On the coradical filtration of Hopf algebras

whose coradical is a Hopf subalgebra

Nicolás Andruskiewitsch and Hans-Jürgen Schneider

Abstract. We offer a new proof of a Theorem of Taft and Wilson on the coradical filtration of pointed Hopf algebras. We conjecture a generalization of this result for Hopf Algebras whose coradical is a Hopf subalgebra; we prove this conjecture in the case of coradically graded Hopf algebras.

§1. Introduction.

A basic Theorem of Taft and Wilson ([TW], [M, Thm. 5.4.1]) describes the coradical filtration of a pointed coalgebra. This Theorem is the key point in the proof of the following result (see e. g. [N, p. 1545], [AS1, Prop. 3.1]): If A is a pointed non-cosemisimple Hopf algebra, with coradical k(Γ) where Γ is a finite abelian group, then there exist g ∈ Γ, a k-character χ of Γ such that χ(g) ≠ 1, and x ∈ A, x ∉ kΓ such that

(1.1)  hxh^{−1} = χ(h)x ∀h ∈ Γ, Δ(x) = x ⊗ g + 1 ⊗ x.

The preceding statement is the starting point in several classification results on pointed Hopf algebras. See [AS2], [CD], [SvO], [CDR]. The proof of the result by Taft and Wilson is not very difficult but somewhat involved [M, Thm. 5.4.1]. In this article we present an alternative natural proof of this Theorem for pointed Hopf algebras and propose a (conjectural) generalization for Hopf algebras whose coradical is a Hopf subalgebra. We prove the validity of the generalization assuming that the Hopf algebra is coradically graded.

Conventions. We shall work over a fixed field k; all the algebras, coalgebras, tensor products and homomorphisms are over k unless explicitly stated. If C is a coalgebra, we denote by G(C) the set of group-like elements of C. If g, h ∈ G(C), then we denote P_{g,h}(C) = P_{g,h} = {x ∈ C : Δ(x) = x ⊗ h + g ⊗ x}; the elements of P_{g,h} are called skew-primitives. When B is a bialgebra, P_{1,1}(B) is just the space P (B) of primitive elements.

If A is an algebra, Alg(A, k) denotes the set of all algebra maps from A to k.

Typeset by AMS-TEX

Key words and phrases. Hopf Algebras.

(N.A.) Forschungsstipendiat der Alexander von Humboldt-Stiftung. Also partially supported by CONICET, CONICOR, SeCYT (UNC) and FAMAF (República Argentina).

§2. Hopf algebras whose coradical is a Hopf subalgebra.

We recall the principle proposed in [AS2], see also [AS3].

Let A be a Hopf algebra over a field k. Let A_{0} ⊆ A_{1} ⊆ ⋯ ⊆ A be the coradical filtration of A and let gr A = ⊕_{n≥0} gr A(n) = ⊕_{n≥0}A_{n}/A_{n−1} (with A_{−1} = 0) be the associated graded coalgebra. If the coradical A_{0} of A is a Hopf subalgebra, then the coradical filtration is in fact a Hopf algebra filtration and gr A is a graded Hopf algebra. See [M, 5.2.8], [Sw, Section 11.2].

Let π : gr A → A_{0} be the unique graded projection; it has a Hopf algebra section γ, namely the inclusion of A_{0} in gr A. Let R = gr A^{co π} = {a ∈ gr A : (id ⊗π)Δ(a) = a ⊗ 1} be the algebra of coinvariants of π. By a result of Radford [R], interpreted in categorical terms by Majid [Mj], R is a braided Hopf algebra and gr A is the bosonization of R and A_{0}: gr A ≃ R#A_{0}. In [AS2], we proposed the following principle to study A: first to study R, then to transfer the information to gr A via bosonization, and finally to lift to A. We recall the relevant formulas within the conventions of [AS1].

