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A Duality for Modules over Monoidal Categories of Representations of Semisimple Hopf Algebras
Journal of Algebra 241, 515᎐547 Ž2001. doi:10.1006rjabr.2001.8771, available online at http:rrwww.idealibrary.com on
A Duality for Modules over Monoidal Categories of Representations of Semisimple Hopf Algebras D. Tambara Department of Mathematical System Science, Hirosaki Uni¨ ersity, Hirosaki 036-8561, Japan Communicated by Susan Montgomery Received January 4, 1999
For a finite dimensional semisimple cosemisimple Hopf algebra A and its dual Hopf algebra B, we set up a natural one-to-one correspondence between categories with actions of the monoidal categories of representations of A and of B. This gives a categorical interpretation of the duality for actions of Hopf algebras on algebras. 䊚 2001 Academic Press
INTRODUCTION Let A be a finite dimensional semisimple cosemisimple Hopf algebra over a field k. Let A be the category of finite dimensional A-modules. As A is a monoidal category, we have a notion of A-modules: A right A-module is a linear category M equipped with a bilinear functor M = A ª M and coherent isomorphisms of associativity and unit. Let B be the dual Hopf algebra of A and B the category of finite dimensional B-modules. The purpose of the paper is to set up a natural one-to-one correspondence between right A-modules and right B-modules with direct summands. Recall the duality theorem for Hopf algebra actions on algebras due to Blattner and Montgomery wBMx, generalizing the duality for group actions due to Nakagami and Takesaki wNTx. It asserts that if R is an algebra on which B acts, the two-step smash product algebra Ž R噛B .噛A is isomorphic to a full matrix algebra over R, without the semisimplicity of A and B assumed. In the case of semisimple Hopf algebras, this is related to the above correspondence as follows. The category R-Mod of R-modules becomes naturally a right A-module. Similarly if S is an algebra with 515 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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A-action, the category S-Mod becomes a right B-module. In our correspondence, the A-module R-Mod is mapped to the B-module R噛B-Mod, and B-module S-Mod is mapped to the A-module S噛A-Mod. The mappings are inverse to each other. So Ž R噛B .噛A-Mod should be equivalent to R-Mod, which is in accord with the duality theorem. The correspondence between A-modules and B-modules is given by categorical analogues of Hom and m functors for usual modules. Let V be the category of finite dimensional k-modules. Then V becomes an Ž A, B .-bimodule and a Ž B, A .-bimodule as well. For an A-module M with direct summands, the corresponding B-module is the category Hom A Ž V , M . of A-linear functors V ª M . This is also equivalent to the B-module M mA V , which is obtained by firstly making the tensor product M mA V and then adjoining direct summands. The duality equivalence M mA V mB V , M is induced by an equivalence of Ž A, A .-bimodules V mB V , A. In addition, the equivalences V mB V , A, V mA V , B can be taken in a coherent way so that the monoidal categories A, B, the Ž A, B .-bimodule V , and the Ž B, A .-bimodule V form a monoidal category of matrix A V ., or in other words a bicategory with two objects. form Ž V B In Section 1 we review basic facts about modules over monoidal categories, including the definition of tensor product categories. In Section 2 we construct the bicategory mentioned above. This gives the bimodule structures on V and the functors V mB V ª A, V mA V ª B. In Section 3 we prove that the induced functors V mB V ª A, V mA V ª B are equivalences, from which the main result follows. In Section 4 we describe the correspondence for categories of modules over rings with Hopf algebra actions in terms of the smash product construction. Notations. For a Hopf algebra A, ⌬ denotes the comultiplication map, and S denotes the antipode of A. For a ring R, R-Mod denotes the category of finitely generated left R-modules.
1. MODULES OVER MONOIDAL CATEGORIES This section consists mainly of the definitions of modules over monoidal categories and tensor products of those modules. Nothing will be new. For generality about monoidal categories see wKx. A k-linear category is a category in which the Hom-sets are k-vector spaces, the compositions are k-bilinear operations and finite direct sums exist. The notion of a k-linear functor C ª D, and a k-bilinear functor C = C ⬘ ª D for k-linear categories C , C ⬘, D will be obvious. Let HomŽ C , D . denote the category of k-linear functors C ª D.
