Frobenius induction for algebras

TIBERIU COCONEŢ Babeş-Bolyai University

Faculty of Economics and Business Administration Str. Teodor Mihali, nr.58-60 400591 Cluj-Napoca, Romania tiberiu.coconet@math.ubbcluj.ro

MARCUS ANDREI BabeşBolyai University

Faculty of Mathematics and Computer Science

Str. Kogălniceanu 1, 400084 Cluj-Napoca, Romania marcus@math.ubbcluj.ro

CONSTANTIN-COSMIN TODEA Technical University of Cluj-Napoca

Department of Mathematics, Str. G. Baritiu 25 Cluj-Napoca 400027, Romania Constantin.Todea@math.utcluj.ro

June 4, 2016

Abstract

Let B → A be a homomorphism of Hopf algebras and let C be an algebra. We consider the induction from B to A of C in two cases: when C is a B-interior algebra and when C is a B-module algebra. Our main results establish the connection between the two inductions. The inspiration comes from finite group representation theory, and some constructions work in even more general contexts.

MSC 16S40, 16S50, 16T05, 16S35, 20C05, 19A22. Keywords Hopf algebra, Frobenius extension, induction, augmented algebra, Hopf module algebra,

smash product, duality.

1 Introduction

In finite group representation theory a notion of Frobenius induction for algebras was introduced by Lluís Puig in [6, Definition 3.3]. If H is a subgroup of the finite group G, k is a field and kH → C is a homomorphism of algebras (C is called a kH-interior algebra), then

Ind_{H}^{G}C := kG ⊗_{kH}C ⊗_{kH}kG is naturally a kG-interior algebra. This construction has many important uses, and it is strongly related to the classical Frobenius induction for modules. If V is a kH-module, then there is a natural isomorphism

Ind_{H}^{G}(End_{k}(V)) ≃ End_{k}(kG ⊗_{kH}V)

of kG-interior algebras. Puig also introduced in [7] a non-injective version, so Ind_{H}^{G}C may be defined for any group homomorphism H → G, and has a similar property. Linckelmann showed in [4] that Puig’s induction may be generalized as follows. If A and B are k-algebras, M is an (A,B)-bimodule and C is a B-interior algebra, then, by definition

Ind_{M}(C) := End_{C^{op}}(M ⊗_{B}C),

which is naturally an A-interior algebra.

On the other hand, if B is a k-algebra acted upon by the subgroup G, Turull defined in [9] the induced G-algebra

Ind_{H}^{G}C := kG ⊗_{kH}C

by regarding C as a kH-module via the given H-action, with multiplication

(g_{1}⊗ c_{1})(g_{1}⊗ c_{1}) = \{ \begin{matrix}g_{1}⊗ a_{1}a_{2}, & if g_{1} = g_{2} \\ 0, & if g_{1}H ≠ g_{2}H, \end{matrix}

and G-action

g_{2}(g_{1}⊗ c) = g_{2}g_{1}⊗ c_{1},

for all c_{1},c_{2} ∈ C and g_{1},g_{2} ∈ G.

In this paper we are concerned with the following two problems. The first is to give conditions on the (A,B)-bimodule M such that the induced algebra Ind_{M}(C) can be expressed in two ways: as an endomorphism algebra, and as a tensor product. The second problem is to find the relationship between the two types of induction.

In Section 2 we give a positive answer to the first question when A is β-Frobenius extension of B as in [3] and [2]. In Section 3 we generalize the surjective version of Puig’s induction to the case of a homomorphism B → A of augmented algebras with some additional conditions. In Section 4 we define Turull’s induction in the situation when B is a Hopf subalgebra of the Hopf algebra A and C is a B-module algebra. We also define an surjective version of Turull’s induction through a homomorphism B → B̄of Hopf algebras.

Our main results are given in Section 5, where we start with a Hopf subalgebra B of A, and a B-module algebra C. Then the smash product C#B is a B-interior algebra, so we may construct Puig’s induction from B to A of C#B, and also the smash product between the Turull’s induced algebra Ind_{B}^{A}C and A. We prove in Theorem 5.4 below that, briefly speaking, induction commutes with the construction of the smash product, and this may also be regarded as a duality theorem. In fact, a particular case of Theorem 5.4 is related to the finite dimensional versions of some results of [5, §9.4]. Finally, Theorem 5.6 is the surjective counterpart of Theorem 5.4.

One might ask which are the applications of these constructions and results. First, we can now generalize other results regarding induction of algebras of Puig and Turull from groups to Hopf algebras. This is the objective of a possible follow-up article. Also, notice that by the methods from this article we can induce new k-algebras starting from a given k-algebra; induction is usually used in module categories. Moreover these induced algebras applied to Hopf module algebras preserves smash products.

Our notations and assumptions are standard. If k a commutative ring and A is a k-algebra, we denote by A-Mod, A^{op}-Modthe category of (unitary) left A-modules, respectively right A-modules. Homomorphisms and subalgebras of k-algebras are unitary. For n a positive integer and a k-algebra A we denote by M_{n}(A) the matrix algebra. We follow [5] for notations and basic facts regarding Hopf algebras, and we recall in each section the needed definitions and results.

2 Injective induction and β-Frobenius extensions

In this section, let k be a commutative ring and let A,B be two k-algebras. A B-interior k-algebra is a kalgebra C for which there is a homomorphism σ : B → C of k-algebras. In this case, we make the convention that C is a (B,B)-bimodule through σ, that is

b_{1}· c · b_{2} = σ(b_{1})cσ(b_{2})

for any b_{1},b_{2} ∈ B and c ∈ C. We denote this by_{B}C_{B}, and implicitly, the action of B is through σ. If M is an (A,B)-bimodule, Linckelmann defined in [4] the induced algebra

Ind_{M}(C) := End_{C^{op}}(M ⊗_{B}C),

which is an A-interior k-algebra with the structural homomorphism

A −→ Ind_{M}(C)

mapping a ∈ A to the C^{op}-endomorphism of M ⊗_{B}C given by left multiplication with a on M ⊗_{B}C. This definition was introduced by Linckelmann in order to generalize Puig’s induction, which was defined for interior algebras given by group algebras. We recall these ideas in the following example.

