An action-free characterization of weak Hopf-Galois extensions

Lars Kadison

Abstract

We define comodule algebras and Galois extensions for actions of bialgebroids. Using just module conditions we characterize the Frobenius extensions that are Galois as depth two and right balanced extensions. As a corollary, we obtain characterizations of certain weak and ordinary Hopf-Galois extensions without reference to action in the hypothesis.

2000 AMS Subject Classification: 13B05, 16W30 Keywords: bialgebroid, depth two, Frobenius extension, Hopf-Galois extension, weak Hopf algebra.

1 Introduction

A finite Hopf H-Galois extension A|B is a Frobenius extension and has many characterizations as an H^{∗}-module algebra A with invariants B satisfying various conditions, or dually, as an H-comodule algebra A with coinvariants B satisfying various conditions [8]; and has many interesting applications [7, 8]. There is recently in [5, 6] a characterization of certain noncommutative Hopf-Galois extensions - those with trivial centralizer (and arising in subfactor theory) - in terms of module-theoretic conditions of depth two on the tensor-square of the extension and “balanced” on the module A_{B}. The main thesis in [6] is that to a depth two ring extension A|B one associates by construction two bialgebroids over the unrestricted centralizer R, a left bialgebroid S := End_{B}A_{B} and a right bialgebroid T := (A ⊗_{B} A)^{B}, the B-central elements with multiplication induced from T ≅ End_{A}(A ⊗_{B} A)_{A}. The bialgebroids S and T are simultaneously each other’s left and right R-dual bialgebroids and they act on A and End_{B}A respectively. If B is trivial, we obtain the two main examples of Lu bialgebroids. If R is trivial, R-bialgebroids are usual bialgebras, and if R is a separable algebra, R-bialgebroids are weak bialgebras: antipodes may be added to create weak Hopf algebras if A|B is additionally a Frobenius extension. If A_{B} is balanced, the invariants A^{S} = B and the endomorphism ring End A_{B} is a smash product of A and S (tensoring over R), which signals a Galois extension. Good definitions of Galois extension in terms of applications have appeared very recently [1, 2]. In this paper we extend the main theorems in [5, 6] to weak Hopf-Galois extension and Galois extensions for bialgebroids.

2 Preliminaries

Let B ⊆ A be an associative not necessarily commutative subring pair sharing 1, also referred to here as a ring extension A|B.

Definition 2.1 A ring extension A|B is depth two (D2) if the tensor-square A ⊗_{B} A is isomorphic both as natural B-A-bimodules (left D2) and as A-B-bimodules (right D2) to a direct summand of a finite direct sum of A with itself. Equivalently, A|B is D2 if there are (left D2 quasibase) elements β_{i} ∈ S, t_{i} ∈ T such that

a ⊗_{B} a′ = ∑_{i} t_{i}β_{i}(a)a′  (1)

and (right D2 quasibase) elements γ_{j} ∈ S, u_{j} ∈ T such that

a ⊗_{B} a′ = ∑_{j} aγ_{j}(a′)u_{j}  (2)

for all a, a′ ∈ A.

Example 2.2 A finite dimensional algebra is D2 with dual bases as a vector space corresponding to D2 quasibases. Given a subgroup of a finite group H < G, the complex group subalgebra pair CH ⊆ CG is D2 iff H ◁ G [4]. Another related example: a normal Hopf subalgebra pair is D2. Yet another is a finite weak Hopf-Galois extension [3, 3.1].

Recall from [6] that a right R′-bialgebroid T′ are two rings R′ and T′ with two maps s̃, t̃ : R′ → T′, a ring homomorphism and anti-homomorphism resp., such that s̃(r)t̃(r′) = t̃(r′)s̃(r) for all r, r′ ∈ R′, (T′, Δ : T′ → T′ ⊗_{R′} T′, ε : T′ → R′) is an R′-coring w.r.t. the R′-R′-bimodule r · x · r′ = xt̃(r)s̃(r′) such that (s̃(r) ⊗ 1)Δ(x) = (1 ⊗ t̃(r))Δ(x), Δ(xy) = Δ(x)Δ(y) (which makes sense thanks to the previous axiom), Δ(1) = 1⊗1, ε(1_{T′}) = 1_{R′} and ε(xy) = ε(s̃(ε(x))y) = ε(t̃(ε(x))y) for all x, y ∈ T′, r, r′ ∈ R′. A left bialgebroid is just a right bialgebroid with three of the axioms transposed [6].

