CO-FROBENIUS HOPF ALGEBRAS AND THE CORADICAL FILTRATION

NICOLÁS ANDRUSKIEWITSCH AND SORIN DĂSCĂLESCU

Abstract. We prove that a Hopf algebra with a finite coradical filtration is co-Frobenius. We also characterize co-Frobenius Hopf algebras with coradical a Hopf subalgebra. Let H be a Hopf algebra whose coradical is a Hopf algebra. Let gr H be the associated graded coalgebra and let R be the diagram of H, c. f. [2]. Then the following are equivalent: (1) H is co-Frobenius; (2) gr H is co-Frobenius;

(3) R is finite dimensional; (4) the coradical filtration of H is finite. This Theorem allows us to give systematically examples of co-Frobenius Hopf algebras, and opens the way to the classification of ample classes of such Hopf algebras.

Introduction

Let H be a Hopf algebra over a field k. Recall that a ”left integral” over H is a linear map ∫ ∈ H^{∗} which is left invariant; that is, α ∫ = α(1) ∫ for all α ∈ H^{∗}. If H is the algebra of regular functions on a compact Lie group G, this is exactly (the restriction of) a left Haar measure on G. It is known that the space of left integrals has dimension ≤ 1. A fundamental problem is to determine the class of Hopf algebras having a non-zero integral. These are called co-Frobenius Hopf algebras.

Examples of co-Frobenius Hopf algebras do not abound in the literature. Finite dimensional Hopf algebras [13] and cosemisimple Hopf algebras [21] are co-Frobenius. The algebra of regular functions on an algebraic group G, in characteristic 0, is co-Frobenius if and only if the group is reductive [19]. A cleft extension of two co-Frobenius Hopf algebras, in particular the tensor product of two co-Frobenius Hopf algebras, is also co-Frobenius [7]. Liftings of finite quantum linear spaces over an abelian group are co-Frobenius [6]. See some more examples in [12]. In all these examples, the coradical filtration is finite. The main theme of this paper is to investigate whether the following conditions on a Hopf algebra H are equivalent:

(a). H is co-Frobenius.

(b). The coradical filtration of H is finite.

In this paper, we prove that (b) always implies (a), see Theorem 1.2. It was shown by Radford [18] that (a) implies (b) under the assumption that the coradical of H is a Hopf subalgebra. We give a different proof of Radford’s theorem.

Assume that the coradical H_{0} of H is a Hopf subalgebra.Then the coradical filtration of H is a Hopf algebra filtration, gr H is a Hopf algebra and there exists a Hopf algebra projection π : gr H → H_{0} which splits the inclusion of H_{0} in gr H as the degree zero component. The subalgebra R of coinvariants of gr H with respect to the coaction of H_{0} via π has a structure of a Hopf algebra in the braided category _{H_{0}}^{H_{0}}YD of Yetter-Drinfeld modules over H_{0}; R is called the diagram of H. The graded Hopf algebra gr H can be reconstructed from R by bosonization, i.e. gr H is isomorphic to Radford’s biproduct R♯H_{0}. See [2] for details.

Date: Final version of January 30, 2002.

Parts of this work were done during visits of the second author to the University of Córdoba, Argentina, in May 2000; and of the first author to the University of Bucharest in November 2000. We thank the FOMEC and CNCSIS (Grant C12) for support to these visits. The first author also thanks ANPCyT, CONICET, Agencia Córdoba Ciencia and Secyt (UNC) for partial support. The second author was partially supported by Grant SM10/01 of the Research Administration of Kuwait University.

Main Theorem. Let H be a Hopf algebra with the coradical H_{0} a Hopf subalgebra. Then the following assertions are equivalent.

(1) H is co-Frobenius.

(2) The associated graded Hopf algebra gr H is co-Frobenius.

(3) The diagram R of H is finite dimensional.

(4) The coradical filtration H_{0}, H_{1}, … of H is finite, i.e. there exists n such that H_{n} = H.