Example 2.1. Let H be a subgroup of a finite group G, and let C be a kH-interior k-algebra. Puig defined in [6] the injective induction from H to G of C as the kG-interior algebra

kG ⊗_{kH}C ⊗_{kH}kG,

with the multiplication given by

(x_{1}⊗ c ⊗ y_{1}) · (x_{2}⊗ d ⊗ y_{2}) = \{ \begin{matrix}x_{1}⊗ cy_{1}x_{2}d ⊗ y_{2}, if y_{1}x_{2} ∈ H \\ 0, if y_{1}x_{2} ∉ H \end{matrix} ,

where x_{1},y_{1},x_{2},y_{2} ∈ G,c,d ∈ C. The identity of this algebra is

∑_{g∈[G/H]}g ⊗ 1_{C}⊗ g^{−1} ,

where [G/H] is a set of representatives of left cosets of H in G. If we set M = kG as (kG,kH)-bimodule, then we have an isomorphism of interior kG-algebras

Ind_{M}(C) ≅ kG ⊗_{kH}C ⊗_{kH}kG.

The objective of this section is to prove that the isomorphism from Example 2.1 is still true in the context of a left β-Frobenius extension of k-algebras B ≤ A, where β a k-algebra automorphism of B. For this we recall some basic results and notations regarding left β-Frobenius extensions from [3].

If M ∈ B-Mod, then_{β}M denotes the left β-twisted B-module with underlying set M and left action

b · m = β(b)m

for any b ∈ B and m ∈ M. One defines similarly the right β-twisted B-module M_{β}, and the β-twisted (B,B)- bimodule_{β}M_{β}.

By [3, Definition 1.1] (see also [2]), the algebra extension B ≤ A is a left β-Frobenius extension if A, as right B-module, is finitely generated and projective, and there is an isomorphism

A ≅ _{β}Hom_{B}(A,B)

of (B,A)-bimodules. Moreover, in this case, by [3, Proposition 1.3], there is a (B,B)-bimodule map φ : A → βB and there are subsets

{r_{i} | i ∈ {1,...,n}}, {l_{i} | i ∈ {1,...,n}}

of A (called dual bases) such that

a = ∑_{i=1}^{n} r_{i}φ(l_{i}a) = ∑_{i=1}^{n} (β^{−1} ∘ φ)(ar_{i})l_{i}  (1)

for all a ∈ A.

We give now the main result of this section, which says that Puig’s injective induction for β-Frobenius extensions and Linckelmann’s generalization agree.

Theorem 2.2. Let B ≤ A be a left β-Frobenius extension of k-algebras, and let C be a B-interior algebra with structural homomorphism σ : B → C. Then A_{β} ⊗_{B}C ⊗_{B}A has an A-interior k-algebra such that we have an isomorphism

Ind_{A_{β}}(C) ≅ A_{β} ⊗_{B}C ⊗_{B}A

of A-interior k-algebras.

Proof. Define the multiplication on A_{β} ⊗_{B}C ⊗_{B}A by

(a_{1}⊗ c_{1}⊗ a′_{1})(a_{2}⊗ c_{2}⊗ a′_{2}) := a_{1}⊗ c_{1}(σ ∘ β^{−1} ∘ φ)(a′_{1}a_{2})c_{2}⊗ a′_{2},

for a_{1},a′_{1},a_{2},a′_{2} ∈ A,c_{1},c_{2} ∈ C. We verify the associativity and the existence of the identity element; the other axioms are obvious. Let a_{1},a′_{1},a_{2},a′_{2},a_{3},a′_{3} ∈ A, and let c_{1},c_{2},c_{3} ∈ C; then

((a_{1}⊗ c_{1}⊗ a′_{1}) (a_{2}⊗ c_{2}⊗ a′_{2})(a_{3}⊗ c_{3}⊗ a′_{3})
= (a_{1}⊗ c_{1}(σ ∘ β^{−1} ∘ φ)(a′_{1}a_{2})c_{2}⊗ a′_{2})(a_{3}⊗ c_{3}⊗ a′_{3})
= a_{1}⊗ c_{1}(σ ∘ β^{−1} ∘ φ)(a′_{1}a_{2})c_{2}(σ ∘ β^{−1} ∘ φ)(a′_{2}a_{3})c_{3}⊗ a′_{3}
= (a_{1}⊗ c_{1}⊗ a′_{1})(a_{2}⊗ c_{2}(σ ∘ β^{−1} ∘ φ)(a′_{2}a_{3})c_{3}⊗ a′_{3})
= (a_{1}⊗ c_{1}⊗ a′_{1})((a_{2}⊗ c_{2}⊗ a′_{2})(a_{3}⊗ c_{3}⊗ a′_{3})) .

The identity element is 1_{A_{β} ⊗_{B}C⊗_{B}A} = ∑_{i=1}^{n} r_{i}⊗ 1_{C}⊗ l_{i}, since we have

(a ⊗ c ⊗ a′)(∑_{i=1}^{n} r_{i}⊗ 1_{C}⊗ l_{i}) = ∑_{i=1}^{n} a ⊗ c(σ ∘ β^{−1} ∘ φ)(a′r_{i})1_{C}⊗ l_{i}

= ∑_{i=1}^{n} a ⊗ c(σ ∘ β^{−1} ∘ φ)(a′r_{i})σ(1_{B}) ⊗ l_{i}

= ∑_{i=1}^{n} a ⊗ c ⊗ (β^{−1} ∘ φ)(a′r_{i})1_{B}l_{i}

= a ⊗ c ⊗ ∑_{i=1}^{n} (β^{−1} ∘ φ)(a′r_{i})l_{i} = a ⊗ c ⊗ a′,

where the last equality is true by (1).

The structural homomorphism of A_{β} ⊗_{B}C ⊗_{B}A as an A-interior algebra is given by

τ : A → A_{β} ⊗_{B}C ⊗_{B}A, a ↦ ∑_{i=1}^{n} ar_{i}⊗ 1_{C}⊗ l_{i}

Indeed, we have

τ(a_{1})τ(a_{2})= ∑_{i=1}^{n} ∑_{j=1}^{n} (a_{1}r_{i}⊗ 1_{C}⊗ l_{i})(a_{2}r_{j} ⊗ 1_{C}⊗ l_{j})

= ∑_{i=1}^{n} ∑_{j=1}^{n} a_{1}r_{i}⊗ σ(1_{B})(σ ∘ β^{−1} ∘ φ)(l_{i}a_{2}r_{j})σ(1_{B}) ⊗ l_{j}

= ∑_{i=1}^{n} ∑_{j=1}^{n} a_{1}r_{i}φ(l_{i}a_{2}r_{j}) ⊗ 1_{C}⊗ l_{j}

= ∑_{j=1}^{n} ( a_{1} ∑_{i=1}^{n} r_{i}φ(l_{i}a_{2}r_{j}) ⊗ 1_{C}⊗ l_{j})

= ∑_{j=1}^{n} a_{1}a_{2}r_{j} ⊗ 1_{C}⊗ l_{j} = τ(a_{1}a_{2}),

where the equality in the last line holds again by (1).