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Ža. A monoidal category over k is a k-linear category A equipped with a k-bilinear functor 䉺: A = A ª A, an object I, and natural isomorphisms
␣ X , Y , Z : X 䉺 Ž Y 䉺Z . ª Ž X 䉺Y . 䉺Z, X : X ª I䉺 X ,
X : X ª X 䉺I
satisfying the identities
Ž ␣ X , Y , Z 䉺W . ␣ X , Y 䉺 Z , W Ž X 䉺 ␣ Y , Z , W . s ␣ X 䉺Y , Z , W ␣ X , Y , Z䉺W , Ž M1. ␣ X , I , Y Ž X 䉺 Y . s X 䉺Y
Ž M2.
for all objects X, Y, Z, W in A. For a monoidal category A, a left A-module is a k-linear category M equipped with a k-bilinear functor 䉺: A = M ª M and natural isomorphisms
␣ X , Y , M : X 䉺 Ž Y 䉺M . ª Ž X 䉺Y . 䉺M, M : M ª I䉺M for X, Y g A, M g M satisfying ŽM1. with Ž X, Y, Z, W . replaced with Ž X, Y, Z, M . and ŽM2. with Ž X, Y . replaced with Ž X, M . for all X, Y, Z g A, M g M . A right A-module is similarly defined. For monoidal categories A and B, an Ž A, B .-bimodule is a k-linear category M equipped with k-bilinear functors 䉺: A = M ª M , 䉺: M = B ª M , and natural isomorphisms
␣ X , Y , M : X 䉺 Ž Y 䉺M . ª Ž X 䉺Y . 䉺M, ␣ X , M , S : X 䉺 Ž M䉺S . ª Ž X 䉺M . 䉺S, ␣ M , S , T : M䉺 Ž S䉺T . ª Ž M䉺S . 䉺T , M : M ª I䉺M,
M : M ª M䉺I
for X, Y g A; M g M ; S, T g B satisfying ŽM1. with Ž X, Y, Z, W . replaced with Ž X, Y, Z, M ., Ž X, Y, M, S ., Ž X, M, S, T ., and Ž M, S, T, U . and ŽM2. with Ž X, Y . replaced with Ž X, M ., Ž M, S . for all X, Y, Z g A; M g M ; S, T, U g B. For left A-modules M and N , an A-linear functor Ž F, .: M ª N consists of a k-linear functor F: M ª N and natural isomorphisms
X , M : F Ž X 䉺M . ª X 䉺F Ž M .
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satisfying the identities
X 䉺Y , M F Ž ␣ X , Y , M . s ␣ X , Y , F Ž M . Ž X 䉺 Y , M . X , Y 䉺 M , I , M F Ž M . s F Ž M . for all X, Y g A, M g M . We write Ž F, . s F occasionally. For A-linear functors Ž F, ., Ž F⬘, ⬘.: M ª N , a morphism Ž F, . ª Ž F⬘, ⬘. is a natural transformation : F ª F⬘ satisfying
XX , M X 䉺 M s Ž X 䉺 M . X , M for all X g A, M g M . With this notion of morphisms, we have the category of A-linear functors M ª N , denoted by Hom A Ž M , N .. For A-linear functors Ž G, .: L ª M and Ž F, .: M ª N , the composition Ž F, .(Ž G, . is the A-linear functor Ž F (G, .: L ª N where
X , L s X , GŽ L. F Ž X , L . for X g A, L g L . We then have the functor Hom A Ž M , N . =Hom A Ž L , M . ª Hom A Ž L , N .
