers, unimodularity, and symmetry. Recall that if V is an irreducible Hmodule then the projective cover P(V ) is the unique (up to isomorphism) indecomposable projective H-module which maps onto V (cf. [CR], p. 131). In the following Lemma, we let  2 H  denote the \distinguished" group-like element of H  such that th = (h)t holds for every left integral t in H and every h 2 H (cf. [Mo], p. 22). The Hopf algebra H is unimodular if every left integral is a right integral for H or, equivalently, if  = ". Lemma. P(|") = P(|). In particular, P(|") is self-dual if and only if H is unimodular. Proof. Put Q = P(|") , an indecomposable projective H-module, by x2.2. It suces to show that Q maps onto |. But Q  = He for some primitive idempotent e 2 H, and He contains a nonzero left integral t of H, because |"  = |" embeds into Q. Thus t = te = (e)t, and so (e) 6= 0 and  = Q ?! He ?! | is epi.  Proposition ([OS]). H is a symmetric algebra precisely if H is unimodular and S 2 is inner. Proof. The conditions are sucient. For, if  2 H  is a nonzero right integral then, putting (h; k) = (hk) for h; k 2 H, one obtains an associative bilinear form ( : ; : ) : H  H ! |which is well-known to be nondegenerate ([LS], Theorem or [R], Cor. 2(b)). Moreover, since H is unimodular, the form also satis es (h; k) = (S 2 (k); h) for h; k 2 H ([LS], Proposition 8 or [R], Theorem 3(a)). Thus, choosing a unit u 2 H so that S 2 (h) = uhu?1 holds for all h 2 H, we can de ne : H H ! |by (h; k) = (uh; k) to obtain the required symmetric form. Conversely, assume that H is symmetric. Then P(V  )  = P(V ) holds for all irreducible H-modules V , by x1.7 (2). In particular, the above Lemma implies that H is unimodular. (For a more elementary proof of this implication, see [H2], Theorem 2.) Thus the associative nondegenerate form (h; k) = (hk) used in the rst paragraph of the proof satis es (h; k) = (S 2 (k); h). On the other hand, by Lemma 1.7(b), (h; k) = (u?1ku; h) for some unit u 2 H. Consequently, S 2 (k) = u?1ku which shows that S 2 is inner.  Remarks and Examples. (1) Finite dimensional commutative Hopf algebras are symmetric, since they are unimodular and involutory. Cocommutative Hopf algebras are symmetric precisely if they are unimodular which is not generally the case (cf. (4) below). Thus symmetry does not pass from a Hopf algebra to its dual. (See also (3) below.)

REPRESENTATIONS OF HOPF ALGEBRAS

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(2) The fact that S 2 is inner for symmetric Hopf algebras H entails in particular that V   =V holds for all H-modules V (x2.2). Furthermore, in view of x1.7 (1), the Proposition implies that S 2 is inner if H is semisimple. For |algebraically closed, this fact has been noted by Larson ([L], Prop. 3.5, see also [R], Theorem 5); the general case is due to Oberst and Schneider [OS]. We also remark that Drinfeld has shown that S 2 is inner for any almost cocommutative Hopf algebra ([Dr], cf. also [Mo], Proposition 10.1.4). (3) The Drinfeld double D(H) of any Hopf algebra H is always symmetric, because the conditions of the Proposition are satis ed by [R2], Theorem 4 and Cor. 2. (For the fact that S 2 is inner, one could also quote Drinfeld's more general theorem about almost commutative Hopf algebras mentioned in (2) above.) Therefore, any H embeds in a symmetric Hopf algebra, and symmetry is not in general inherited by Hopf subalgebras. Moreover, the dual D(H) is unimodular precisely if both H and H  are, and similarly for the property of the square of the antipode being inner ([R2], Cor. 4 and Prop. 8). Consequently, D(H) is symmetric if and only if both H and H  are. (4) Let H = u(g) denote the restricted enveloping algebra of the nite dimensional restricted p-Lie algebra (g; [p]) over a eld |of characteristic p > 0, and let adg (x) 2 End (g) be de ned as usual by adg(x)(y) = [x; y] for x; y 2 g. By [LS], Cor. on p. 91 (cf. also [Sch]), symmetry (or unimodularity) of H is equivalent to tr(adg (x)) = 0 for all x 2 g. This condition is certainly satis ed if g is nilpotent or if g = [g; g] and so, in particular, if g is simple. The condition is also satis ed for the diamond algebra, for example, but not for the 2-dimensional solvable p-Lie algebra (cf. x4.3).