Conventions. We refer to [20, 16, 9] for notation and facts about coalgebras, comodules and Hopf algebras. Throughout, C is a coalgebra. The category of right (respectively left) comodules over C is denoted by M^{C} (respectively ^{C}M). If M ∈ M^{C} then we denote the comodule structure map by ρ : M → M ⊗ C. If M is a right (or left) C-comodule, the injective envelope of a comodule M is denoted by E(M). The coradical filtration of C is denoted by C_{0} ⊆ C_{1} ⊆ . . .. If H is a Hopf algebra, the left and the right rational parts of H^{∗} are equal, and they are denoted by H^{∗rat}. The category of left-left Yetter-Drinfeld modules over a Hopf algebra H (with bijective antipode) is denoted by _{H}^{H}YD.

1. Proof of the main theorem

In this Section, we prove the main theorem. Since some of the implications hold without the assumption on the coradical, we state them separately. We will need the following well-known characterizations of a Hopf algebra with non-zero integral.

A coalgebra C is called left co-Frobenius if there exists a monomorphism of left C^{∗}-modules from C to C^{∗}. Also, C is called left semiperfect if the injective envelope E(S) of any simple right C-comodule S is finite dimensional; see [11, 14]. A left co-Frobenius coalgebra is left semiperfect, while the converse does not hold in general. However, for a Hopf algebra H, the following are equivalent: (a) H is left co-Frobenius; (b) H is left semiperfect; (c) H has non-zero left integrals; (d) H^{∗rat} ≠ 0; (e) all the same properties with ”left” replaced by ”right”. See for instance [14] or [9, Ch. 5].

We first recall the definition of the Loewy series of M ∈ M^{C} [20, p. 183], [11, p. 148]: M_{0} = s(M), the socle of M, i.e. the sum of all simple subcomodules of M, and M_{n+1}/M_{n} = s(M/M_{n}), for n ≥ 0. The sequence M_{0} ⊆ M_{1} ⊆ … is a chain of C-subcomodules of M. Since M is the sum of all its finite dimensional subcomodules, M is the union of all M_{n}’s. There is an equivalent definition of the Loewy series, implicit in [11]: M_{n} = ρ^{−1}(M ⊗C_{n}), for any n ≥ 0. The Loewy series of a comodule generalizes the coradical filtration of a coalgebra. The graded object associated to the coradical filtration is a graded coalgebra denoted by gr C [20, Section 11.1]. Similarly, the graded object associated to the Loewy series of M is a graded comodule over gr C, denoted by gr M. For this, it is enough to observe that

(1.1)  ρ(M_{n}) ⊆ ⨁_{i=0}^{n} M_{i} ⊗ C_{n−i},  n ≥ 0.

(1.1) can be proved by repeating word-by-word the argument in [20, pp. 190–191]; note that Lemma 9.1.5 in loc. cit. generalizes to the tensor product of two filtered vector spaces.

Proposition 1.1. (i). Let S be a simple right C-comodule. Then gr E(S) is isomorphic to the injective envelope Egr (S) of the simple right gr C-comodule S.

(ii). C is left semiperfect if and only if so is gr C.

Proof. (i). Since the socle of a direct sum of comodules is the sum of the socles of these comodules, we see that for any family (M(λ))_{λ∈Λ} of right C-comodules we have gr (⊕_{λ∈Λ}M(λ)) ≃ ⊕_{λ∈Λ}gr M(λ) as right gr C-comodules. In particular if we write C_{0} = Soc(C) = ⊕_{S∈S} S for some family S of simple right C-subcomodules of C, then C = ⊕_{S∈S} E(S) (see [9, Theorem 2.4.16]), so then gr C = ⊕_{S∈S} gr E(S) as right gr C-comodules. In particular gr E(S) is injective as a right gr C-comodule. Since S = E(S)_{0} = gr E(S)(0) ⊆ gr E(S), we obtain that Egr (S) ⊆ gr E(S). On the other hand (gr C)_{0} = gr C(0) = C_{0} = ⊕_{S∈S} S as right gr C-comodules, so then gr C = ⊕_{S∈S} Egr (S), showing that Egr (S) = gr E(S) for any S ∈ S. (ii). Given a simple S, E(S) is finite dimensional if and only if so is gr E(S) ≃ Egr (S). ⊓ ⊔

Theorem 1.2. Let C be a coalgebra such that C = C_{n} for some n ≥ 0. Then the rational part of the right (or left) C^{∗}-module C^{∗} is non-zero. In particular, a Hopf algebra with a finite coradical filtration is co-Frobenius.

Proof. If C = C_{0}, then C is cosemisimple, therefore it is left and right co-Frobenius (see [9, Exercise 3.3.17]). In particular the rational part of C^{∗} as a right (or left) C^{∗}-module is non-zero. If C is not cosemisimple, let n be such that C_{n−1} ≠ C_{n} = C. Then C/C_{n−1} is a semisimple right C-comodule, in particular it contains a maximal subcomodule. This induces a maximal right C-subcomodule X of C. The proof goes now as in [9, Proposition 3.2.2]. The natural projection C ⟶ C/X produces an injective morphism of right C^{∗}-modules (C/X)^{∗} ↪ C^{∗}. But C/X is simple, so it is finite dimensional and rational as a left C^{∗}-module, and then (C/X)^{∗} is a rational right C^{∗}-module (see [9, Lemma 2.2.12]). This shows that the rational part of the right C^{∗}-module C^{∗} is non-zero. ⊓ ⊔

Proof of the Main Theorem. (1) ⇔ (2) follows from Proposition 1.1, and (4) ⇒ (1) follows from Theorem 1.2.

We consider now (2) ⇔ (3). Write, as in the proof of Proposition 1.1, H_{0} = ⊕_{S∈S} S for some family S of simple left H-subcomodules of H. It is enough to prove that for any S ∈ S, the injective envelope Egr (S) of S as a left gr H-comodule is Egr (S) = R♯S.

We identify gr H and R♯H_{0}, then gr H = ⊕_{S∈S} R♯S. We first show that R♯S is an injective left gr H-subcomodule of gr H for any S ∈ S. The comultiplication of the biproduct is Δ(r♯h) = ∑ r^{1}♯(r^{2})_{−1}h_{1} ⊗ (r^{2})_{0}♯h_{2}, where r ↦ ∑ r^{1} ⊗ r^{2} is the comultiplication of the braided Hopf algebra R, and r ↦ ∑ r_{−1} ⊗ r_{0} is the left coaction of H_{0} on R. This implies that Δ(R♯S) ⊆ gr H ⊗ (R♯S). The injectivity follows from the fact that R♯S is a direct summand of the left gr H-comodule gr H. The claim follows now immediately if we take into account that S is contained in R♯S, and that for any S, T ∈ S, S ≠ T, (R♯S) ∩ (R♯T) = 0.

We prove (3) ⇒ (4). Since R is finite dimensional there exists a positive integer n such that R = R(0) + … + R(n). Then gr H = gr H(0) + … + gr H(n), implying that H_{n} = H. ⊓ ⊔

Here is an alternative proof of (4) ⇒ (1). If the coradical filtration is finite, then the graded braided Hopf algebra R has a finite grading, i.e. R = ⊕_{0≤n≤N}R(n), with R(N) ≠ 0. Since R(0) = k, any non-zero element t ∈ R(N) is an integral in R. Hence R is finite dimensional by [10, Cor. 5.8]. Remark 1.3. The referee informs us that yet another proof of (4) ⇒ (1) was found by W. Chin using only injective modules and the Fundamental Theorem of Hopf algebras.

2. Examples from the Lifting Method

We now explain how to obtain new examples of co-Frobenius Hopf algebras via the main theorem. We rely on the Lifting method [2, 3]; a detailed exposition is [5].

Recall that a graded coalgebra C = ⊕_{i≥0}C(i) is coradically graded if its coradical filtration is given by C_{n} = ⊕_{i≤n}C(i) for any n ≥ 0.

Let K be a Hopf algebra with bijective antipode. Let V be a Yetter-Drinfeld module over K; let c : V ⊗V → V ⊗V be the braiding in this category. There is a remarkable graded braided Hopf algebra 𝔅(V) = ⊕_{n≥0}𝔅^{n}(V) in _{K}^{K}YD called the Nichols algebra of V, with 𝔅^{0}(V) = k. It is characterized by the following properties:

(i). The space of primitive elements is P (𝔅(V)) = 𝔅^{1}(V) ≃ V.

(ii). 𝔅(V) is generated by 𝔅^{1}(V) as algebra.

Condition (i) implies that 𝔅(V) is coradically graded. Assume that K is cosemisimple. Then G := 𝔅(V)#K is coradically graded [2, Lemma 2.3]. We then conclude from the main theorem:

If 𝔅(V) is finite-dimensional and H is any Hopf algebra with gr H ≃ G as graded Hopf algebras, then H is co-Frobenius. In particular, G is co-Frobenius.

Now, 𝔅(V) depends only on the braiding c as an algebra and coalgebra. Conversely, given a finitedimensional vector space V provided with an invertible solution c of the braid equation, one can construct 𝔅(V).

To provide new examples of co-Frobenius Hopf algebras, one needs:

(a). To find pairs (V, c) with dim 𝔅(V) < ∞.

(b). To find cosemisimple Hopf algebras K such that (V, c) in _{K}^{K}YD, (V, c) as in (a).

(c). To find Hopf algebras H with gr H ≃ 𝔅(V)#K as graded Hopf algebras, (V, c) as in (b).

Let us recall that c is of diagonal type if V has a basis x_{1}, …, x_{θ} such that c(x_{i} ⊗ x_{j}) = q_{ij}x_{j} ⊗ x_{i}, for all 1 ≤ i, j ≤ θ, where q_{ij} are some scalars. Example 2.1. Let (V, c) be a braided vector space of Cartan type; that is, c is of diagonal type, q_{ii} ≠ 1, for all i, all the entries are roots of one, q_{ij}q_{ji} = q_{ii}^{a_{ij}}, for all 1 ≤ i ≠ j ≤ θ, where a_{ij} are some non-positive integers [3]. Set a_{ii} = 2 for all i. The integers a_{ij} can be choosen such that −ord q_{ii} < a_{ij} ≤ 0; then the matrix (a_{ij}) is a generalized Cartan matrix.

Let K be a cosemisimple Hopf algebra provided with a family of one-dimensional representations χ_{1}, …, χ_{θ} of K, and a family of central group-likes g_{1}, …, g_{θ} ∈ G(K) ∩ Z(K), such that χ_{j}(g_{i}) = q_{ij}, 1 ≤ i, j ≤ θ. For instance, K could be the group algebra of ℤ^{θ}. If (a_{ij}) is actually a finite Cartan matrix and the orders of the q_{ij}’s are odd and not divisible by 3, when the matrix (a_{ij}) has a component of type G_{2}, then 𝔅(V) has finite dimension, which can be explicitly computed. See [3]. Therefore G := 𝔅(V)#K is a co-Frobenius Hopf algebra. Example 2.2. Keep the notation of the preceding example. Assume for simplicity that K is the group algebra of an abelian group. Let (λ_{ij}) be a linking datum for q_{ij}, χ_{1}, …, χ_{θ}, g_{1}, …, g_{θ} as in [4]. Let H be the Hopf algebra constructed as in [4, Th. 1.1]. Then gr H ≃ G := 𝔅(V)#K; therefore H is co-Frobenius. Example 2.3. Keep the notation of the first example. Assume q_{ij}q_{ji} = 1 for all i ≠ j. Then 𝔅(V) is called a finite quantum linear space; the examples in [6] are exactly liftings of quantum linear spaces over an arbitrary abelian group.

The question of the classification of all co-Frobenius Hopf algebras whose coradical is a Hopf subalgebra arises naturally. The Lifting method [2, 3, 4, 5] seems to be the pertinent tool to deal with this very hard problem.

3. More examples of co-Frobenius Hopf algebras

We now present a method to check that a Hopf algebra whose coradical is not a Hopf subalgebra is co-Frobenius, based on Theorem 1.2. Lemma 3.1. Let H be a Hopf algebra and let K be a Hopf subalgebra such that

(3.1) KH_{0} ⊆ H_{0} (in particular, K ⊆ H_{0}). (3.2) H is a finitely generated left K-module.

Then the coradical filtration of H is finite; and H is co-Frobenius.

Proof. By (3.1), each term H_{n} of the coradical filtration is a K-submodule; by (3.2), the coradical filtration is then finite. The last claim follows from Theorem 1.2. ⊓ ⊔

In representation-theoretic terms, condition (3.1) means the following. If V is a simple K-comodule and W is a simple H-comodule then the H-comodule V ⊗ W is completely reducible. Indeed, let Ĥ denote the set of (isomorphism classes of) simple comodules over a Hopf algebra H. We may write a representative for its class without danger of confusion. If U is any finite dimensional H-comodule, then let C_{U} ⊂ H denote the subcoalgebra of U. We have C_{U⊗W} = C_{U}C_{W}. Also K = ∑_{V ∈K̂} C_{V}, H_{0} = ∑_{W ∈Ĥ} C_{W}. We are led to the following definition.

Definition 3.2. Let H be a Hopf algebra with bijective antipode. Let Ĥsoc be the subset of Ĥ consisting of all simple comodules V such that

(3.3)  V ⊗ W and W ⊗ V are completely reducible

for all W ∈ Ĥ. The Hopf socle of H is Hsoc = ∑_{V ∈Ĥsoc} C_{V}.

Lemma 3.3. Let H be a Hopf algebra with bijective antipode. The Hopf socle of H is a cosemisimple Hopf subalgebra of H. If H is a finitely generated left Hsoc -module, then H is co-Frobenius.

Proof. It is clear that Hsoc is a subcoalgebra of H_{0}. We prove that Hsoc is a subalgebra of H. Let V, U ∈ Ĥsoc . Then V ⊗ U = ⊕_{1≤j≤r}W_{j}, where W_{j} are simple comodules. We have to show that W_{j} ∈ Ĥsoc , 1 ≤ j ≤ r. Let W ∈ Ĥ. Then V ⊗ U ⊗ W is completely reducible; since W_{j} ⊗ W is a subcomodule of V ⊗ U ⊗ W, it is also completely reducible. By an analogous argument, W ⊗ W_{j} is completely reducible. Therefore, W_{j} ∈ Ĥsoc . We finally prove that Hsoc is stable under the antipode S. If V is a finite dimensional comodule, we denote by V^{∗} the comodule structure on the dual defined via S, and by ^{∗}V the comodule structure on the dual defined via S^{−1}. We know that (V ⊗ W)^{∗} ≃ W^{∗} ⊗ V^{∗}, ^{∗}(V ⊗ W) ≃ ^{∗}W ⊗ ^{∗}V for any two finite dimensional comodules V and W. If V ∈ Ĥsoc and W ∈ Ĥ, then ^{∗}(V^{∗} ⊗ W) ≃ ^{∗}W ⊗ V is completely reducible, therefore V^{∗} ⊗ W is completely reducible. Similarly, W ⊗ V^{∗} is completely reducible. Thus, V^{∗} ∈ Ĥsoc , as needed. The last statement is an immediate consequence of Lemma 3.1. ⊓ ⊔

The argument in the proof of the Lemma shows that the definition of Hsoc is symmetric; that is, it can be defined starting from right comodules instead of left comodules. We do not know if Hsoc is a maximal Hopf subalgebra of H_{0}. Example 3.4. We assume that k is algebraically closed and has characteristic 0. Let 𝔤 be a simple finite dimensional Lie algebra of rank n, and let G be the corresponding simply-connected algebraic group. Let q ∈ k be a root of 1 of odd order ℓ ≥ 3, ℓ not divisible by 3 if 𝔤 contains a component of type G_{2}; and let U_{q}(𝔤) be the quantized enveloping algebra as defined in [15]. Recall the definition of modules of type 1 [15]. The following facts were proved by Lusztig:

• The set of isomorphism classes of simple U_{q}(𝔤)-modules of type 1 is parametrized by (ℤ_{≥0})^{n} [15, Prop. 6.4]. If λ ∈ (ℤ_{≥0})^{n}, the corresponding simple module L(λ) has highest weight λ.

• If λ ∈ (ℤ_{≥0})^{n}, decompose it as λ = λ′ + ℓλ′′, where λ′, λ′′ ∈ (ℤ_{≥0})^{n} with each entry of λ′ living in the interval [0, ℓ − 1]. Then L(λ) ≃ L(λ′) ⊗ L(ℓλ′′) (Steinberg-type Theorem, [15, Th. 7.4]).

• There is an epimorphism of Hopf algebras U_{q}(𝔤) → U(𝔤). If λ′′ ∈ (ℤ_{≥0})^{n}, the simple U_{q}(𝔤)- module L(ℓλ′′) inherits a structure of U(𝔤)-module; and as such, it is simple with highest weight λ′′ [15, Prop. 7.5]. Denote this last module by \underline{L}(λ′′).

Let k_{q}[G] ⊂ U_{q}(𝔤)^{∗} be the Hopf algebra of matrix coefficients of modules of type 1. This is the algebra of functions on a quantum group at a root of one, as in [8]. The coradical of k_{q}[G] is the span of the matrix coefficients of the simple modules L(λ), λ ∈ (ℤ_{≥0})^{n}. It is known that it is not a Hopf subalgebra. The subalgebra of k_{q}[G] spanned by the matrix coefficients of the simple modules L(ℓλ′′), λ′′ ∈ (ℤ_{≥0})^{n}, is a Hopf subalgebra of k_{q}[G], isomorphic to the algebra k[G] of regular functions on G. It is known that it is central [8]. It is also known that the injective envelopes of the simple modules of type 1, which are exactly the injective envelopes of simple k_{q}[G]-comodules, are finite-dimensional [1]. Thus k_{q}[G] is semiperfect, and hence as noted in Section 1, it is co-Frobenius.

Here is an alternative proof based on Lemma 3.1.

Proof. Let H = k_{q}[G], K = k[G]. (3.2) holds by [8]. Condition (3.1) is a consequence of the ”Steinberg-type” decomposition theorem quoted above. Indeed, let V = L(ℓμ), W = L(λ) with μ, λ ∈ (ℤ_{≥0})^{n}. Decompose λ as before, so that L(λ) ≃ L(λ′) ⊗ L(ℓλ′′), λ′, λ′′ ∈ (ℤ_{≥0})^{n}. By Weyl’s Theorem, \underline{L}(λ′′) ⊗ \underline{L}(μ) is completely reducible, say isomorphic to ⊕_{j=1,...,M} \underline{L}(σ_{j}). Then

W ⊗ V ≃ L(λ′) ⊗ L(ℓλ′′) ⊗ L(ℓμ) ≃ L(λ′) ⊗ (⊕_{j=1,...,M}L(ℓσ_{j}))
≃ ⊕_{j=1,...,M}L(λ′) ⊗ L(ℓσ_{j}) ≃ ⊕_{j=1,...,M}L(λ′ + ℓσ_{j}).

Therefore, KH_{0} = H_{0}K ⊆ H_{0}. The claim follows now from Lemma 3.1. ⊓ ⊔

In conclusion, we conjecture that the converse of Theorem 1.2 holds. That is, we conjecture that the coradical filtration of a co-Frobenius Hopf algebra is finite. In this direction, we do not know if the converse of Lemma 3.3 is true; that is, if any co-Frobenius Hopf algebra is of finite type over Hsoc .

Acknowledgement

We thank the referee for the valuable comments on the first version of this paper.

References

[1] H. H. Andersen, P. Polo and K. Wen, Representations of quantum algebras, Invent. Math. 104 (1991), 1-59.

[2] N. Andruskiewitsch, H. J. Schneider, Liftings of quantum linear spaces and pointed Hopf algebras of order p^{3}, J. Algebra 209 (1998), 658-691.

[3] , Finite quantum groups and Cartan matrices, Adv. Math. 154, 1-45 (2000).

[4] , Finite quantum groups over abelian groups of prime exponent, Ann. Sci. Ec. Norm. Super., to appear.

[5] , Pointed Hopf algebras, to appear in ”New directions in Hopf algebras”, MSRI series, Cambridge U. Press.

[6] M. Beattie, S. Dăscălescu and L. Grünenfelder, Constructing pointed Hopf algebras by Ore extensions, J. Algebra 225 (2000), 743–770.

[7] M. Beattie, S. Dăscălescu, L. Grünenfelder and C. Năstăsescu, Finiteness conditions, co-Frobenius Hopf algebras, and quantum groups, J. Algebra 200 (1998), 312–333.

[8] C. De Concini, V. Lyubashenko, Quantum function algebra at roots of 1, Adv. Math. 108, 205–262 (1994).

[9] S. Dăscălescu, C. Năstăsescu, Ş. Raianu, Hopf algebras: an introduction, Marcel Dekker (2000).

[10] D. Fischman, S. Montgomery, H.-J. Schneider, Frobenius extensions of subalgebras of Hopf algebras, Trans. Amer. Math. Soc. 349 (1997), 4857-4895.

[11] J. Green, Locally finite representations, J. Algebra 41 (1976), pp. 137–191.

[12] P. H. Hai,The integral on quantum supergroups of type A_{r|s}, math.QA/9812036.

[13] R. Larson and M. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. of Math. 91 (1969), pp. 75–94.

[14] B. I. P. Lin, Semiperfect coalgebras, J. Algebra 49 (1977), 357-373.

[15] G. Lusztig, Modular representations and quantum groups, Contemp. Math. 82 (1989), pp. 59–77.

[16] S. Montgomery, Hopf algebras and their actions on rings, CBMS Lecture Notes 82, Amer. Math. Soc., 1993.

[17] W.D. Nichols, Bialgebras of type one, Commun. Alg. 6 (1978), pp. 1521–1552.

[18] D. E. Radford, Finiteness conditions for a Hopf algebra with non-zero integrals, J. Alg. 46 (1977), 189-195.

[19] J. Sullivan, Affine group schemes with integrals, J. Algebra 22 (1972), 546-558.

[20] M. Sweedler, Hopf algebras, Benjamin, New York, 1969.

[21] , Integrals for Hopf algebras. Ann. of Math. (2) 89 (1969), 323-335.

N. A.: Facultad de Matemática, Astronomía y Física, CIEM–CONICET, Universidad Nacional de Córdoba, (5000) Ciudad Universitaria, Córdoba, Argentina

   E-mail address: andrus@mate.uncor.edu

S. D.: Facultatea de Matematica, University of Bucharest, Str. Academiei 14, RO-70109 Bucharest 1, Romania

Kuwait University, Faculty of Science, Det. Mathematics & Computer Sc., PO Box 5969, Safat 13060, Kuwait

   E-mail address: sdascal@al.math.unibuc.ro, sdascal@mcs.sci.kuniv.edu.kw