Explicitly, the requested isomorphism is given by

Ψ : A_{β} ⊗_{B}C ⊗_{B}A → Ind_{A_{β}}(C), a ⊗ c ⊗ a′↦ Ψ(a ⊗ c ⊗ a′) = Ψ_{a⊗c⊗a′} ,

where

Ψ_{a⊗c⊗a′} (b ⊗ d) = a ⊗ c(σ ∘ β^{−1} ∘ φ)(a′b)d

for any b ∈ A,d ∈ C.

We first verify that Ψ is a homomorphism of k-algebras; for this let a_{1},a_{2},a′_{1},a′_{2},b ∈ A and c_{1},c_{2},d ∈ C; then we have

Ψ((a_{1}⊗ c_{1}⊗ a′_{1})(a_{2}⊗ c_{2}⊗ a′_{2})) = Ψ(a_{1}⊗ c_{1}(σ ∘ β^{−1} ∘ φ)(a′_{1}a_{2})c_{2}⊗ a′_{2})

= Ψ_{a_{1}⊗c_{1}(σ∘β^{−1} ∘φ)(a′_{1}a_{2})c_{2}⊗a′_{2}} , where

Ψ_{a_{1}⊗c_{1}(σ∘β^{−1} ∘φ)(a′_{1}a_{2})c_{2}⊗a′_{2}} (b ⊗ d) = a_{1}⊗ c_{1}(σ ∘ β^{−1} ∘ φ)(a′_{1}a_{2})c_{2}(σ ∘ β^{−1} ∘ φ)(a′_{2}b)d.

On the other hand, we have that

Ψ(a_{1}⊗ c_{1}⊗ a′_{1}) ∘ Ψ(a_{2}⊗ c_{2}⊗ a′_{2}) = Ψ_{a_{1}⊗c_{1}⊗a′_{1}} ∘ Ψ_{a_{2}⊗c_{2}⊗a′_{2}} ,

where

Ψ_{a_{1}⊗c_{1}⊗a′_{1}} (Ψ_{a_{2}⊗c_{2}⊗a′_{2}} (b ⊗ d)) = Ψ_{a_{1}⊗c_{1}⊗a′_{1}} (a_{2}⊗ c_{2}(σ ∘ β^{−1} ∘ φ)(a′_{2}b)d)
= a_{1}⊗ c_{1}(σ ∘ β^{−1} ∘ φ)(a′_{1}a_{2})c_{2}(σ ∘ β^{−1} ∘ φ)(a′_{2}b)d.

Clearly, Ψ_{a⊗c⊗a′} is a homomorphism of C^{op}-modules. Moreover, Ψ is an homomorphism of A-interior algebras, since it is easy to verify the commutativity of the diagram

\xymatrix{A \ar[r]^{^{Id^{A}}} \ar[d]^{τ} & A \ar[d] \\ A_{β} ⊗_{B}C ⊗_{B}A \ar[r]^{Ψ} & Ind_{A_{β}}(C)},

where the right-hand side arrow is the structural homomorphism from Linckelmann’s definition. Its inverse is given by

Ψ^{−1} : Ind_{A_{β}}(C) → A_{β} ⊗_{B}C ⊗_{B}A, f ↦ ∑_{i=1}^{n} f(r_{i}⊗ 1_{C}) ⊗ l_{i}.

Indeed, for each i ∈ {1,...,n} we may write

f(r_{i}⊗ 1_{C}) = ∑_{j∈J} m_{j,r_{i}} ⊗ n_{j,r_{i}} ∈ A_{β} ⊗_{B}C,

where m_{j,r_{i}} ∈ A,n_{j,r_{i}} ∈ C for any j ∈ J, where J is a finite set of indices. Let a ∈ A, and c ∈ C. Then we have

(Ψ ∘ Ψ^{−1} )(f)(a ⊗ c) = ∑_{i=1}^{n} ∑_{j∈J} Ψ(m_{j,r_{i}} ⊗ n_{j,r_{i}} ⊗ l_{i})(a ⊗ c)

= ∑_{i=1}^{n} ∑_{j∈J} m_{j,r_{i}} ⊗ n_{j,r_{i}}(σ ∘ β^{−1} ∘ φ)(l_{i}a)c

= ∑_{i=1}^{n} f(r_{i}⊗ 1_{C})(σ ∘ β^{−1} ∘ φ)(l_{i}a)c

= ∑_{i=1}^{n} f(r_{i}⊗ (σ ∘ β^{−1} ∘ φ)(l_{i}a)c)

= f ( ∑_{i=1}^{n} r_{i}φ(l_{i}a) ⊗ c) = f(a ⊗ c),

where the fourth equality holds since (σ ∘ β^{−1} ∘ φ)(l_{i}a)c is in C, while the sixth equality is true by (1).

3 Augmented algebras and the non-injective Puig induction

The first aim of this section is to define a non-injective induction through a homomorphism of augmented algebras which generalizes Puig’s non-injective induction [7, Section 3] through a homomorphism of group algebras. We will show that it coincides with Linckelmann’s generalization discussed in Section 2, if we choose a suitable bimodule. The second aim is related to the context of Section 5, since Hopf algebras are augmented algebras.

Let k be a commutative ring. Recall that a k-algebra A is augmented if there is a homomorphism

α_{A}: A → k

of k-algebras; we denote it by (A,α_{A}). In this case we can give to k a structure of trivial left (right) Amodule (and also of trivial (A,A)-bimodule) through α_{A}. We denote these by _{α_{A}}k, k_{α_{A}}, respectively _{α_{A}}k_{α_{A}}. A homomorphism ϕ : B → A between two augmented k-algebras (A,α_{A}),(B,α_{B}) is a homomorphism of k-algebras satisfying α_{A}∘ ϕ = α_{B}.