Example 2.3 [6, section 5] Given a D2 extension A|B, the ring T = (A ⊗_{B} A)^{B} is a right bialgebroid over the centralizer R := C_{A}(B) with maps s̃(r) = 1 ⊗ r, t̃(r) = r ⊗ 1, Δ(x) = ∑_{j}(x^{1} ⊗_{B} γ_{j}(x^{2})) ⊗_{R} u_{j} and ε(x) = x^{1}x^{2} for all x = x^{1} ⊗ x^{2} ∈ T. Note then that _{R}T_{R} is given by r · x · r′ = rx^{1} ⊗ x^{2}r′ and _{R}T is finite projective from eq. (2).

The dual left R-bialgebroid is S, there being two R-valued nondegenerate pairings of S and T; e.g., ⟨α, t⟩ = t^{1}α(t^{2}) for each α ∈ S, t ∈ T. The left bialgebroid structure is given by \overline{s}(r) = λ_{r}, left multiplication by r ∈ R, \overline{t}(r) = ρ_{r}, right multiplication, Δ(α) = ∑_{i} α(?t_{i}^{1})t_{i}^{2} ⊗ β_{i} and ε_{S}(α) = α(1).

The R-bialgebroid S acts on A by α▷a = α(a) (where R acts as a subring of A) with invariants A^{S} = {a ∈ A|α ▷ a = ε_{S}(α)a ∀α ∈ S} ⊇ B, and if A_{B} is balanced, A^{S} = B. A is thereby a left S-module algebra (or algebroid [6, 2.1]). We need the dual notion:

Definition 2.4 Let T′ be a right R′-bialgebroid (T′, s̃, t̃, Δ, ε). A (right) T′- comodule algebra A′ is a ring A′ with ring homomorphism R′ → A′ together with a coaction δ : A′ → A′ ⊗_{R′} T′, where values δ(a) are denoted by the Sweedler notation a_{(0)} ⊗ a_{(1)}, such that A′ is a right T′-comodule over the R′-coring T′ [1, 18.1], δ(1_{A′}) = 1_{A′} ⊗1_{T′}, ra_{(0)} ⊗a_{(1)} = a_{(0)} ⊗t̃(r)a_{(1)} for all r ∈ R′, and δ(aa′) = δ(a)δ(a′) for all a, a′ ∈ A′. The subring of coinvariants is A′^{co T′} := {a ∈ A′|δ(a) = a ⊗ 1_{T′}}. Consequently R′ and A′^{co T′} commute in A′.

For example, T′ is a comodule algebra over itself. A D2 extension A|B has T -comodule algebra A [3, 5.1], indeed a T -Galois extension, which we define as follows.

Definition 2.5 Let T′ be a left finite projective right R′-bialgebroid. A T′- comodule algebra A′ is a (right) T′-Galois extension of its coinvariants B′ if the (Galois) mapping β : A′ ⊗_{B′} A′ → A′ ⊗_{R′} T′ defined by β(a ⊗ a′) = aa′_{(0)} ⊗ a′_{(1)} is bijective.

3 D2 characterization of Galois extensions

In this section we provide characterizations of generalized Hopf-Galois extensions in analogy with the Steinitz characterization of Galois extension of fields as being separable and normal.

Theorem 3.1 Let A | B be a Frobenius extension. The extension A|B is T -Galois for some left finite projective right bialgebroid T over some ring R if and only if A|B is D2 with A_{B} balanced.

Proof. (⇒) Since _{R}T ⊕∗ ≅ _{R}R^{t} for some positive integer t, we apply to this the functor A ⊗_{R} − from left R-modules into A-B-bimodules which results in _{A}A⊗_{B}A_{B} ⊕ ∗ ≅ _{A}A_{B}^{t}, after using the Galois A-B-isomorphism A ⊗_{B} A ≅ A ⊗_{R} T. Hence, A|B is right D2, and left D2 since A|B is Frobenius [6, 6.4].

Let E := End A_{B}. The module A_{B} is balanced iff the natural bimodule EA_{B} is faithfully balanced, which we proceed to show based on the following claim. Let R be a ring, M_{R} and _{R}V modules with _{R}V finite projective. If ∑_{j} m_{j}ϕ(v_{j}) = 0 for all ϕ in the left R-dual ^{∗}V, then ∑_{j} m_{j} ⊗_{R} v_{j} = 0. This claim follows immediately by using dual bases f_{i} ∈ ^{∗}V, w_{i} ∈ V.

Given F ∈ End_{E}A, it suffices to show that F = ρ_{b} for some b ∈ B. Since λ_{a} ∈ E, F ∘λ_{a} = λ_{a}∘F for all a ∈ A, whence F = ρ_{F (1)}. Designate F (1) = a. If we show that a_{(0)} ⊗a_{(1)} = a⊗1 after applying the right T -valued coaction on A, then a ∈ A^{co T} = B. For each α ∈ ^{∗}(_{R}T), define \overline{α} ∈ End A_{B} by \overline{α}(x) = x_{(0)}α(x_{(1)}). Since ρ_{r} ∈ E for each r ∈ R by 2.4,

aα(1_{T}) = F (\overline{α}(1)) = \overline{α}(F (1)) = a_{(0)}α(a_{(1)})

for all α ∈ ^{∗}T. By the claim a_{(0)} ⊗_{R} a_{(1)} = a ⊗ 1_{T}.

(⇐) Let T be the left projective right bialgebroid (A ⊗_{B} A)^{B} over R = C_{A}(B). Using a right D2 quasibase, we give A the structure of a right T - comodule algebra via a_{(0)} ⊗ a_{(1)} := ∑_{j} γ_{j}(a) ⊗ u_{j} ∈ A ⊗_{R} T, the details of which are in [3, 5.1]. The D2 condition ensures that θ : A ⊗_{R} T → A ⊗_{B} A defined by θ(a ⊗ t) = at^{1} ⊗ t^{2} is an isomorphism.

Note that for each b ∈ B

b_{(0)} ⊗ b_{(1)} = ∑_{j} γ_{j}(b) ⊗_{R} u_{j} = b ⊗ ∑_{j} γ_{j}(1)u_{j} = b ⊗ 1_{T}

so B ⊆ A^{co ρ}. The converse: if ρ(x) = x ⊗ 1_{T} = ∑_{j} γ_{j}(x) ⊗ u_{j} applying θ we obtain x ⊗_{B} 1 = 1 ⊗_{B} x. Since A_{B} is balanced, we know A^{S} = B under the action ▷ of S on A [6, 4.1]. Applying μ(α ⊗ id) for each α ∈ S, where μ is multiplication, we obtain α ▷ x = α(x) = α(1)x, whence x ∈ B.

The Galois mapping β : A ⊗_{B} A → A ⊗_{R} T given by

β(a ⊗ a′) := aa′_{(0)} ⊗_{R} a′_{(1)}  (3)

is an isomorphism since θ is an inverse by eq. (2). Q.e.d.

Corollary 3.2 Let k be a field and A|B be a Frobenius extension of kalgebras with centralizer R a separable k-algebra. The extension A|B is weak Hopf-Galois iff A|B is D2 with A_{B} balanced.

The proof of ⇐ depends first on recalling that the right R-bialgebroid T is a weak bialgebra since R has an index-one Frobenius system (ϕ : R → k, e_{i}, f_{i} ∈ R) where ∑_{i} e_{i}f_{i} = 1_{A} and ∑_{i} ϕ(re_{i})f_{i} = r = ∑_{i} e_{i}ϕ(f_{i}r) for all r ∈ R, whence Δ(t) = ∑_{i} t_{(1)}e_{i} ⊗_{k} f_{i}t_{(2)} and ε(t) = ϕ(t^{1}t^{2}) satisfy the axioms of a weak bialgebra [6, (96)]. Since A|B is Frobenius, the dual bases tensor is a nondegenerate right integral in T, whence T is weak Hopf algebra by the Larson-Sweedler-Vecsernyes theorem. The coaction on A has values in A⊗_{R} T ≅ (A⊗_{k} T)Δ(1), given by a⊗t ↦ ∑_{i} ae_{i} ⊗f_{i}t, an isomorphism of the Galois A-corings in [3, 5.1] and in [2, 2.1]. The proof of ⇒ follows from the fact that a weak Hopf-Galois extension is D2 and an argument that A_{B} is balanced like the one above.

Example 3.3 A separable field extension \overline{k}|k is a weak Hopf-Galois extension, since \overline{k} is a separable k-algebra.

The theorem provides another proof and extends the theorems [6, 8.14] and [5, 6.6] as we see below. We define an irreducible k-algebra extension to be an extension where the centralizer is the trivial k1.

Corollary 3.4 Let A|B be an irreducible extension. The extension A|B is finite Hopf-Galois ⟺ A|B is a D2, right balanced extension.

The proof of ⇒ does not require the centralizer to be trivial. The Frobenius condition may be dropped here from the proof of ⇐ since the bialgebra T acts Galois implies it is a Hopf algebra [9]. If the characteristic of k is zero, the Larson-Radford theorem permits the condition “right balanced” to be replaced by “separable extension” [3, 4.1].

References

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Matematiska Institutionen, Göteborg University, S-412 96 Göteborg, Sweden, lkadison@c2i.net, September 20, 2004.

[AMA - Algebra Montpellier Announcements - 01-2005] [April 2005]
Received September 2004.