Ž Ž F , . , Ž G, . . ¬ Ž F , . ( Ž G, . . The composition is strictly associative. Thus we have a 2-category in which the objects are left A-modules, the Hom-categories are Hom A Ž M , N ., and the horizontal compositions are the above compositions. This 2-category is denoted by A-Mod. An A-linear functor F: M ª N is called an equivalence if there is an A-linear functor G: N ª M with isomorphisms F (G ( 1 in Hom A Ž N , N . and G( F ( 1 in Hom A Ž M , M .. Then G is called a quasi-inverse of F. It should be noted that F is an equivalence of A-modules if it is simply an equivalence of categories. Indeed, any quasi-inverse of F, as categories, can be made into an A-linear functor giving a quasi-inverse of F as A-modules. Let V be the monoidal category of finite dimensional vector spaces over k. Any k-linear category C becomes a left V-module by setting k n m X s X n, the n-fold direct sum. Žb. For a right A-module M , a left A-module N , and a k-linear category L , an A-bilinear functor Ž F, ␣ .: M = N ª L consists of a k-bilinear functor F: M = N ª L and natural isomorphisms
␣ M , X , N : F Ž M, X 䉺N . ª F Ž M䉺 X , N .
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satisfying F Ž ␣ M , X , Y , N . ␣ M , X 䉺Y , N F Ž M, ␣ X , Y , N . s ␣ M 䉺 X , Y , N ␣ M , X , Y 䉺 N ,
␣ M , I , N F Ž M, N . s F Ž M , N . for all M g M , N g N , X, Y g A. With an obvious definition of morphisms, we have the category of A-bilinear functors M = N ª L , denoted by BiHom A Ž M , N ; L .. We will construct a k-linear category M mA N and an A-bilinear functor M = N ª M mA N inducing an equivalence HomŽ M mA N , L . ª BiHom A Ž M , N ; L . for any k-linear category L . As a k-linear category, M mA N has the following presentation by generators and relations. Objects are finite direct sums of the symbols w M, N x for M g M , N g N . Generators for morphisms are the symbols
w f , g x : w M, N x ª w M⬘, N⬘ x for morphisms f : M ª M⬘ in M and g: N ª N⬘ in N , and the symbols
␣ M , X , N : w M, X 䉺N x ª w M䉺 X , N x , X ␣M , X , N : w M䉺 X , N x ª w M, X 䉺N x
for objects M g M , X g A, and N g N . Relations among them are Ži. ᎐ Žv. below. Ži. ŽLinearity .
w f q f ⬘, g x s w f , g x q w f ⬘, g x ,
w f , g q g ⬘x s w f , g x q w f , g ⬘x ,
w af , g x s aw f , g x s w f , ag x for morphisms f, f ⬘: M ª M⬘ in M ; g, g ⬘: N ª N⬘ in N ; and a g k. Žii. ŽFunctoriality.
w f ⬘ f , g ⬘g x s w f ⬘, g ⬘ xw f , g x , w 1 M , 1 N x s 1w M , N x for morphisms f : M ª M⬘, f ⬘: M⬘ ª M⬙ in N⬘ ª N⬙ in N . Žiii. ŽIsomorphism. X ␣M , X , N ␣M , X , N s 1,
M and g: N ª N⬘, g ⬘:
X ␣M , X , N ␣ M , X , N s 1.
Živ. ŽNaturality.
␣ M ⬘, X ⬘, N ⬘ w f , u䉺 g x s w f 䉺u, g x ␣ M , X , N
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for morphisms f : M ª M⬘ in M , u: X ª X ⬘ in A, and g: N ª N⬘ in N . Žv. ŽPentagon and triangle.
w ␣ M , X , Y , N x ␣ M , X 䉺Y , N w M, ␣ X , Y , N x s ␣ M 䉺 X , Y , N ␣ M , X , Y 䉺 N , ␣ M , I , N w M, N x s w M , N x , where wy, N x s wy, 1 N x, w M, yx s w1 M , yx. The bilinear functor T : M = N ª M mA N is defined by T Ž M, N . s w M, N x for objects and T Ž f , g . s w f , g x for morphisms. The isomorphisms ␣ M , X , N then give T a structure of an A-bilinear functor. From this construction, it will be obvious that for any k-linear category L , the functor Hom Ž M mA N , L . ª BiHom A Ž M , N ; L . G ¬ G(T is an equivalence. Žc. If M is a right A-module, N is an Ž A, B .-bimodule and L is a right B-module, then Hom B Ž N , L . has a structure of a right A-module and M mA N has a structure of a right B-module. The action of an object X g A on an object Ž F, . g Hom B Ž N , L . is given by
Ž F , . 䉺 X s Ž F⬘, ⬘ . , where F⬘ Ž N . s F Ž X 䉺N . and F⬘Ž N䉺Y . s F Ž X 䉺 Ž N䉺Y .. F Ž ␣ X, N, Y .