3.1 The surjective case.

Lemma 3.1. Let ϕ : B → A be a surjective homomorphism of augmented k-algebras. Assume that K is a subalgebra of B such that

Kerϕ ≤ (Kerα_{B}∩ K)B,

and let C be a interior B-algebra with structural homomorphism σ : B → C. Then there is an isomorphism

A_{ϕ} ⊗_{B}C ≅ k_{α_{B}} ⊗_{K}C

of C^{op}-modules.

Proof. From the isomorphism B/Kerϕ ≅ A of k-algebras we get that

A_{ϕ} ≅ B/Kerϕ

as B^{op}-modules. It follows that we have an isomorphism

A_{ϕ} ⊗_{B}C ≅ B/Kerϕ ⊗_{B}C

of C^{op}-modules. Consider the map

ψ : B/Kerϕ ⊗_{B}C → k_{α_{B}} ⊗_{K}C, ψ(b̄⊗ c) = 1 ⊗ σ(b)c,,

for b ∈ B, c ∈ C, and b̄= b + Kerϕ. Then ψ is a well-defined homomorphism of C^{op}-modules with respect to choosing a representative of b̄, since if b̄_{1} = b̄_{2} for some b̄_{1},b̄_{2} ∈ B/Kerϕ, then there is m ∈ Kerα_{B}∩ K and b′∈ B such that b_{2} = b_{1}+ mb′, thus

1 ⊗ σ(b_{2})c = 1 ⊗ σ(b_{1})c + 1 ⊗ σ(m)σ(b′)c = 1 ⊗ σ(b_{1})c.

The fact that ψ is an isomorphism of C^{op}-modules with its inverse

ψ^{−1} : k_{α_{B}} ⊗_{K}C → B/Kerϕ ⊗_{B}C, 1 ⊗ c ↦ ¯1⊗ c

is easy to check.

Proposition 3.2. Let ϕ : B → A be a surjective homomorphism of augmented k-algebras. Let C be a interior B-algebra with structural homomorphism σ : B → C. Then the following statements hold.

a) If K is a subalgebra of B such that

Kerα_{B}∩ K ≤ Kerϕ,

then there is a structure of K^{op}-module on k_{α_{B}} ⊗_{K}C such that (k_{α_{B}} ⊗_{K}C)^{K} is a interior A-algebra;

b) If K is a subalgebra of B such that

Kerα_{B}∩ K ≤ Kerϕ ≤ (Kerα_{B}∩ K)B

then

Ind_{A_{ϕ}}(C) ≅ (k_{α_{B}} ⊗_{K}C)^{K}

as interior A-algebras.

Proof. a) We have that k_{α_{B}} ⊗_{K}C is naturally a K^{op}-module with the action

(1 ⊗ c)x = 1 ⊗ cσ(x), c ∈ C, x ∈ K.

Since K is augmented, recall that

(k_{α_{B}} ⊗_{K}C)^{K} = {1 ⊗ c ∈ k_{α_{B}} ⊗_{K}C | 1 ⊗ cσ(x) = 1 ⊗ cα_{B}(x), ∀x ∈ K} .

We claim that (k_{α_{B}} ⊗_{K}C)^{K} is a k-algebra with the multiplication

(1 ⊗ c)(1 ⊗ d) = 1 ⊗ cd, 1 ⊗ c, 1 ⊗ d ∈ (k_{α_{B}} ⊗_{K}C)^{K}.

Indeed, let 1 ⊗ c,1 ⊗ d ∈ (k_{α_{B}} ⊗_{K}C)^{K}; then we have

((1 ⊗ c)(1 ⊗ d)) x= 1 ⊗ cdσ(x) = (1 ⊗ c)(1 ⊗ dσ(x)) = (1 ⊗ c)(1 ⊗ dα_{B}(x))
= (1 ⊗ c)(1 ⊗ d)(1 ⊗ α_{B}(x)1_{C}) = ((1 ⊗ c)(1 ⊗ d)) α_{B}(x).

Next, it is easy to verify that the above multiplication is well-defined, associative and distributive. We define the map

σ′: B/Kerϕ → k_{α_{B}} ⊗_{K}C,  b̄ ↦ 1 ⊗ σ(b),

which is an homomorphism of k-algebras. We verify that

Imσ′⊆ (k_{α_{B}} ⊗_{K}C)^{K}.

Indeed, let x ∈ K; then we have that

x − α_{B}(x) ∈ Kerα_{B}∩ K

hence, for any b ∈ B we obtain that

bx − bα_{B}(x) ∈ B(Kerα_{B}∩ K).

But since Kerα_{B}∩ K ≤ Kerϕ, we get that \overline{bx}= \overline{bα_{B}(x)}. Since σ′is also a homomorphism of B^{op}-modules, we deduce that σ′(b̄)x = σ′(b̄)α_{B}(x), and thus

σ′(b̄) ∈ (k_{α_{B}} ⊗_{K}C)^{K}.

Since A ≅ B/Kerϕ as B^{op}-modules we deduce that (k_{α_{B}} ⊗_{K}C)^{K} is an interior A-algebra through the composition of σ′with this isomorphism.

b) The isomorphisms

Ind_{A_{ϕ}}(C) ≅ End_{C^{op}}(k_{α_{B}} ⊗_{K}C) ≅ Hom_{K^{op}}(k,k_{α_{B}} ⊗_{K}C) ≅ (k_{α_{B}} ⊗_{K}C)^{K}

hold, where the first isomorphism is given by Lemma 3.1.

Proposition 3.2 allow us to state the next definition, and then notice that Linckelmann’s generalization agree with our generalization of Puig induction through homomorphisms of augmented k-algebras.

Definition 3.3. Let ϕ : B → A be a surjective homomorphism of augmented k-algebras. Assume that K is a subalgebra of B such that

Kerα_{B}∩ K ≤ Kerϕ ≤ (Kerα_{B}∩ K)B.

Let C be a interior B-algebra with structural homomorphism σ : B → C. The surjective induction of C through ϕ is the A-interior algebra

IndP_{ϕ}(C) := (k_{α_{B}} ⊗_{K}C)^{K}.

Example 3.4. Let k be a field and let B be a finite dimensional Hopf k-algebra. Let K be a normal Hopf subalgebra of B, and set

A:= B/K^{+} B.

In this case BK^{+} = K^{+} B. By considering the homomorphism

ϕ : B → A, b ↦ b̄:= b + BK^{+} ,

of Hopf algebras, we are in the situation of Proposition 3.2.

3.2 The general case.

Definition 3.5. Let ϕ : B → A be a homomorphism of augmented k-algebras. Assume that K be is subalgebra of B such that

Kerα_{B}∩ K ≤ Kerϕ ≤ (Kerα_{B}∩ K)B,

and let C be a B-interior algebra with structural homomorphism σ : B → C. The induction of C through ϕ is

IndP_{ϕ}(C) := Ind_{M}(C),

where we denoted M := A_{ϕ}, regarded as an (A,B)-bimodule.

Remark 3.6. Note that if we write ϕ = i∘ϕ̄, where the map i is the inclusion from ϕ(B) to A, ϕ̄: B → ϕ(B), and M_{1} := A regarded as an (A,ϕ(B))-bimodule, then

IndP_{ϕ}(C) = Ind_{M_{1}}(IndPϕ̄(C)).

In particular, if in the above definition, the algebra extension ϕ(B) ≤ A is a left β-Frobenius extension, then by Definition 3.3 and Theorem 2.2 we have an isomorphism

IndP_{ϕ}(C) ≅ A_{β} ⊗_{ϕ(B)}(k_{α_{B}} ⊗_{K}C)^{K}⊗_{ϕ(B)}A

of interior A-algebras. In the case of group algebras, we deduce [4, Example 1.4].

4 Induction for Hopf module algebras

In [9, Definition 8.1] A. Turull defined an induction of a H-algebra from a subgroup H to a finite group G. We will generalize this to the context of Hopf module algebras. In addition, we will also define a surjective version of Turull’s induction.

4.1 The injective case.

In this subsection let k be a field, let A be a finite dimensional Hopf algebras, and let B be a Hopf subalgebra of A. The counit of A is denoted by ε, the comultiplication is denoted by Δ and the antipode is S. We will use the Sweedler notation

Δ(a) = ∑ a_{(1)}⊗ a_{(2)}

for any a ∈ A.

Let F be the k-algebra considered in [8]; as a k-vector space F consists of all right B-linear maps f : A → k, that is,

F = {f ∈ A^{×} | f(ab) = f(a)ε(b), for all a ∈ A, b ∈ B};

the product is given by

(f · f′)(a) = ∑ f(a_{(2)})f′(a_{(1)}), f,f′∈ F.

The algebra F has identity ε, and it is a left A-module with action

(af)(a′) = f(S(a)a′), a,a′∈ A.

Next, let C_{1} := A ⊗_{B}k, which is an A-module coalgebra with comultiplication

C_{1} → C_{1}⊗ C_{1}, a ⊗ 1 ↦ ∑ (a_{((1)}⊗ 1) ⊗ (a_{(2)}⊗ 1).

It is well known that

C_{1} ≅ A/AB^{+}

as A-module coalgebras. Moreover, from the proof of [8, Lemma 1.1] we know that

F ≅ (C_{1}^{×} )^{op},

that is, F is essentially the opposite of the k-dual algebra of A ⊗_{B}k.

The following lemma is probably well-known, but for completeness we include here its proof.

Lemma 4.1. With the above notations, F is a left A-module algebra.

Proof. From the above we know that F is a left A-module. First we verify that for any a ∈ A,f,f′∈ F we have

a(f · f′) = ∑ (a_{(1)}f) · (a_{(2)}f′).

Indeed, for any a′∈ A, we have

[a(f · f′)](a′) = (f · f′)(S(a)a′)∑ f((S(a)a′)_{(2)})f′((S(a)a′)_{(1)});

on the other hand, we have

[∑ (a_{(1)}f) · (a_{(2})f′)] (a′) = ∑ ∑ (a_{(1)}f)(a′_{(2)})(a_{(2)}f′)(a′_{(1)})
= ∑ ∑ f(S(a_{(1)})a′_{(2)})f′(S(a_{(2)})a′_{(1)}).

The last sums from the above equalities are equal, since Δ is a homomorphism of k-algebras, and since Δ satisfies [5, Proposition 1.5.10].

By using Lemma 4.1, we can give the following definition.

Definition 4.2. Let B be a Hopf subalgebra of the finite dimensional Hopf algebra A. Let C be a B-module algebra. Then the induction of C from B to A is the A-module algebra

IndT_{B}^{A}(C) := F ⊗_{k}C,

with multiplication given by

(f ⊗ c)(f′⊗ c′) = f · f′⊗ cc′, f,f′∈ F, c,c′∈ C,

and A-module algebra structure given by

a(f ⊗ c) = af ⊗ c, a ∈ A, f ∈ F, c ∈ C.

Example 4.3. Let H be a subgroup of a finite group G, let B = kH and let A = kG. In this case the product in F is given by

(f · f′)(g) = f(g)f′(g), g ∈ G.

Moreover, we have an isomorphism of left kG-module coalgebras

A ⊗_{B}k ≅ k[G/H],

where [G/H] is a set of representatives of left cosets of H in G, and a (non-canonical) isomorphism of k-spaces between k[G/H] and its k-dual k[G/H]^{×} . We obtain an isomorphism of kG-module algebras

F ⊗ C ≅ (k[G/H]^{∗})^{op}⊗ C ≅ kG ⊗_{kH}C,

where kG ⊗_{kH}C is the induced algebra of C, introduced by Turull [9, Definition 8.1].

4.2 The surjective case.

We may define a surjective variant of Turull’s induction just by taking a certain subalgebra of invariants.

Definition 4.4. Let B be a finite dimensional Hopf algebra, and let K be a normal Hopf subalgebra in B. Set \overline{B}:= B/BK^{+} (recall that in this case BK^{+} = K^{+} B) and let ϕ : B → \overline{B}be the canonical projection.

Let C be a B-module algebra. The surjective induction of the B-module algebra C through ϕ is the B̄-module subalgebra of K-invariant elements of C, that is,

IndT_{ϕ}(C) := C^{K}.

5 The connection between Puig’s induction and Turull’s induction

We keep the notations and assumptions of Section 4, that is, k is a field and B is a Hopf subalgebra of the finite dimensional Hopf algebra A.

If we take B = k in Lemma 4.1, then F is identified to (A^{×} )^{op} as an A-module algebra, where A^{×} = Hom_{k}(A,k) is the k-dual of A. We also denote by

A^{∗} := Hom_{B}(A,B)

the B-dual of A, which is naturally an (A,B)-bimodule.

Remark 5.1. Notice that (End_{B}(A))^{op} is an interior A-algebra with the structural homomorphism

A → (End_{B}(A))^{op}, a ↦ (x ↦ xa),

and End_{B^{op}}(A^{∗})is a A-interior algebra with the structural homomorphism

A → End_{B^{op}}(A^{∗}), a ↦ (θ ↦ aθ), (aθ)(x) = θ(xa),

for any a,x ∈ A and θ ∈ A^{∗}. Lemma 5.2. a) Regarding (A^{×} )^{op} as an A-module algebra, there is an anti-algebra isomorphism

ρ^{op} : (A^{×} )^{op}#A → End_{k}(A), ρ^{op}(f#a)(x) = ∑ f(S(x_{(1)}))x_{(2)}a,

where f ∈ A^{×} and a,x ∈ A;

b) We have that ρ^{op}(F #A) ⊆ End_{B}(A);

c) There is an isomorphism of A-interior algebras

(End_{B}(A))^{op} ≅ End_{B^{op}}(A^{∗}).

Proof. a) We prove that ρ^{op} is an anti-algebra homomorphism. Indeed, let f,f′∈ A^{×} and a,a′,x ∈ A; then we have

ρ^{op}((f#a)(f′#a′))(x) = ∑ ∑ (f · (a_{(1)}f′))(S(x_{(1)}))x_{(2)}a_{(2)}a′
= ∑ ∑ ∑ f(S(x_{(1)})_{(2)})(a_{(1)}f′))(S(x_{(1)})_{(1)})x_{(2)}a_{(2)}a′
= ∑ ∑ ∑ f(S(x_{(1)_{(1)}})(a_{(1)}f′))(S(x_{(1)_{(2)}}))x_{(2)}a_{(2)}a′
= ∑ ∑ ∑ f(S(x_{(1)_{(1)}})f′(S(a_{(1)})S(x_{(1)_{(2)}}))x_{(2)}a_{(2)}a′

On the other hand, we have

ρ^{op}(f′#a′)(ρ^{op}(f#a)(x)) = ρ^{op}(f′#a′)(∑ f(S(x_{(1)}))x_{(2)}a)
= ∑ f(S(x_{(1)}))ρ^{op}(f′#a′)(x_{(2)}a)
= ∑ ∑ f(S(x_{(1)}))f′(S((x_{(2)}a)_{(1)}))(x_{(2)}a)_{(2)}a′
= ∑ ∑ ∑ f(S(x_{(1)}))f′(S((x_{(2)_{(1)}}a_{(1)}))x_{(2)_{(2)}}a_{(2)}a′
= ∑ ∑ ∑ f(S(x_{(1)}))f′(S(a_{(1)})S((x_{(2)_{(1)}}))x_{(2)_{(2)}}a_{(2)}a′,

and the above last sums are equal by coassociativity.

Next, it is enough to prove that ρ^{op} is injective. For this we define two maps

ρ′: (A^{×} )^{op}#A → End_{k}(A)

and

ψ : End_{k}(A) → End_{k}(A)

such that ρ′= ψ ∘ ρ^{op} and ρ′is injective, thus ρ^{op} is injective. These maps are defined as follows. ρ′(f#a)(x) := f(S(x))a, and

ψ(ζ)(x) := ∑ S(x_{(2)})ζ(x_{(1)}),

where f ∈ A^{×} , ζ ∈ End_{k}(A) and a,x ∈ A. The injectivity of ρ′is easy to check, since it follows by the simple argument that ρ′ takes a k-basis into a k-basis. We verify that ρ′= ψ ∘ ρ^{op}. Indeed, for any f ∈ A^{×} and a,x ∈ A we have that

(ψ ∘ ρ^{op})(f#a)(x) = ∑ S(x_{(2)})ρ^{op}(f#a)(x_{(1)})
= ∑ ∑ S(x_{(2)})f(S(x_{(1)_{(1)}}))x_{(1)_{(2)}}a
= ∑ ∑ S(x_{(2)})x_{(1)_{(2)}}f(S(x_{(1)_{(1)}}))a
= ∑ ∑ S(x_{(2)_{(2)}})x_{(2)_{(1)}}f(S(x_{(1)}))a
= ∑ ε(x_{(2)})f(S(x_{(1)}))a
= ∑ f(S(x_{(1)}ε(x_{(2)}))a
= f(S(ax))a = ρ′(f#a)(x).

b) Let f ∈ F, b ∈ B and a,x ∈ A; then we have

ρ^{op}(f#a)(bx) = ∑ f(S(b_{(1)}x_{(1)}))b_{(2)}x_{(2)}a
= ∑ f(S(x_{(1)})S(b_{(1)}))b_{(2)}x_{(2)}a
= ∑ f(S(x_{(1)}))ϵ(S(b_{(1)}))b_{(2)}x_{(2)}a
= ∑ f(S(x_{(1)}))ϵ(b_{(1)})b_{(2)}x_{(2)}a
= ∑ bf(S(x_{(1)}))x_{(2)}a.

c) Consider the map

Φ : (End_{B}(A))^{op} → End_{B^{op}}(A^{∗}),  f ↦ f^{∗},  f^{∗}(θ) = θ ∘ f,

where θ ∈ A^{∗}. We have that A^{∗} is a right B-module satisfying

(θ · b)(a) = θ(a)b,

for any a ∈ A and b ∈ B. Consequently

f^{∗}(θ · b)(a) = ((θ · b) ∘ f)(a) = θ(f(a))b = (f^{∗}(θ) · b)(a),

hence Φ is well defined. It is easy to check that Φ is an homomorphism of interior A-algebras, with respect to the structure of interior A-algebras given in Remark 5.1.

Since we are dealing with finite dimensional Hopf algebras, we may choose a basis

{e_{i} | i = 1,...,n},

of A as a left B-module. Further, for any i,j ∈ {1,...,n}, the left B-linear maps

f_{i,j} : A → A, f_{i,j}(e_{k}) = δ_{i,k}· e_{j},

form a right B-module basis of End_{B}(A), where End_{B}(A) is a right B-module by transporting the structure of right B-module of (M_{n}(B))^{op} through the isomorphism

End_{B}(A) ≅ (M_{n}(B))^{op};

here (M_{n}(B))^{op} is a right B-module by multiplying each element of a matrix on the left hand side with the same element of B.

Dually, we have that the set

{e_{i}^{∗} | i = 1,...,n},

where

(e_{i}^{∗}· b)(e_{j}) = e_{i}^{∗}(e_{j})b,

is a basis of A^{∗} as a right B-module. Similarly, the maps

f_{i,j}^{∗} : A^{∗} → A^{∗}, f_{i,j}^{∗}(e_{k}^{∗}) = δ_{i,k}· e_{j}^{∗}

form a basis of End_{B^{op}}(A^{∗})as right B-module, the structure being obtained as above by transporting the right B-module structure of (M_{n}(B))^{op} through an isomorphism

End_{B^{op}}(A^{∗}) ≅ (M_{n}(B))^{op}.

Now, we have that

(Φ(f_{i,j})(e_{k}^{∗}))(e_{m}) = (e_{k}^{∗}∘ f_{i,j})(e_{m}) = e_{k}^{∗}(f_{i,j}(e_{m}))

= \{ \begin{matrix}e_{k}^{∗}(e_{j}), if i = m \\ 0, if i ≠ m \end{matrix}

= \{ \begin{matrix}1, if k = j and i = m \\ 0, otherwise m \end{matrix}

= (δ_{j,k} · e_{i}^{∗})(e_{m}) = (f_{j,i}^{∗}(e_{k}^{∗}))(e_{m}),

hence Φ maps a B-basis of End_{B}(A^{∗}) bijectively onto a B-basis of End_{B^{op}}(A^{∗}).

Remark 5.3. By Lemma 5.2 i), ii) we may define the injective homomorphism of k-algebras

ρ_{F}^{op} : F #A → (End_{B}(A))^{op} ρ_{F}^{op}(f#a) = ρ^{op}(f#a),

where f ∈ F, and a ∈ A. It is easy to check that ρ_{F}^{op} is an homomorphism of A-interior algebras, and since

dim_{k}(F #A) = dim_{k}End_{B}(A) = (dim_{k}A)^{2}/dim_{k}B,

we deduce that ρ_{F}^{op} is actually an isomorphism.

Recall that if B is a Hopf subalgebra of a finite dimensional Hopf algebra A, then, by [3, Theorem 1.7], there is β ∈ Aut_{k}(B) such that B ≤ A is a left β-Frobenius extension. We may now state the injective version of our main result.

Theorem 5.4. Let B be a Hopf subalgebra of a finite dimensional Hopf algebra A, and let C be a B-module algebra. Then there is an isomorphism

IndT_{B}^{A}(C)#A → Ind_{A_{β}}(C#B)

of A-interior algebras.

Proof. We will construct the isomorphisms of A-interior algebras

\xymatrix{(F ⊗ C)#A \ar[r]^{Φ} & (End_{B}(A))^{op}⊗ C \ar[r] & End_{B^{op}}(A^{∗}) ⊗ C}

and

\xymatrix{End_{B^{op}}(A^{∗}) ⊗ C \ar[r]^{Ψ} & End_{(C#B)^{op}}(A^{∗}⊗_{B}C#B) \ar[r] & End_{(C#B)^{op}}(A_{β} ⊗_{B}C#B)}

in a sequence of steps.

Step 1. Let

Φ : (F ⊗ C)#A → (End_{B}(A))^{op}⊗ C, (f ⊗ c)#a ↦ ρ_{F}^{op}(f#a) ⊗ c,

for any f ∈ F,a ∈ A,c ∈ C. By Remark 5.3 we know that ρ_{F}^{op} is an isomorphism of A-interior algebras, hence Φ is an isomorphism of A-interior algebras; the structure of A-interior algebra of (F ⊗ C)#A is given by

A → (F ⊗ C)#A, a ↦ (ε ⊗ 1)#a,

while the A-interior structure for (End_{B}(A))^{op}⊗ C is obtained by composing the structural homomorphism from Remark 5.1 with the homomorphism of k-algebras

(End_{B}(A))^{op} → (End_{B}(A))^{op}⊗ C, η ↦ η ⊗ 1.

Step 2. By Lemma 5.2 c) we obtain the isomorphism

(End_{B}(A))^{op}⊗ C ≅ End_{B^{op}}(A^{∗}) ⊗ C

of A-interior algebras.

Step 3. Define the map

Ψ : End_{B^{op}}(A^{∗}) ⊗ C → End_{(C#B)^{op}}(A^{∗}⊗_{B}C#B)

by

Ψ(f^{∗}⊗ c)(θ ⊗ c′#b′) = f^{∗}(θ) ⊗ cc′#b′,

for f^{∗} ∈ End_{B^{op}}(A^{∗}), θ ∈ A^{∗}, and b′∈ B,c,c′∈ C. This is clearly a homomorphism of A-interior algebras.

We only need verify that the domain and the codomain have the same dimension as k-vector spaces; then, by using arguments similar to those from the end of proof of Lemma 5.2, we deduce that Ψ is bijective. Recall that

End_{B^{op}}(A^{∗}) ≅ M_{n}(B)^{op} ≅ (M_{n}(k) ⊗ B)^{op}

as k-algebras, where n is the number of elements of a basis of A as left B-modules. It follows that

End_{B^{op}}(A^{∗}) ⊗ C ≅ (M_{n}(k) ⊗ B)^{op}⊗ C

as k-algebras, and in particular

dim_{k}(End_{B^{op}}(A^{∗}) ⊗ C) = n^{2}· dim_{k}B · dim_{k}C.

By Theorem 2.2 we have that

dim_{k}(End_{(C#B)^{op}}(A^{∗}⊗_{B}C#B)) = dim_{k}(A_{β} ⊗_{B}C#B ⊗_{B}A);

but the last term is equal to

dim_{k}((A ⊗_{B}k) ⊗ C ⊗ B ⊗_{B}A) = dim_{k}(F ⊗ C ⊗ A) = n^{2}· dim_{k}B · dim_{k}C,

where the last equality is true since dim_{k}F = dim_{k}A/dim_{k}B = n (see the remarks from the beginning of Section 4 and [5, Corollary 3.2.1, Theorem 3.3.1]).

Step 4. Next, by [3, Remark 1.2, b)] it follows that B ≤ A is also right β^{−1} -Frobenius extension, hence by [3, Definition 1.1], there is an isomorphism

A ≅ Hom_{B}(A,B)_{β^{−1}}

of (A,B)-bimodules. Clearly, this isomorphism induces the isomorphism

A_{β} ≅ A^{∗}

of (A,B)-bimodules, and hence the isomorphism

End_{(C#B)^{op}}(A^{∗}⊗_{B}C#B) ≅ End_{(C#B)^{op}}(A_{β} ⊗_{B}C#B)

of A-interior algebras. We compose the isomorphisms from these four steps and we are done.

Combining this with Theorem 2.2, we obtain the following corollary which generalizes [1, Theorem 1].

Corollary 5.5. There is an isomorphism of A-interior algebras

(F ⊗ C)#A → A_{β} ⊗_{B}C#B ⊗_{B}A.

Finally, in the surjective case, we have a result analogous to Theorem 5.4.

Theorem 5.6. Let B be a finite dimensional Hopf algebra and let K be a normal Hopf subalgebra of B. Set \overline{B}:= B/BK^{+} and let ϕ : B → \overline{B}be the canonical projection.

Let C be a B-module algebra. Then there is an isomorphism

IndT_{ϕ}(C)#B̄≅ IndP_{ϕ}(C#B)

of B̄-interior algebras.

Proof. More explicitly, the requested isomorphism is

C^{K}#\overline{B}≅ (k_{ε} ⊗_{K}C#B)^{K},

where the smash prooduct C#B is the B-interior algebra with the structural homomorphism

B → C#B, b ↦ 1#b.

Recall that the left action of K on C#B is given by

x(c#b) = ∑ x_{(1)}c#x_{(2)}b,  (2)

for all x ∈ K, c ∈ C and b ∈ B. For the proof we introduce another left action of K on C#B, by

x(c#b) = c#xb;  (3)

notice that if c ∈ C^{K}, then

x(c#b) = ∑ x_{(1)}c#x_{(2)}b = ∑ c#ε(x_{(1)})x_{(2)}b = c#xb.

Note also that the right action of K on B is

(c#b)x = c#bx,

and it comes from the B-interior algebra structure.

We show that there is an isomorphism of right K-modules

Φ : C#\overline{B}−→ k_{ε} ⊗_{K}C#B, c#b̄ ↦ 1 ⊗ c#b,

for any c ∈ C, b ∈ B. Here C#\overline{B}is a right K-module with the right action

(c#b̄)x = S(x)c#b̄

for any c ∈ C, b ∈ B and x ∈ K. It is clear that this action is well defined, since \overline{B}is a trivial right K-module. Indeed, Φ is a well-defined map, because if b̄′= b̄∈ B̄, then

b − b′= ∑_{i∈I} x_{i}b′′_{i},

where x_{i} ∈ K^{+} , b′′_{i}∈ B for any i ∈ I where I is a finite set of indices; by using the equality (3), we get that

1 ⊗ c#b = 1 ⊗ c#(b′+ ∑_{i∈I} x_{i}b′′_{i}) = 1 ⊗ c#b′+ ∑_{i∈I} 1 ⊗ c#x_{i}b′′_{i}= 1 ⊗ c#b′.

Next, Φ is a homomorphism of right K-modules, because we have

Φ((c#b̄)x) = Φ(S(x)c#b̄) = Φ(S(x)c#\overline{(ε(S(x)))^{−1} bx})
= 1 ⊗ ε(S(x)))^{−1} S(x)c#bx
= ε(ε(S(x)))^{−1} S(x)) ⊗ c#bx
= ε(S(x)))^{−1} ε(S(x))) ⊗ c#bx = 1 ⊗ c#bx
= (1 ⊗ c#b)x = Φ(c#b̄)x.

Finally, it is easy to see that the map

Ψ : k_{ε} ⊗_{K}C#B → C#\overline{B}, 1 ⊗ c#b ↦ c#b̄,

is the inverse of Φ.

Since Φ is an isomorphism of K-modules, it restricts to an isomorphism

(C#\overline{B})^{K} ≅ (k_{ε} ⊗_{K}C#B)^{K}.

of vector spaces between the subspaces K-invariants. We have the obvious isomorphism

(C#B)^{K} ≅ C^{K}#\overline{B},

and from this we obtain an isomorphism, still denoted by Φ, of vector spaces

Φ : C^{K}#\overline{B}→ (k_{ε} ⊗_{K}C#B)^{K}.

This isomorphism allows us to make the identification

(k_{ε} ⊗_{K}C#B)^{K} = (k_{ε} ⊗_{K}C^{K}#B)^{K},

but but the left actions (2) and (3) coincide on C^{K}#B, we deduce that Φ is the isomorphism between the vector spaces from the statement of the theorem. The fact that Φ is actually an homomorphism of \overline{B}-interior algebras follows by a straightforward verification.

In the particular case of a group G acting on the k-algebra C, the smash product C#kG is just the skew group algebra C ∗ G, and we immediately deduce the following result.

Corollary 5.7. Let G be a finite group, let ϕ : G → Ḡ be an group epimorphism, and denote K := Ker(ϕ). If C is a G-algebra, then there is a isomorphism of kḠ-interior algebras

C^{K}∗ Ḡ≃ (k ⊗_{kK}C ∗ G)^{K},

mapping c ∗ ḡto 1 ⊗ c ∗ g, for any c ∈ C^{K} and any ḡ∈ Ḡ, where g ∈ G such that ϕ(g) = ḡ.

References

[1] T. Coconeţ, Remarks on induction of G-algebras and skew group algebras, Mathematica Tome 51 74 (2), 135–142 (2009).

[2] Y. Doi, A note on Frobenius extensions in Hopf algebras, Commun. Algebra, 25, 3699–3710 (1997).

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

[4] M. Linckelmann, Induction for interior algebras, Quart. J. Math. 53, 195–200 (2002).

[5] S. Montgomery, Hopf algebras and their actions on rings, American Mathematical Society, 1993.

[6] L. Puig, Pointed Groups and Construction of Characters, Math. Z. 176, 265–292 (1981)

[7] L. Puig, On the Local Structure of Morita and Rickard Equivalences between Brauer Blocks, Progress in Mathematics (Boston, Mass.), Vol. 178, Birkhäuser, Basel, 1999.

[8] K.-U. Ulbrich, On modules induced or coinduced from Hopf subalgebras, Math. Scand. 67, 177–182 (1990).

[9] A. Turull, Reduction theorem for Clifford classes, J. Group Theory 9, 27–47 (2006).