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3. The Grothendieck ring

The notations introduced at the beginning of x2 remain in eect. Furthermore, as in x1, V1 ; : : : ; Vt denotes a full set of nonisomorphic irreducible H-modules, Pi = P(Vi ) are their projective covers, and J = rad H is the Jacobson radical of H. 3.1 The Grothendieck ring. As is well-known (e.g., [Ser]), the Grothendieck group G0(H) is endowed with the structure of an augmented Z-algebra with identity element 1 = [|"]: The augmentation is dimk : G0(H) ! Zand multiplication is given by [V ]
[W] = [V W]. We will refer to G0 (H) as the Grothendieck ring of H. Putting [V ] = [V  ] we obtain an anti-automorphism  of the ring G0(H) which permutes the basis f[Vi ]g of G0(H), and  is an involution of G0(H) if S 2 is inner (xx2.1, 2.2). Similarly, [P] = [P ] yields an automorphism of the group K0 (H) which permutes the basis f[Pi]g and has order 2 if S 2 is inner. Putting [P]
[V ] = [P V ] induces a right G0(H)-module structure on K0 (H). The fact that P V is projective if P is can be seen as follows. Clearly, it suces to prove that H V is projective for any V . Now H V is a (left) H-Hopf module (cf. [Mo], Example 1.9.3), and so H V is in fact free as left H-module, by the Fundamental Theorem of Hopf Modules ([Mo], Theorem 1.9.4). | The Cartan map c : K0 (H) ! G0(H) is a -equivariant map of G0(H)-modules. With respect to the G0(H)-modules structures and duality maps  on K0 (H) and G0 (H), the bilinear maps h : ; : i : K0 (A)  G0(A) ! Z and f : ; : g : K0 (H)  K0 (H) ! Z of x1.2 satisfy the identities hax; yi = ha; yx i ; fa; bg = fb; a g for x; y 2 G0(H) and a; b 2 K0 (H). The rst follows from the |-linear isomorphisms HomH (P V; W)  = (W V  P  )H  = HomH (P; W

  V  ), and the second from the isomorphism HomH (V; W)  Hom (W ; V ) (cf. xx 2.1, 2.2). We point = H out two consequences of these formulas.  The G0(H)-module K0 (H) is faithful: Indeed, if x 2 G0 (H) satis es K0 (H)x = 0 then 0 = hK0 (H)x; 1i = hK0 (H); x i. In view of the orthogonality relations, the latter condition is equivalent with x = 0, and hence with x = 0.

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 The left and right radical of the form f : ; : g are both equal to the kernel of the Cartan map c: Since Ker(c) is -invariant and fa; bg = fb ; ag holds for all a; b 2 K0 (H), it suces to show that the right radical of f : ; : g equals Ker(c). But, as above, fK0 (H); ag = hK0 (H); c(a)i = 0

is equivalent to c(a) = 0. Recall also that the form f : ; : g has matrix C 0, and it is symmetric if H is symmetric. Remarks and Examples. (1) If I is an ideal of H that is contained in J = rad H, then in ation of  = modules along the canonical map  : H ! H=I gives a group isomorphism G0(H=I) ! G0 (H) (cf. [Ba1], p. 455). In case I is a Hopf ideal of H, this is an isomorphism of rings. This applies in particular if J is a Hopf ideal of H. See x3.3 below for a detailed discussion of this case. (2) If H is semisimple then all H-modules are projective. Thus K0 (H) = G0 (H) and the Cartan map c is the identity map in this case. For example, let H = (|G) denote the dual of the group algebra |G of the nite group G, so H  = |jGj as |-algebras. Here, G0(H)  = ZG, the integral group ring of G. Up to a nite separable eld extension, this example covers all semisimple commutative Hopf algebras (cf. [Mo], Theorem 2.3.1). (3) Let H be almost cocommutative in the sense of Drinfeld [Dr]. Then S 2 is inner and V W  = W V holds for all H-modules V and W ([Dr], cf. also [Mo], Lemma 10.1.2 and Proposition 10.1.4). Hence G0 (H) is a commutative ring and  is an involution in this case. 3.2 Semiprimeness. Recall a subgroup X of K0(H) is said to be isotropic with respect to form f : ; : g if fX; X g = 0. In case f : ; : g is symmetric, this condition is equivalent with fx; xg = 0 for all x 2 X. Proposition. K0 (H) contains no nonzero -invariant isotropic G0(H)-submodules if and only if G0(H) is a semiprime ring and the Cartan map c is injective. Proof. First assume that K0 (H) contains no nonzero -invariant isotropic G0 (H)-submodule. Then, in particular, Ker(c) = 0 and so c is injective. Let N denote the nilpotent radical of the (Noetherian) ring G0(H) and suppose, by way of contradiction, that N 6= 0. Choose n so that I = N n 6= 0, but I 2 = 0, and note that I is -invariant, because N certainly is. Also, since c is injective, c Q is bijective, and so I \ Im(c) 6= 0. Consequently, letting X denote the preimage of I in K0 (H) under c, we have I  c(X) and X is a nonzero -invariant G0(H)-submodule of K0(H). Since I 2  c(XI), we deduce that c(XI) = 0, and hence XI = 0. On the other hand, hXI; 1i = hX; I i  fX; X g = 6 0, by assumption on the form f : ; : g. This contradiction shows that we must have N = 0, and so G0(H) is semiprime. Conversely, assume that G0(H) is a semiprime ring and the Cartan map c is injective. If X is a nonzero -invariant isotropic G0(H)-submodule of K0 (H), then I = c(X) is a nonzero -invariant right ideal of G0 (H) (and so I is actually a two-sided ideal of G0(H)). Moreover, 0 = fX; X g = hX; I i = hXI; 1i = hXIG0 (H); 1i = hXI; G0 (H)i, and hence the orthogonality relations imply that XI = 0. Therefore, I 2 = c(XI) = 0, contradicting semiprimeness of G0(H).  Examples. (1) Semisimple Hopf algebras H: In this case, C is the identity matrix and C 0 is a diagonal matrix with positive entries. Therefore the quadratic form Q(x) = fx; xg is positive de nite and no nonzero isotropic submodules of K0 (H) can exist. Therefore, G0 (H) is semiprime if H is semisimple. | More generally, this shows that G0(H) is a semiprime ring if J is a Hopf ideal of H, because G0(H)  = G0 (H=J) holds in this case. (2) Group algebras over suciently large elds: For group algebras |G of nite groups G, the Grothendieck ring G0 (|G) can be explicitly described in terms of (Brauer or ordinary) characters, and semiprimeness is evident from this description. However, it also follows from the Proposition, in view of the fact that the Cartan matrix C (= C 0, because |is assumed large) is known to be invertible and to have the form C = DT D, where D is an integer matrix, the so-called decomposition matrix

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11

(cf. [CR], Cor. 18.10 and Theorem 18.25). Consequently, C is positive de nite, and semiprimeness of G0(|G) follows as in (1). (3) The restricted enveloping algebra H = u(sl(2; |)) provides an example of a Grothendieck ring G0(H) which is not semiprime. See x4.4. 3.3 The case when J = rad H is a Hopf ideal. We discuss the case where J = rad H is a Hopf ideal of H in some detail. Some properties of this case have already been noted in x3.1 Remarks and Examples (1) and in x3.2 Examples (1). Since "(J) = 0 and S(J)  J are automatic, J = rad H is a Hopf ideal if and only if
(J)  H J + J H : Several further equivalent conditions are given in the following Lemma. In this connection, see also [PQ] and the remarks in [Mo], p. 62/63. Lemma. The following conditions are equivalent. (1) J is a Hopf ideal of H. (2) Tensor products of semisimple H-modules are semisimple. (3) If V , W are H-modules, then J
(W V )  JW V + W JV . (4) If V , W are H-modules, with V semisimple, then J
(W V ) = JW V . (5) Same as (4), but with V semisimple and W projective. (6) If V , W are H-modules, with V semisimple, then P(W V )  = P(W) V . (7) Same as (6), but with V and W both semisimple. Proof. (1) , (2): (2) is equivalent with the equation J
(H=J H=J) = 0 which in turn is equivalent with
(J)  H J + J H. (1) ) (3) is clear, and (3) ) (1) follows by taking V = W = H. (3) ) (5): Write W  Q  = H r for suitable Q and r. Then we have H-module isomorphisms (W V )  (Q V )  = Hr V  = H r dim|V ;

the last isomorphism being a consequence of the Fundamental Theorem of Hopf Modules (cf. x3.1). Hence, as |-vector spaces, J
(W V )  J
(Q V )  = Jr V  = (JW  JQ) V  = (JW V )  (JQ V ) : = J r dim|V 

But (3) implies that J
(W V )  JW V , and similarly for Q, because V is semisimple. Thus, comparing dimensions, we obtain (5). (5) ) (4) and (6): We write head X = X=JX for all H-modules X. Now let V , W be H-modules, with V semisimple. Then head P(W)  = head W, a general property of projective covers, and (5) implies that head(P(W) V )  = (head P(W)) V . Thus (head W) V  = head(P(W) V ) is semisimple, and so head(W V ) canonically maps onto (head W) V . On the other hand, the canonical epimorphism P(W) V  W V entails an epimorphism (head W) V  = head(P(W) V )  head(W V ). So (head W) V and head(W V ) are in fact isomorphic which proves (4). For (6), consider the composite epimorphism P(W V )  W V  head(W V )  = (head W) V  = head(P(W) V ) ; where the rst two epimorphisms are the canonical ones and the last isomorphism was established above. Using the Nakayama Lemma we deduce that P(W V ) maps onto P(W) V . Similarly, the canonical epimorphism P(W) V  W V implies that P(W) V maps onto P(W V ). Inasmuch

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as P(W) V and P(W V ) are both nite dimensional, the latter two epimorphism are in fact isomorphisms, thereby proving (6). (6) ) (7) is trivial. (7) ) (2): Let V and W be semisimple H-modules and put X = W V . By two applications of (7), P(X)  = P(W) V  = P(|") X : On the other hand, by general properties of projective covers, P(X)  = P(head X), and (7) applied to head Xfurther implies that P(X)  P( | )

head X. Therefore, P( |") X  = = P(|") head X. " Comparing dimensions, we conclude that the canonical map X  head X is an isomorphism and so X is semisimple. Since (4) clearly implies (5), the proof is complete.  Corollary. Assume that J = rad H is a Hopf ideal of H. Then, for any H-module V , the projective cover P(V ) is given by P(V )  = P(|") (V=JV ). Consequently, K0 (H) is a free G0(H)-module of rank 1 with generator [P(|")]. Furthermore, dim P(|") is an eigenvalue of the Cartan matrix C, and hence a divisor of det C in Z. Proof. In view of the general isomorphism P(V )  = P(V=JV ), the isomorphism P(V )  = P(|")

(V=JV ) follows from (7) in the Lemma applied to V=JV . In particular, we have Pi  = P(|") Vi for i = 1; : : : ; t which can be written as a matrix equation over G0(H) as follows: 0 [V ] 1 0 [V ] 1 1 1 [P(|")]
B @ ... CA = C
B@ ... CA : [Vt] [Vt] The assertion concerning dim P(|") now follows by applying the augmentation dim : G0 (H) ! Zto this equation.  Remarks and Examples. (1) If all simple H-modules are 1-dimensional (equivalently, H=J  = |r as |-algebras for some r) then all tensor products of simple H-modules are 1-dimensional as well, and hence condition (2) of the Lemma is clearly satis ed. Thus J is a Hopf ideal in this case. By x3.1 Remarks and Examples (1) and (2), we conclude that

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G0 (H)  = ZG ; where G is the group of group-like elements of H  . For explicit examples of this kind, see xx4.1, 4.2, 4.3 below. (2) If H = |G is a nite group algebra then J is a Hopf ideal precisely if G has a normal Sylow p-subgroup ([M]). Furthermore, Brockhaus ([Br]) has shown that if P(V )  = P(|") V holds for all semisimple |G-modules V then J is a Hopf ideal of |G. This is a substantial strengthening of the implication (6) ) (1) in the Lemma for group algebras. I do not know to what extent this fact generalizes to Hopf algebras. The present proof of Brockhaus' theorem depends on the classi cation of nite simple groups. (3) The example H = u(sl(2; |)) shows that all assertions of the Corollary are false in general: K0 (H) is no longer generated by [P(|")] in this case, and dim P(|") = 2p is not an eigenvalue of the Cartan matric C. See x4.4. 3.4 The Cartan matrix. In this subsection, we determine the rank of the |-linear map

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REPRESENTATIONS OF HOPF ALGEBRAS

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Note that rank cfH equals the rank of the Cartan matrix C of H in case p = 0, and the rank of the reduction of C mod p when p > 0. We will need the right adjoint action / of H on H which is de ned by X h / k = S(k2 )hk1 (h; k 2 H) : Theorem. Suppose that |is a splitting eld for H. Then rank cfH = dim (H / t) ; where t is any nonzero left integral of H. Moreover, if S 2 is inner then the following are equivalent. (1) H is semisimple; (2) C = Id; (3) p does not divide det C. Note that, for p = 0, condition (3) just says that det C 6= 0. Proof. By Theorem 1.6, we have a commutative diagram cfH K^ 0 (H) 0 (H) ????! G^

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?

e? y=

e? y=

T(H)reg ????t! C(H)reg

:

(:)

So rank cfH = rank( : )t.

By [R], Proposition 7, the map ( : )t has the following description: Let t be a left integral for H and  a right integral for H  such that (t) = 1. Then, for any h 2 H, ([h])t = (h / t) *  ; where * is de ned by (h * f)(k) = f(kh) for h; k 2 H and f 2 H  . Furthermore,  is a free H-basis of (H  ; *) ([LS], p. 83 or [R], Corollary 2). Therefore, rank( : )t = dim (H / t), as claimed. The implications (1) ) (2) ) (3) are trivial. For (3) ) (1), note that (3) says that cfH is injective, and hence so is the map ( : )t in the above diagram. Now suppose that S 2 is inner, say S 2 (h) = uhu?1 for some unit u 2 H. Since "(u) 6= 0, we have [u] 6= 0 in T(H)reg, and so ([u])t 6= 0. Therefore, by the above formula for ( : )t , u / t 6= 0. The computation X X X u / t = S(t2 )ut1 = S(t2 )S 2 (t1 )u = S( S(t1 )t2 )u = "(t)u now shows that "(t) 6= 0, and so H is semisimple.  P Remarks and Examples. (1) Finite group algebras H = |G: Taking t = g2G g we have g / t = jC G (g)jcg for g 2 G, where C G (g) denotes the centralizer of g in G and cg 2 |G is the class sum of g, that is, the sum of the elements in the conjugacy class of g. Therefore, |G/ t is the |-linear span of all class sums cg such that p does not divide jC G (g)j. (These classes are in particular p-regular, that is, they consist of elements having order not divisible by p.) Consequently, rank cgG = #fconj. classes C of G such that jCjp = jGjpg ; where j : jp denotes the p-part of the size (j : j0 = 1). (2) If H is involutory then the implication (3) ) (1) above follows more simply from Theorem ?!|Z Z=pZfactors through the 2.3(b) which asserts that, for H not semisimple, the map G0(H) dim cokernel of the Cartan map c. Now just use the fact that if det C 6= 0 then det C is the size of this cokernel. (3) If K is a local Hopf subalgebra of H (or, more generally, a local subalgebra so that H is free over K) then dim K divides det C. Indeed, since G0 (K) = h[|"K ]i and projective H-modules are projective, and hence free, over K, the restriction map induces an epimorphism G0 (H)=cK0(H) Res  G0(K)=cK0 (K)  = Z=dim K
Z:

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3.5 Trace spaces for Hopf algebras. [R], Lemma 1 gives two alternative descriptions of the space

of |-valued traces C(H). Namely, C(H) is the set of invariants with respect to the left coadjoint action of H on H  that is given by X (h . f)(k) = f(S(h2 )kh1 ) for h; k 2 H, f 2 H  . (This is the transpose of the action / that was used in x3.4.) Alternatively, one can characterize C(H) as the set CocomH  of cocommutative elements of H  . To summarize, C(H) = CocomH  = (H  )H : Note that ff 2 H  : f vanishes on J g is the coradical (H  )0 of H  (cf. [Mo], Proposition 5.2.9). Therefore, C(H)reg can be viewed as the set Cocom(H  )0 of cocommutative elements of the coradical of H  . Since the coradical (H  )0 is clearly mapped to itself under ., we have C(H)reg = Cocom(H  )0 = (H  )H0 : From the foregoing, we see in particular that C(H) is a |-subalgebra of H  , because CocomH  clearly is. In Corollary 3.7 we will show that if H=J is separable then C(H)reg is also a |-subalgebra of H  or, equivalently, [H; H] + J is a coideal of H. Note further that C(H) and C(H)reg are both stable under the antipode S  of H  , since [H; H] and J = rad H are stable under S. Examples. (1) H = (|G) for G a nite group: Here C(H)reg = C(H) = H  = |G. Since |G need not be commutative, this example shows in particular that the isomorphism Z(H)  = C(H) of Lemma 1.7 is not in general an isomorphism of |-algebras. (2) H = |G for G a nite group: Here, (h . f)(g) = f(gh ) for g; h 2 G and f 2 H  , where h g = h?1 gh is conjugation in G. Thus C(H) is canonically isomorphic with the |-algebra of all |valued functions on the set of conjugacy classes of G, with \pointwise" multiplication of functions. Furthermore, [H; H] + J is the |-linear span of the elements g ? gph0 for g; h 2 G, where gp0 denotes the p0 -part of g (= g if p = 0). Therefore, T(H)reg can be identi ed with the |-vector space with basis the set of p-regular conjugacy classes of G. Under this identi cation, the image [g] 2 T(H)reg of g 2 G becomes the conjugacy class of p0 -part of g. Furthermore, C(H)reg can be viewed as the algebra of |-valued functions on the set of p-regular conjugacy classes of G: A trace f 2 C(H) belongs to C(H)reg precisely if it satis es f(g) = f(gp0 ) for all g 2 G. See [P], p. 56{58 for all this. 3.6 Proposition. The character map  : G0(H) ! C(H) is a ring homomorphism which satis es V  = S  (V ) and V (1) = dim (V ). If |is a splitting eld for H then C(H)reg is a |-subalgebra of H  and  induces an isomorphism of |-algebras  = e = Id  : G^ 0 (H) = | G0 (H) ?! C(H)reg : Proof. First,  " = ", and so  respects the identity elements of G0(H) and C(H). Let V and W be H-modules. Then, by x2.1, there are |-isomorphisms End (V W)  = = V W W V   End (V ) End (W). Under this identi cation, trV W becomes trV trW trV W : End (V W)  | | = End (V ) End (W) ??????! = |; and X hV W = (h1 )V (h2)W (h 2 H) ; where h 7! hV denotes the structure map H ! End (V ),Petc.. Therefore, the factPthat  is a ring homomorphism follows from the computation V W (h) = trV (h1)V trW (h2 )W = V (h1 )W (h2 ) = (V W ) (h). The formula V  = S  (V ) follows similarly, since the obvious |-linear isomorphism End (V  )  = End (V ) sends hV  7! S(h)V . The formula V (1) = dim (V ) is trivial. The remaining assertions now follow from the fact that, for |a splitting eld, the map e is an isomorphism between G^ 0(H) and C(H)reg, by Theorem 1.6(b). 

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3.7 Corollary. If H=J is separable over |then C(H)reg is a |-subalgebra of H . Proof. Choose a eld extension K=|so that K is a splitting eld for He = K H. By Proposition e reg is a K-subalgebra of He  = K H . By separability, rad He = K J and so C(H) e reg = 3.6, C(H)  e K C(H)reg. Therefore, C(H)reg = C(H)reg \ H is closed under multiplication, since both H  and e reg are.  C(H) Remark. Of course, H=J is separable in case the base eld |is perfect. In addition, H=J is known to be separable if either J = 0 (cf. [Mo], Cor. 2.2.2(1)), or if H = |G is the group algebra of a nite group G ([CR], Theorem 7.10), or if H = u(g) is the restricted enveloping algebra of a \classical" p-Lie algebra g ([Se], p. 101). I am not aware of an example where H=J is not separable. 4. Examples

4.1 The Sweedler algebra. Let H = H4 = |h1; g;x; gxj g2 = 1; x2 = 0; xg = ?gxi and assume that |has characteristic p = 6 2. This algebra is quasitriangular but not unimodular (cf. [Mo], 1.5.6, 2.1.2,  |hgi =  (|hgi). Thus J is a Hopf ideal, and so we obtain ring 10.1.17). Here, J = xH and H=J =  Zhgi. The two irreducible H-modules are |" and |, where  : g 7! ?1; x 7! 0 isomorphisms G0(H) =

is the distinguished group-like element of H  . Their projective covers, P(|") and P(|), are dual to each other, by Lemma 2.5. Therefore, P(|") has socle isomorphic to |, and the socle of P(|) is isomorphic to |". Thus, in G0(H), we have c([P(|")]) = c([P(|)]) = [|"] + [|], and so the Cartan matrix is   C = 11 11 :

Note that the foregoing applies verbatim whenever H has only two irreducible modules, |" and |, where  is the distinguished group-like element. A cocommutative example of this form is the restricted enveloping algebra H = u(g) of the restricted 2-dimensional solvable 2-Lie algebra g in characteristic p = 2 (cf. x4.3 below). 4.2 Radford's algebra. The following Hopf algebra H is taken from [R3], Example 1 which in turn is based on an earlier construction due to Taft [T]. Fix n > 1 and assume that |contains a root of unity ! of order n. Then H is generated by elements x, y, g subject to the relations gn = 1; xn = yn = 0; xg = !gx; gy = !yg; xy = !yx : Furthermore, g is group-like, and x and y are g; 1-primitive. The Hopf algebra has |-basis fg` xr ys : 0  `; r; s < ng, H is unimodular, and S 2 (h) = ghg?1 holds for all h 2 H (see [R3]). Thus H is symmetric. The radical of H is J = xH + yH, a Hopf ideal of H, and H=J  = |hgi. The simple H-modules are: |i (i = 0; : : : ; n ? 1), where  2 G(H  ) is given by (g) =P!, (x) = (y) = 0, and the map [|i ] 7! gi L gives a ring isomorphism G0(H) ! Zhgi. Putting e = n1 gi 2 H we have e = e2 and P(|") = He = 0r;s 2. Thus g = |f  |h  |e, with [h; f] = ?2f, [h; e] = 2e, [e; f] = h, and the p-map is given by e[p] = f [p] = 0, h[p] = h. Let V = |a  |b denote the canonical g-module that is given by the matrices

1 0  0 1 0 0 hV = 0 ?1 ; eV = 0 0 ; fV = 1 0 :

The symmetric group Sm acts on the m-fold tensor product V m by permuting positions, and this action commutes with the H-action on V m , since H is cocommutative. Thus the Sm - xed points form an H-submodule of V m which we denote by V (m):

?
V (m) = V m Sm :

By [Bou], Proposition 4 on p. IV.44,

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The modules V (m) (0  m < p) form a complete set of irreducible H-modules, and they are in fact absolutely irreducible (cf. [C] or [SF], p. 207). Furthermore, all V (m) (0  m < p) are self-dual, by uniqueness of dimension. Note that V (0) = |" and V (1) = V . In the following, we will write vm = [V (m)] 2 G0(H) and v = [V ] 2 G0(H). Thus v0 = 1 and G0(H) =

p?1 M m=0

Zvm  = Zp

:

The ring structure of G0(H). Identifying Sm?1 with the subgroup StabSm (m)  Sm we have (V m )Sm?1 = V (m ? 1) V which shows that V (m) is a submodule of V (m ? 1)P V for all? m. Further
more, if m  2 then V (m ? 1) V contains the element x = (m ? 1)!a m?1 b ? 2Sm?1 b a m?1  which satis es hx = (m ? 2)x, ex = 0, and f m?1 x = 0. It follows that, for 2  m < p+2, Hx is the simple module V (m ? 2). For m < p, the submodules V (m ? 2) and V (m) of V (m ? 1) V intersect trivially, and so V (m ? 1) V = V (m)  V (m ? 2) if 2  m < p, by counting dimensions. For m = p, the element y = a p 2 V (p) satis es hy = py = 0 = ey, and the element x belongs to V (p), since (p ? 1)! = ?1 mod p. Therefore, V (p) has V (p ? 2) and V (0) as composition factors, and there must be another copy of V (0), for dimension reasons. Since V (p)  V (p ? 1) V , we conclude that V (p ? 1) V has 2 copies of V (0) and at least one copy of V (p?2) as composition factors. There actually is a second copy of V (p?2). For, HomH (V (p?1) V; V (p?2))  = HomH (V (p?1); V (p?2) V ) = HomH (V (p?1); V (p?1)V (p?3)) shows that there is an epimorphism  : V (p ? 1) V  V (p ? 2), and  does not split, because V (p ? 1) V is projective while V (p ? 2) is not (see below). Summarizing, we have shown that the following equations hold in G0(H):

v0
v = v1; vm?1
v = vm + vm?2 (2  m < p) and vp?1
v = 2vp?2 + 2v0 : P P It follows easily by induction that mi=0 Zvi = mi=0 Zvi holds for all m = 0; : : : ; p ? 1. In particular, G0 (H) = Z[v]  = Z[X]=(g(X)) ;

where g(X) is the characteristic polynomial of the p  p-matrix of the endomorphism of G0 (H) = Zp that is given by multiplication with v. One can show that g(X) = (2 ? X)fp (X)2 ; where

X

(p?1)=2

k + ap;k 

X k ; ap;k = b 41 (p ? 1 ? 2k)c : a p;k k=0 Consequently, G0(H) is not a semiprime ring. We also remark that fp (X) is irreducible and satis es fp (X) = (X ? 2) p?2 1 mod p. In particular: fp (X) =

(?1)ap;k

C(H)reg  = k[X]= ((X ? 2)p )  = kCp ;

where Cp is the cyclic group of order p.

The principal indecomposable modules. We now discuss the projective covers

P(m) = P(V (m)) (0  m < p):

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By Theorem 2.3(b), we know that dim P(m) = p`m for some `m 2 N. In particular, if m < p ? 1 then V (m) is not projective, and consequently P(m) has at least 2 constituents V (m), one in the head and

REPRESENTATIONS OF HOPF ALGEBRAS

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19

another one in the socle. (Recall that H is a symmetric algebra.) This forces dim P(m)  2(m + 1). Thus, if p?2 1  m < p ? 1 then we must have `m  2. Furthermore, using the isomorphism V (m) V  = V (m + 1)  V (m ? 1) for 1  m < p ? 1, we see that the projective module P(m) V maps onto  = V (m + 1)  V (m ? 1), and hence P(m + 1)  P(m ? 1) is a direct summand of P(m) V . This implies `m+1 + 1  2`m , and it follows by induction that `m  2 holds for all m < p ? 1. On the other hand, since |is a splitting eld for H, we have an isomorphism H =

p?1 M

m=0

P(m)m+1 :

Counting dimensions, we conclude that `m = 2 for m < p ? 1 and `p?1 = 1. Thus dim P(m) = 2p (0  m < p ? 1) and P(p ? 1)  = V (p ? 1) : In particular, V (p ? 1) is the only projective module of dimension p. Since the projective module V (p ? 1) V maps onto V (p ? 2), we have P(p ? 2)  = V (p ? 1) V : Using the fact that P(m + 1)  P(m ? 1) is a direct summand of P(m) V for 1  m < p ? 1, we obtain the following isomorphisms P(m) V  = P(m + 1)  P(m ? 1) (1  m < p ? 2) and P(p ? 2) V  = P(p ? 3)  V (p ? 1)2 : Put am = [P(m)] 2 K0 (H), so a0; a1; : : : ; ap?1 is the canonical Z-basis of K0 (H). An easy induction shows that ap?1 = [V (p ? 1)] generates K0 (H) as G0(H)-module, and so K0 (H) is free over G0(H). The Cartan map. We claim that the Cartan map c : K0 (H) ! G0 (H) is given by c(ap?1 ) = vp?1 and c(am ) = 2vm + 2vp?2?m (0  m < p ? 1): Indeed, the rst formula is clear from P(p ? 1)  = V (p ? 1), and the second formula for m = p ? 2 follows from P(p ? 2)  = V (p ? 1) V which implies c(ap?2) = vp?1 v = 2vp?2 +2v0 . For m = p ? 3, we use the isomorphism P(p ? 2) V  = P(p ? 3)  V (p ? 1)2 to obtain c(ap?3 ) = c(ap?2 )v ? 2vp?1 = (2vp?2 + 2v0)v ? 2vp?1 = 2(vp?1 +vp?3 )+2v ? 2vp?1 = 2vp?3 +2v1, as required. The assertion for the remaining 0  m < p ? 3 now follows by induction, based on the isomorphism P(m+1) V  = P(m+2)  P(m), or c(am ) = c(am+1 )v ? c(am+2 ). Therefore, the Cartan matrix has the form 02 0 ::: 0 2 01 BB 0 2 : : : 2 0 0 CC C=B BB 0: : : ::2 : ::::::: : :2: : :0:: :0: CCC : @2 0 ::: 0 2 0A 0 : : : : : : : : : : : : 1 pp Loewy series. The Loewy series of the principal indecomposable modules P(m) (0  m < p ? 1) follows easily. Indeed, the head P(m)=P(m)J and the socle annP (m) (J) are both isomorphic to V (m), and the core P(m)J=socle P(m) consists of two copies of V (p ? 2 ? m). It follows that ExtH (V (m); V (m)) = 0 for 0  m < p ? 1 (and of course also for the projective module V (p ? 1)), and hence the core of P(m) must actually be isomorphic with V (p ? 2 ? m)  V (p ? 2 ? m). Thus, for 0  m < p ? 1, we have V (m) P(m) = V (p ? 2 ? m)  V (p ? 2 ? m) : V (m) Consequently, the radical J = rad H has nilpotence index 3. This could of course also be checked using the explicit generators ep?1 (h + 1) and (h + 1)f p?1 for J that are given in [Se], p. 99.

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20 [Ba1] [Ba2] [BGR] [Bou] [Br] [C] [CR] [Dr] [H] [H2] [K] [La] [L] [LR] [LR2] [LS] [M] [Mo] [OS] [P] [PQ] [PQ2] [Po] [R] [R2] [R3] [Sch] [Se] [Ser] [SF] [Sw] [T] [V] [Zhu]

MARTIN LORENZ

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