6
X N, Y
F ŽŽ X 䉺N . 䉺Y . X 䉺 N, Y
6
6
F⬘Ž N . 䉺Y s F Ž X 䉺N . 䉺Y for N g N , Y g B. The associativity of this action is given by the isomorphisms F Ž ␣ X , X ⬘, N . : F Ž X 䉺 Ž X ⬘ 䉺N . . ª F Ž Ž X 䉺 X ⬘ . 䉺N . for X, X ⬘ g A, N g N .
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The action of an object Y g B on an object w M, N x g M mA N is given by
w M, N x 䉺Y s w M, N䉺Y x , and the action on the morphism ␣ M , X , N is given by w M, X 䉺N x 䉺Y sw M, Ž X 䉺N . 䉺Y x w M, ␣ y1 X, N, Y x
6
␣ M, X , N 䉺Y
w M, X 䉺 Ž N䉺Y .x ␣ M, X , N 䉺Y
6
6
w M䉺 X, N x 䉺Y s w M䉺 X, N䉺Y x. The associativity of this action is given by the isomorphisms
w M, ␣ N , Y , Y ⬘ x : M, N䉺 Ž Y 䉺Y⬘ . ª M, Ž N䉺Y . 䉺Y ⬘ for Y, Y ⬘ g B. We have an equivalence of k-categories Hom B Ž M mA N , L . ª Hom A Ž M , Hom B Ž N , L . . G ¬ Ž M ¬ Ž N ¬ G Ž w M, N x . . . This follows from the equivalence at the end of Žb.. Žd. A bicategory E consists of a set J; a collection of k-linear categories Ei j for i, j g J; bilinear functors 䉺 i jk : Ei j = Ejk ª Ei k for i, j, k g I; objects Ii g Eii ; and natural isomorphisms
␣ X , Y , Z : X 䉺i jl Ž Y 䉺jk l Z . ª Ž X 䉺i jk Y . 䉺i k l Z, X : X ª Ii 䉺ii j X ,
X : X ª X 䉺i j j I j
for X g Ei j , Y g Ejk , and Z g Ek l satisfying identities analogous to ŽM1. and ŽM2.. See wBx. Each category Eii becomes a monoidal category and Ei j becomes an Ž Eii , Ej j .-bimodule. Moreover, 䉺i jk : Ei j = Ejk ª Ei k becomes an Ej j-bilinear functor and hence induces a functor Ei j mE j j Ejk ª Ei k . This in turn becomes an Ž Ei i , Ek k .-linear functor. Že. A k-linear category C is said to have direct summands if any idempotent endomorphism e: X ª X in C is a projection to a direct summand of X. The envelope C of C is the category defined as follows: An object of C is a pair Ž X, e . of an object X g C and an idempotent
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e g End X. A morphism Ž X, e . ª Ž X ⬘, e⬘. is a morphism f : X ª X ⬘ in C such that fe s f s e⬘ f. Then C has direct summands, and the functor C ª C : X ¬ Ž X, 1. has the following universality: For any k-linear category D with direct summands, the induced functor Hom Ž C , D . ª Hom Ž C , D . is an equivalence. Indeed, an idempotent f g EndŽ X, e . gives rise to the decomposition Ž X, e . ( Ž X, f . [ Ž X, e y f . so that f is isomorphic to the projection to the summand Ž X, f .. Any functor F : C ª D extends to F: C ª D which takes Ž X, e . to the direct summand of F Ž X . corresponding to the idempotent F Ž e . and takes a morphism f : Ž X, e . ª Ž X ⬘, e⬘. to the component of F Ž f .. See wGV, p. 413x or wFS, p. 15x. For a monoidal category A, A-Modk denotes the 2-category whose objects are left A-modules with direct summands and whose Hom-categories are categories of A-linear functors as in A-Mod. If M is an A-module, then M becomes an A-module by setting X 䉺 Ž M, e . s Ž X 䉺M, X 䉺e .. For a right A-module M and a left A-module N , the envelope of M mA N is denoted by M mA N . 2. THE BICATEGORY ASSOCIATED WITH THE DUAL PAIR Ž A, B . Let A be a finite dimensional semisimple cosemisimple Hopf algebra. Actually, semisimplicity implies cosemisimplicity in characteristic zero wLR2x. We need the result that the square of the antipode S of A is the identity, due to Larson and Radford wLR1x in characteristic zero and to Etingof and Gelaki wEGx in positive characteristic. Let B s A*, the dual Hopf algebra. Put A s A-Mod, B s B-Mod, and V s k-Mod. In this section we construct a bicategory E , in the sense of Section 1Žd., with index set J s 1, 24 such that
ž
E11 E21
E12 A s E22 V
/
ž
V . B
/
The canonical pairing between A and B is denoted by ² ᎐, ᎐ :. After Sweedler’s book wSx, the left action © and the right action £ of A on B are defined by a©bs
Ý b1² a, b2 : , b £ a s Ý ² a, b1 : b 2
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for a g A, b g B with ⌬Ž b . s Ýb1 m b 2 , so that ² a⬘, a © b : s ² a⬘a, b : , ² a⬘, b £ a: s ² aa⬘, b : . Then the left action
and the right action a
b s b £ S Ž a. ,
b
a s S Ž a . © b.
of A on B are defined by
We need to choose linear isomorphisms A ª B, B ª A in a special way. PROPOSITION 2.1. There exist linear isomorphisms : A ª B, : B ª A such that
Ž a⬘a . s a⬘
Ž a. ,
Ž b⬘ © a . s b⬘ Ž a . , Ž b⬘b . s b⬘
Ž aa⬘ . s Ž a . Ž a £ b⬘ . s Ž a . b⬘,
Ž b. ,
Ž a⬘ © b . s a⬘ Ž b . ,
a⬘,
Ž bb⬘ . s Ž b .
b⬘,
Ž b £ a⬘ . s Ž b . a⬘
for all a, a⬘ g A, b, b⬘ g B, and that S s 1,
S s 1.
Such a pair Ž , . is unique modulo the relation Ž , . ; Ž , y1 . for scalars / 0. Proof. Take integrals ⌳ g A, ⌫ g B. Since A, B are unimodular, left integrals and right integrals are the same and SŽ ⌳ . s ⌳, SŽ ⌫ . s ⌫ wLS, Corollary to Proposition 4 and the first corollary to Proposition 8x. Since A is involutory, we have ² aa⬘, ⌫ : s ² a⬘a, ⌫ : wLS, the second corollary to Proposition 8x. It follows that a ⌫s⌫ a. Define : A ª B by
Ž a. s a
⌫s⌫
a.
Then is an isomorphism of two-sided Hopf A-modules, that is, the first four identities in the statement hold for all a, a⬘ g A, b⬘ g B wS, p. 96x. Conversely, any map A ª B satisfying those identities comes from an integral in B in this way, hence it is unique up to scalar. Similarly, define : B ª A by
Ž b. s b
⌳s⌳
Then satisfies the next four identities.
b.
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Using an identity b © S Ž a. s S Ž a £ S Ž b . . and the fact that SŽ ⌫ . s ⌫, S 2 s 1, one has S Ž a . s S Ž a
⌫. s SŽ ⌫.
Sy1 Ž a . s ⌫
S Ž a. s S Ž a. .
Thus S s S. And S s S as well. Now the computations
² S Ž a. , b: s ² S Ž a.
⌳ , b: s ² ⌳ £ Ž a . , b:
s ² ⌳ , Ž a . b: s ² ⌳ , Ž a £ b .: s² ⌳ , Ž a £ b.
⌫: s ² S Ž a £ b . ⌳ , ⌫:
s ² ⑀ S Ž a £ b . ⌳ , ⌫: s ²² a, b: ⌳ , ⌫: s ² a, b :² ⌳ , ⌫ : yield S s ² ⌳, ⌫ :1. We can take ⌳, ⌫ so that ² ⌳, ⌫ : s 1. Then S s 1. Thus S s 1, S s 1. We fix such a choice of : A ª B, : B ª A throughout the paper. For X g A, Y g B we have the maps
X : X m A ª X m A xma¬
Ý a1 x m a2 ,
X : A m X ª A m X amx¬
Ý a1 m a2 x,
X , Y : X m Y ª X m Y xmy¬
Ý ai x m yi s Ý x j m bj y,
␥Y : A m Y ª Y m A amy¬
Ý yi m aai ,
where ⌬ Ž a. s
Ý a1 m a2 ,
Ž y. s
Ý yi m a i ,
Ž x. s
Ý x j m bj ,
and : Y ª Y m A, : X ª X m B are the right comodule structures coming from the left module structures.
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525
These are all bijections with inverses given by
y1 X : xma¬
Ý Sy1 Ž a1 . x m a2 , y1 X : a m x ¬ Ý a1 m S Ž a 2 . x, y1 X , Y : x m y ¬ Ý S Ž a i . x m yi s Ý x j m S Ž bj . y, y1 ␥y1 Ž a i . m yi . Y : y m a ¬ Ý aS Replacing the roles of A and B, we have the similar maps Y , Y ,  Y , X , ␥ X for X g A, Y g B. Now we define the bicategory E as follows. The index set is 1, 24 . Let E11 s A , E12 s V , E21 s V ,
E22 s B .
The composition functors 䉺 i jk : Ei j = Ejk ª Ei k for i, j, k s 1, 2 are given by X 䉺111 X ⬘ s X m X ⬘,
Y 䉺222 Y⬘ s Y m Y ⬘,
X 䉺112 V s X m V ,
Y 䉺221 V s Y m V ,
V 䉺211 X s V m X ,
V 䉺122 Y s V m Y ,
V 䉺121 V⬘ s V m A m V⬘,
V 䉺212 V⬘ s V m B m V⬘
for X, X ⬘ g A; Y, Y⬘ g B; V, V ⬘ g V . Here the module structures of X m X ⬘, Y m Y ⬘ are the usual ones. In X m V, V m X, Y m V, V m Y the module structures of X, Y are forgotten. In V m A m V⬘, V m B m V ⬘, we regard A, B as the left regular modules. The units I g E11 , I g E22 are the trivial modules k. Next we define the natural transformations of associativity
␣ i jk l : 䉺i jl ( Ž 1 E i j = 䉺jk l . ª 䉺i k l ( Ž 䉺 i jk = 1 E k l . . ␣ 1111 , ␣ 2222 , ␣ 1112 , ␣ 2111 , ␣ 2221 , ␣ 1222 are the identity. ␣ 1122 , ␣ 1121 , ␣ 1211 , ␣ 1221 , ␣ 1212 are given by
Ž X 䉺V . 䉺Y
6
5 XmVmY
Ž X, Y .
Ž X 䉺V . 䉺V⬘
6
␣ 1121
6
X 䉺 Ž V 䉺V⬘ . 5 X m V m A m V⬘
␣ 1122
6
X 䉺 Ž V 䉺Y . 5 X 䉺V 䉺Y
5 X m V m A m V⬘
Ž y1 X .
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D. TAMBARA
Ž .
Ž V 䉺V⬘ . 䉺V⬙
6
␣ 1212
5 V m Y m A m V⬘
6
Ž␥ Y .
Ž V 䉺Y . 䉺V⬘
6
␣ 1221
5 V m A m V⬘ m X
6
V 䉺 Ž V⬘ 䉺V ⬙ . 5 V m V⬘ m B m V⬙
Ž X .
Ž V 䉺V ⬘ . 䉺 X
6
V 䉺 Ž Y 䉺V . 5 V m A m Y m V⬘
␣ 1211
6
V 䉺 Ž V⬘ 䉺 X . 5 V m A m V⬘ m X
5 V m A m V ⬘ m V⬙
for X g A; Y g B; V, V⬘, V ⬙ g V . Here Ž  X , Y ., Ž␥ X ., . . . stand for the maps induced by  X , Y , ␥ X , . . . in an obvious way. The remaining ␣ 2211 , ␣ 2212 , ␣ 2122 , ␣ 2112 , ␣ 2121 are defined by interchanging A and B, and and . Finally, the natural isomorphisms for unit
i j : 1 E i j ª Ii 䉺ii j Ž ᎐ . ,
i j : 1 E i j ª Ž ᎐ . 䉺i j j I j
are given by the maps x ¬ 1 m x, x ¬ x m 1. THEOREM 2.2. The data Ei j , 䉺i jk , Ii , ␣ i jk l , i j , i j constitute a bicategory E. Proof. The triangle identities for i j , i j are obvious. We prove that ␣ i jk l satisfy the pentagon identity X 䉺 Ž Y 䉺 Ž Z䉺W .. X 䉺 Ž Y 䉺 Ž Z䉺W .. ␣ i jkm
6 X 䉺 ŽŽ Y 䉺Z . 䉺W .
6
␣ i jlm
s Ž X 䉺Y . 䉺 Ž Z䉺W . ␣ iklm
6
Ž X 䉺 Ž Y 䉺Z .. 䉺W ␣ i jkl 䉺W
6
X 䉺 ␣ jklm
ŽŽ X 䉺Y . 䉺Z .. 䉺W
6
ŽŽ X 䉺Y . 䉺Z .. 䉺W for X g Ei j , Y g Ejk , Z g Ek l , and W g El m .
ŽPi jk l m .
DUALITY FOR MODULES
527
ŽPi jk l m . is clear for Ž ijklm. s Ž11111., Ž11112., Ž12222., and for these with 1 and 2 interchanged, because all ␣ involved are the identity maps. ŽP11122 . Let X, X ⬘ g A; V g V ; Y g B. The left-hand side of ŽP11122 . for Ž X, X ⬘, V, Y . becomes 䢇
䢇
X 䉺 Ž X ⬘ 䉺 Ž V 䉺Y .. s X m X ⬘ m V m Y X 䉺 ␣ 1122
Ž  X ⬘, Y .
6
6
X 䉺 ŽŽ X ⬘ 䉺V . 䉺Y . s X m X ⬘ m V m Y ␣ 1122
Ž X, Y .
6
6
Ž X 䉺 Ž X ⬘ 䉺V .. 䉺Y s X m X ⬘ m V m Y ␣ 1112 䉺Y
1
6
6
ŽŽ X 䉺 X ⬘. 䉺V . 䉺Y sX m X ⬘ m V m Y, while the right-hand side becomes X 䉺 Ž X ⬘ 䉺 Ž V 䉺Y .. s X m X ⬘ m V m Y ␣ 1112
1
6
6
Ž X 䉺 X ⬘. 䉺 Ž V 䉺Y . s X m X ⬘ m V m Y ␣ 1122
Ž  Xm X ⬘, Y .
6
6
ŽŽ X 䉺 X ⬘. 䉺V . 䉺Y sX m X ⬘ m V m Y. So we are reduced to showing the identity X m X⬘ m Y Ž  X ⬘, Y . 23
Ž  X, Y .13
s
 Xm X ⬘, Y
6
6
X m X⬘ m Y
X m X⬘ m Y
X m X ⬘ m Y.
Ž2.3.
6
X m X⬘ m Y Here Ž  X , Y .13 is the map operating as  X , Y on the Ž1, 3. position of the tensor product X m X ⬘ m Y. