New York Journal of Mathematics

New York J. Math. 6 (2000) 325–329.

Doi-Koppinen Hopf Modules Versus Entwined Modules

Peter Schauenburg

Abstract. A Hopf module is an A-module for an algebra A as well as a C-comodule for a coalgebra C, satisfying a suitable compatibility condition between the module and comodule structures. To formulate the compatibility condition one needs some kind of interaction between A and C. The most classical case arises when A = C =: H is a bialgebra. Many subsequent variants of this were unified independently by Doi and Koppinen; in their version an auxiliary bialgebra H, over which A is a comodule algebra and C a module coalgebra, governs the compatibility. Another very general type of interaction between A and C is an entwining map as studied by Brzeziński — without an auxiliary bialgebra.

Every Doi-Koppinen datum induces an entwining structure, so Brzeziński’s notion of an entwined module includes that of a Doi-Koppinen Hopf module. This paper investigates whether the inclusion is proper.

By work of Tambara, every entwining structure can be obtained from a suitable Doi-Koppinen datum whenever the algebra under consideration is finite dimensional.

We show by examples that this need not be true in general.

Contents

1. The definitions 325 2. The finite dimensional case 327 3. Counterexamples 327 References 328

1. The definitions

Throughout the paper k is a field, and all algebras, coalgebras etc. are meant to be over k. We denote the multiplication map of an algebra A by ∇: A ⊗ A → A, and the comultiplication of a coalgebra C by Δ: C → C ⊗ C, using Sweedler’s notation in the form Δ(c) = c_{(1)} ⊗ c_{(2)}. We refer to [4] for the general background on Hopf algebra theory.

Received December 6, 2000. Mathematics Subject Classification. 16W30. Key words and phrases. Hopf algebra, Hopf module, entwining structure.

325^{ISSN 1076-9803/00}

Let H be a Hopf algebra. A (right-right) Hopf module over H is a right H-module as well as right H-comodule M with the property that the comodule structure map ρ: M → M ⊗H is an H-module map; here M ⊗H is endowed with the diagonal right H-module structure. This means that (mh)_{(0)}⊗(mh)_{(1)} = m_{(0)}h_{(1)}⊗m_{(1)}h_{(2)} holds for m ∈ M and h ∈ H, when we write the comodule structure as ρ(m) = m_{(0)}⊗m_{(1)}. Equivalently, the H-module structure μ: M ⊗H → M is an H-comodule map. This ur-notion of Hopf module has seen far-reaching generalizations. First of all, it makes perfect sense to replace the H-module structure by an A-module structure for an H-comodule algebra A, or, dually, to replace the H-comodule structure by a Ccomodule structure for an H-module coalgebra C. Moreover, one has reason to study Hopf modules which are A-modules for an H-comodule algebra A, but only \overline{H}-comodules for some quotient coalgebra and right module \overline{H} of H. To unify all these situations Doi [2] and Koppinen [3] have introduced what we will call Doi-Koppinen Hopf modules with respect to a Doi-Koppinen datum. By definition, a Doi-Koppinen datum is a triple (A, C, H) in which H is a bialgebra, A is a right Hcomodule algebra, and C is a right H-module coalgebra whose module structure we denote by a dot. A Doi-Koppinen Hopf module with respect to the Doi-Koppinen datum (A, C, H) is a right A-module as well as right C-comodule M satisfying the compatibility condition ρ(ma) = m_{(0)}a_{(0)} ⊗ m_{(1)} · a_{(1)} for all a ∈ A and m ∈ M. Again, we may formulate this as the condition that the module structure is a Ccolinear map, or, equivalently, the comodule structure is an A-module map, by making the obvious definitions.

As Brzeziński [1] realized, all that is really needed to write down a Hopf modulelike compatibility between a right A-module and right C-comodule structure is a so-called entwining structure: This is by definition a triple (A, C, ψ) in which ψ is a map ψ: C ⊗ A → A ⊗ C satisfying

(A ⊗ Δ)ψ = (ψ ⊗ C)(C ⊗ ψ)(Δ ⊗ A): C ⊗ A → A ⊗ C ⊗ C,
ψ(C ⊗ ∇) = (∇ ⊗ C)(A ⊗ ψ)(ψ ⊗ A): C ⊗ A ⊗ A → A ⊗ C,

(A ⊗ ε)ψ = ε ⊗ A, and ψ(c ⊗ 1) = 1 ⊗ c. An entwined module with respect to the entwining structure ψ is by definition a right A-module and right C-comodule M such that

ρμ = (μ ⊗ C)(M ⊗ ψ)(ρ ⊗ A): M ⊗ A → M ⊗ C.

Every Doi-Koppinen datum induces an entwining structure, namely

ψ: C ⊗ A ∋ c ⊗ a ↦ a_{(0)} ⊗ c · a_{(1)} ∈ A ⊗ C.

The entwined modules with respect to this entwining structure are precisely the Doi-Koppinen Hopf modules for the Doi-Koppinen datum in consideration.

While entwining structures are sometimes harder to cope with notationally, they offer a clear conceptual advantage over Doi-Koppinen data: The auxiliary bialgebra that is needed to formulate the compatibility condition for a Doi-Koppinen Hopf module is no longer needed for entwining structures. However, the obvious question arises whether entwining structures are a truly more general notion than Doi-Koppinen data, or whether in fact every entwining structure is induced by a suitable Doi-Koppinen datum.

2. The finite dimensional case

If the algebra A in an entwining structure (A, C, ψ) is finite dimensional, then work of Tambara [5] shows that there is in fact a suitable bialgebra H and Doi-Koppinen datum (A, C, H) inducing the entwining map ψ.

Let us rephrase the results of Sections 1 and 2 of [5] that are relevant for our question: Let A be an algebra. A transition map through A is a pair (V, ψ) in which V is a vector space and ψ: V ⊗ A → A ⊗ V is a linear map satisfying

(∇ ⊗ V)(A ⊗ ψ)(ψ ⊗ A) = ψ(V ⊗ ∇): V ⊗ A ⊗ A → A ⊗ V

and ψ(v ⊗ 1) = 1 ⊗ v. Transition maps through A form a monoidal category with (V, ψ) ⊗ (W, ϕ) := (V ⊗ W, (ψ ⊗ W)(V ⊗ ϕ)). Now if A is finite dimensional, then there exists a bialgebra e(A) = a(A, A) and an e(A)-comodule algebra structure A → A ⊗ e(A) with the following property: A category equivalence between the category of right e(A)-modules and the category of transition maps through A is given by assigning to an e(A)-module V the transition map (V, ψ) with ψ(v ⊗ a) = a_{(0)} ⊗ v · a_{(1)}.

From the results of Tambara we have thus summed up we conclude immediately:

Proposition 2.1. If the algebra A has finite dimension, then every entwining structure (A, C, ψ) is induced by a Doi-Koppinen datum.

In fact we need only add the observation that an entwining structure (A, C, ψ) is the same as a coalgebra in the monoidal category of transition maps through A, hence the same as a coalgebra in the monoidal category of e(A)-modules; thus (A, C, ψ) is induced by a Doi-Koppinen datum (A, C, e(A)).

3. Counterexamples

There exist entwining structures (A, C, ψ) that cannot be obtained from a Doi-Koppinen datum (A, C, H); this can even happen for finite dimensional C.

The examples rely on the following easy observation.

Lemma 3.1. Let (A, C, ψ) be an entwining structure. Fix c ∈ C and γ ∈ C^{∗}, and define the vector space endomorphism T_{c,γ} of A by T_{c,γ}(a) = (A ⊗ γ)ψ(c ⊗ a).

If the entwining structure (A, C, ψ) is induced by a Doi-Koppinen datum, then every a ∈ A is contained in a finite dimensional T_{c,γ}-invariant subspace of A.

Proof. If (A, C, ψ) is induced by the Doi-Koppinen datum (A, C, H), then we find T_{c,γ}(a) = a_{(0)}γ(c · a_{(1)}). From this formula it is obvious that every H-subcomodule of A is a T_{c,γ}-invariant subspace. But every a ∈ A is contained in a finite dimensional H-subcomodule of A. □

Example 3.2. Let C be the two-dimensional coalgebra k1 ⊕ kt with grouplike element 1 and (1, 1)-primitive t. Let A be the free algebra on generators X_{i} for i ∈ ℤ. Define the entwining map ψ: C ⊗ A → A ⊗ C by ψ(1 ⊗ a) = a ⊗ 1 for all a ∈ A and

ψ(t ⊗ X_{i_{1}}X_{i_{2}} … X_{i_{n}}) = X_{i_{1}+1}X_{i_{2}+1} … X_{i_{n}+1} ⊗ t.

Then (A, C, ψ) is an entwining structure that cannot be obtained from any Doi-Koppinen datum.

Proof. It is easy to check directly that ψ is an entwining map; see below for a more conceptual idea. Choose γ ∈ C^{∗} with γ(t) = 1. Then we have T_{t,γ}(X_{i}) = X_{i+1}. Thus the T_{t,γ}-invariant subspace of A generated by X_{0} is infinite dimensional. We conclude that (A, C, ψ) cannot be obtained from any Doi-Koppinen datum. □

While our counterexample cannot be obtained from a Doi-Koppinen datum, it can be derived from something very similar, which we will call an alternative Doi-Koppinen datum. This is by definition a triple (A, C, H) consisting of a bialgebra H, a left H-module algebra A, and a left H-comodule coalgebra C; we will write the comodule structure of the latter as c ↦ c_{[−1]} ⊗ c_{[0]}. One can check that every alternative Doi-Koppinen datum induces an entwining structure in a fashion very similar to the case of a Doi-Koppinen datum: One defines ψ: C ⊗ A → A ⊗ C by ψ(c ⊗ a) = c_{[−1]} · a ⊗ c_{[0]}. We omit the necessary verifications as they are very similar to the ones for Doi-Koppinen data. Our example above can be obtained from an alternative Doi-Koppinen datum (C, A, kℤ): The necessary comodule structure on C corresponds to the ℤ-grading of C in which 1 and t have degrees 0 and 1, respectively, and the module algebra structure on A is that for which the generator 1 ∈ ℤ shifts the indices in all the free generators X_{i} by one.

Remark 3.3. If the coalgebra C has finite dimension, then every entwining structure (A, C, ψ) is induced by an alternative Doi-Koppinen datum.

We will not supply any details of the proof, which is very similar to that of Proposition 2.1; we only remark that Tambara’s a(C^{∗}, C^{∗}) can serve as the necessary bialgebra.

The question remains whether there exist entwining structures (A, C, ψ) that is induced neither by a Doi-Koppinen datum, nor by an alternative Doi-Koppinen datum.

Example 3.4. Let C = k1 ⊕ ⨁_{i∈ℤ} kt_{i} where 1 is grouplike and each t_{i} is (1, 1)- primitive, and we let A = k[τ_{i} | i ∈ ℤ]/(τ_{i}τ_{j} | i, j ∈ ℤ). An entwining structure ψ: C ⊗ A → A ⊗ C can be defined by ψ(1 ⊗ τ_{j}) = τ_{j} ⊗ 1, ψ(t_{i} ⊗ 1) = 1 ⊗ t_{i}, and ψ(t_{i} ⊗ τ_{j}) = τ_{j+1} ⊗ t_{i+1}.

If γ ∈ C^{∗} satisfies γ(t_{1}) = 1, then we find T_{t_{0},γ}(τ_{j}) = τ_{j+1} for all j ∈ ℤ, so that the T_{t_{0},γ}-invariant subset of A generated by τ_{0} is infinite dimensional. Thus ψ is not induced by a Doi-Koppinen datum. A similar argument using the endomorphism T′: C → C defined by T′(c) = (α ⊗ C)ψ(c ⊗ τ_{0}) with α ∈ A^{∗} satisfying α(τ_{1}) = 1 shows that ψ is not induced by an alternative Doi-Koppinen datum. We omit checking that ψ is an entwining map.

References

[1] T. Brzeziński, On modules associated to coalgebra Galois extensions, J. Algebra 215 (1999), 290–317, MR 2000c:16047, Zbl 936.16030.

[2] Y. Doi, Unifying Hopf modules, J. Algebra 153 (1992), 373–385, MR 94c:16048, Zbl 782.16025.

[3] M. Koppinen, Variations on the smash product with applications to group-graded rings, J. Pure Appl. Algebra 104 (1995), 61–80, MR 96j:16028, Zbl 838.16035.

[4] S. Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, AMS, Providence, Rhode Island, 1993, MR 94i:16019, Zbl 793.16029.

[5] D. Tambara, The coendomorphism bialgebra of an algebra, J. Fac. Sci. Univ. Tokyo, Sect.IA, Math. 37 (1990), 425–456, MR 91f:16048, Zbl 717.16030.

Doi-Koppinen Hopf Modules Versus Entwined Modules 329

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