The action of A_{0} on R is given by the adjoint representation composed with γ. The coaction is δ_{R} = (π ⊗ id)Δ. These two structures are related by the Yetter-Drinfeld condition:

δ_{R}(h.r) = h_{(1)}r_{(−1)} S(h_{(3)}) ⊗ h_{(2)}.r_{(0)}.

Hence, R is an object of the category _{A_{0}}^{A_{0}}YD of Yetter-Drinfeld modules over A_{0}. Moreover, R is a subalgebra of gr A and a coalgebra with comultiplication Δ_{R}(r) = r_{(1)}γπ S(r_{(2)}) ⊗ r_{(3)}; the counit is the restriction of the counit of gr A. To avoid confusions, we denote here the comultiplication of R in the following way: Δ_{R}(r) = ∑ r^{(1)} ⊗r^{(2)}, or even we omit sometimes the summation sign. The multiplication m and the comultiplication Δ of R satisfy Δm = (m ⊗ m)(id ⊗c ⊗ id)(Δ ⊗ Δ). Here c is the commutativity constraint of _{A_{0}}^{A_{0}}YD; explicitly c_{M,N}(m⊗n) = m_{(−1)}.n⊗m_{(0)}, for M, N ∈ _{A_{0}}^{A_{0}}YD, m ∈ M, n ∈ N.

It is not difficult to see that R is a braided Hopf algebra in _{A_{0}}^{A_{0}}YD. Note that gr A can be reconstructed from R and A_{0} in the following way: the multiplication gives rise to an isomorphism between gr A and tensor product R ⊗ A_{0} and the Hopf algebra structure is given via the smash product and smash coproduct:

(2.1)  \begin{aligned}(r#h)(s#f) & = r(h_{(1)}.s)#h_{(2)}f, \\ Δ(r#h) & = r^{(1)}#(r^{(2)})_{(−1)}h_{(1)} ⊗ (r^{(2)})_{(0)}#h_{(2)}. \end{aligned}

Note also that R inherits the grading from gr A: R = ⊕_{n≥0}R(n), where R(n) = R ∩ gr A(n). Moreover, it is a braided graded Hopf algebra with respect to this grading. Hence gr A(n) = R(n)#A_{0} and R_{0} = R(0) = k1. Following [CM], we say that a graded coalgebra C = ⊕_{n≥0}C(n) is coradically graded if the coradical filtration coincides with the filtration deduced from the grading: C_{m} = ⊕_{n≤m}C(n). It is known that gr A is coradically graded [AS2, Lemma 2.3]. It follows that

(2.3)  R is also a coradically graded coalgebra;

in particular

(2.4)  R_{0} = k1 = R(0) and P (R) = R(1).

Therefore, gr A_{1} = A_{0} ⊕ [P (R)#A_{0}].

§3. The Theorem of Taft and Wilson for pointed Hopf algebras.

The Theorem of Taft and Wilson gives information about the coradical filtration of a pointed coalgebra.

Theorem 3.1. ([TW]; see [M, Thm. 5.4.1]). Let C be a pointed coalgebra. (i) If n ≥ 1, the n-th term of the coradical filtration can be decomposed as

(3.1)  C_{n} = ∑_{g,h∈G(C)} C_{n}(g, h),

where

C_{n}(g, h) = {x ∈ C : Δ(x) = x ⊗ h + g ⊗ x + u, for some u ∈ C_{n−1} ⊗ C_{n−1}}.

(ii) The first term of the coradical filtration can be expressed as

(3.2)  C_{1} = kG(C) + (⊕_{g,h∈G(C)}P_{g,h}). □

The inclusions ⊇ in (3.1) and (3.2) are clear; the point of the Theorem is the other inclusions.

Let A be a Hopf algebra whose coradical A_{0} is a Hopf subalgebra. Let gr A be the associated graded Hopf algebra and R the associated braided graded Hopf algebra as in §2.

Lemma 3.2. Theorem 3.1 holds for R.

Proof. Let r ∈ R_{n}. By [AS2, Lemma 2.4], R_{n} = ⊕_{0≤j≤n}R(j); so, we can assume r ∈ R(n). Since R is a graded coalgebra and R(0) = k1, Δ_{R}(r) = r_{1} ⊗1+1⊗r_{2} +u, where r_{1}, r_{2} ∈ R(n), u ∈ ⊕_{0<s<n}R(s) ⊗ R(n − s). Applying id ⊗ε, resp. ε ⊗ id,we see that r = r_{1}, resp. r = r_{2}. The equality (3.1) follows; the equality (3.2) is part of [AS2, Lemma 2.4] as explained above– see (2.4). □

We assume now that A is pointed. Let us denote Γ = G(A), G = gr A. So A_{0} = kΓ and G is coradically graded.

Lemma 3.3. Theorem 3.1 holds for gr A.

Proof. Let x ∈ G_{n}. By[AS2, Lemma 2.4], we can asume that x ∈ G(n) = R(n)#kΓ. Now we can decompose R(n) in isotypical components as left comodule over kΓ:

R(n) = ⊕_{g∈Γ}R(n)_{g}, where R(n)_{g} = {r ∈ R(n) : δ(r) = g ⊗ r}.

We can assume that x = r#h, where r ∈ R(n)_{g} and g, h ∈ Γ. We know by Lemma 3.2 that Δ_{R}(r) = r ⊗ 1 + 1 ⊗ r + u, where u ∈ ⊕_{0<s<n}R(s) ⊗ R(n − s). We apply (2.1) to get

Δ_{G}(x) = r^{(1)}#(r^{(2)})_{(−1)}h ⊗ (r^{(2)})_{(0)}#h
= r#h ⊗ 1#h + 1#gh ⊗ r#h + ũ
= x ⊗ h + gh ⊗ x + ũ,

where ũ ∈ ⊕_{0<s<n}R(s)#kΓ ⊗ R(n − s)#kΓ. This proves (3.1). The proof of (3.2) is exactly the same; just use that R(1) = P (R). □

Lemma 3.4. Theorem 3.1 holds for a pointed Hopf algebra A.

Proof. The proof of (3.1) for A is a direct consequence of Lemma 3.3. Indeed, let x ∈ A_{n} and let \overline{x} be its class in gr A(n) = A_{n}/A_{n−1}. We can assume, by Lemma 3.3, that

Δ(\overline{x}) = \overline{x} ⊗ h + g ⊗ \overline{x} + u,

where g, h ∈ Γ, u ∈ ⊕_{i+j<n} gr A(i) ⊗ gr A(j). Hence there exist a, b ∈ A_{n−1}, v ∈ A_{n−1} ⊗ A_{n−1} such that

Δ(x) = (x + a) ⊗ h + g ⊗ (x + b) + v = x ⊗ h + g ⊗ x + (a ⊗ h + g ⊗ b + v) ;

(3.1) follows.

To prove (3.2), we have to lift some obstruction. Let g, h ∈ Γ and let A_{g,h} be the space of all x ∈ A such that

(3.3)  Δ(x) = x ⊗ h + g ⊗ x + u,

for some u ∈ kΓ ⊗ kΓ. By (3.1), which we already proved,

A_{1} = ∑_{g,h∈Γ} A_{g,h}.

Therefore, we have to prove the somewhat more precise result

(3.4)  A_{g,h} = P_{g,h} + kΓ.

Clearly, the inclusion ⊇ is true. Let x ∈ A_{g,h} and let u be as in (3.3). By the coassociativity condition, (3.3) implies the equality

(3.5)  u ⊗ h + (Δ ⊗ id)(u) = g ⊗ u + (id ⊗Δ)(u).

We look for a ∈ kΓ such that x − a ∈ P_{g,h}. This is equivalent to

(3.6)  u = a ⊗ h + g ⊗ a − Δ(a).

The Lemma will follow from the next result. □

Lemma 3.5. Given u ∈ kΓ ⊗ kΓ satisfying (3.5), there exists a ∈ kΓ such that (3.6) holds.

Proof. This is a particular case of Lemma A.1, cf. the Appendix. □

§4. The first term of the coradical filtration of a Hopf algebra. Now we propose a generalization of the Theorem of Taft and Wilson for arbitrary Hopf algebras. We assume in this section that the ground field k is algebraically closed. Let A be a Hopf algebra and let Â be the set of isomorphy classes of irreducible left comodules. The space C_{τ} of matrix coefficients of τ ∈ Â is a simple subcoalgebra of A and A_{0} = ⊕_{τ∈Â}C_{τ}.

Conjecture 4.1. The first term of the coradical filtration of A is given by

(4.1)  A_{1} = A_{0} + ∑_{τ∈Â}(C_{τ} ∧ k1).A_{0}

= A_{0} + ∑_{τ,μ∈Â} C_{τ} .C_{μ} ∧ C_{μ}. (4.2)

We observe that (4.1) implies (4.2): it is clear that

A_{1} ⊇ A_{0} + ∑_{τ,μ∈Â} C_{τ} .C_{μ} ∧ C_{μ} ⊇ A_{0} + ∑_{τ∈Â}(C_{τ} ∧ k1).A_{0};

the second inclusion follows from a direct computation.

We first check that the conjecture, for pointed Hopf algebras, is nothing but the second part of the Theorem of Taft and Wilson. Indeed, if A is pointed, then Â is identified with G(A) and the conjecture reads

(4.3)  A_{1} = A_{0} + ∑_{g∈G(A)}(kg ∧ k1).A_{0}

= A_{0} + ∑_{g,h∈G(A)} kg ∧ kh, (4.4)

after a suitable change of variables in (4.4). Now we claim that

(4.5)  kg ∧ kh = P_{g,h} + kg = P_{g,h} + kh.

The second equality is evident since g − h ∈ P_{g,h}. Also, the inclusion ⊇ in the first equality follows by definition. Now, if x ∈ kg ∧ kh, then

Δ(x) = g ⊗ x + h ⊗ x − ϵ(x)g ⊗ h.

Therefore x − ϵ(x)g ∈ P_{g,h} and (4.5) follows. We see, via (4.5), that (4.4) is equivalent to (3.2).

We assume now that the coradical A_{0} is a Hopf subalgebra of A. Let gr A, R be as in §2.

Lemma 4.2. The conjecture 4.1 holds for gr A.

Proof. By Lemma 2.3, gr A_{1} = A_{0} ⊕ gr A_{1} = A_{0} ⊕ P (R)#A_{0}. Now let P (R) = ⊕_{τ∈Â}P (R)_{τ} be the decomposition of P (R) into isotypical components as A_{0}- comodule. If r ∈ P (R)_{τ}, then by (2.1)

Δ_{gr A}(r) = r ⊗ 1 + r_{(−1)} ⊗ r_{(0)} ∈ gr A ⊗ 1 + C_{τ} ⊗ gr A;

that is, r ∈ C_{τ} ∧ k1. Hence (4.1) holds. □

Appendix. The co-Hochschild cohomology for Hopf algebras. In this Appendix we prove a result needed in the text, in the context of co-Hochschild cohomology for Hopf algebras; cf. [GS]. See also [S] for applications to finiteness results for semisimple Hopf algebras.

Let C be a coalgebra and let V be a (C, C)-bicomodule, with structure maps σ : V → V ⊗ C, ρ : V → C ⊗ V. We denote as usual σ(v) = v_{(−1)} ⊗ v_{(0)}, ρ(v) = v_{(0)} ⊗ v_{(1)}; then the compatibility condition between the two coactions justifies the notation

v_{(−1)} ⊗ v_{(0)} ⊗ v_{(1)} := v_{(−1)} ⊗ ρ (v_{(0)}) = σ (v_{(0)}) ⊗ v_{(1)}.

Analogous to the usual Hochschild complex of an algebra with coefficients in a bimodule, the co-Hochschild complex of the coalgebra C with coefficients in the bicomodule V is

0 ⟶ Hom(V, k) ⟶^{δ^{0}} Hom(V, C) ⟶^{δ^{1}} Hom(V, C ⊗ C) ⟶^{δ^{2}} Hom(V, C ⊗ C ⊗ C) ⟶^{δ^{3}} …

where, if for f ∈ Hom(V, C^{⊗n}) we denote f(v) = v^{1} ⊗ v^{2} ⊗ ⋯ ⊗ v^{n}, then

(A.1) δ^{n}(f)(v) := v_{(−1)} ⊗ f (v_{(0)})

+ ∑_{i=1}^{n}(−1)^{i}v^{1} ⊗ v^{2} ⊗ ⋯ ⊗ v^{i−1} ⊗ Δ(v^{i}) ⊗ v^{i+1} ⊗ ⋯ ⊗ v^{n}

+ (−1)^{n+1}f(v_{(0)}) ⊗ v_{(1)}.

It is routinary to verify that δ^{n+1}δ^{n} = 0 for all n ≥ 0.

The coalgebra C itself is a (C, C)-bicomodule with σ = ρ = Δ. We regard C ⊗C as a (C, C)-bicomodule with left coaction in the first factor and right coaction in the second factor.

Definition. A coalgebra C is coseparable iff the comultiplication Δ : C → C ⊗ C has a retraction π : C ⊗ C → C of (C, C)-bicomodules. Equivalently, if C is an injective (C, C)-bicomodule.

For instance, let X be a set and let C = kX be the ”set coalgebra”; it has a basis (x)_{x∈X} and the comultiplication is given by Δ(x) = x ⊗ x. Let π : C ⊗ C → C be the map

π(x ⊗ y) = δ_{x,y}x, x, y ∈ X.

Then π is a bicomodule retraction of Δ. That is, C is coseparable.

Let C be coseparable and let π be as in the preceding definition. Let

s_{n} : Hom(V, C^{⊗n}) → Hom(V, C^{⊗n−1}), n ≥ 1

be the map given by

s_{n}(f)(v) := ϵπ (v_{(−1)}(v_{(0)})^{1}) (v_{(0)})^{2} ⊗ ⋯ ⊗ (v_{(0)})^{n},

with the same convention as in (A.1). Then

(A.2)  δ^{n−1}s_{n} + s_{n+1}δ^{n} = id,  ∀n ≥ 1.

In particular, the n-th cohomology groups H^{n}(C, V) of the co-Hochschild complex are 0 for n ≥ 1.

Lemma A.1. Let X be a set, C = kX the set coalgebra, g, h ∈ X. Given u ∈ kX ⊗ kX satisfying (3.5), there exists a ∈ kX such that (3.6) holds.

Proof. Let V be a one dimensional vector space with base {v}. We regard V as a (C, C)-bicomodule via

σ(v) = g ⊗ v, ρ(v) = v ⊗ h.

Given u ∈ kX ⊗ kX, a ∈ kX, we define f ∈ Hom(V, C^{⊗2}), g ∈ Hom(V, C) by

f(v) = u, g(v) = a.

Then (3.5) means that δ^{2}(f) = 0, (3.6) that δ^{1}(g) = f, and the Lemma that H^{2}(C, V) = 0. This follows from the above considerations. One could check, explicitly, that a = (ϵπ ⊗ id)(g ⊗ u) satisfies (3.6) whenever u satisfies (3.5). □

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N. Andruskiewitsch: FAMAF. UNC. (5000) C. Universitaria. Córdoba. Argentina E-mail address: andrus@mate.uncor.edu

H.-J. Schneider: Mathematisches Seminar der Universität München. Theressienstr. 39. (80333) München. Germany

E-mail address: hanssch@rz.mathematik.uni-muenchen.de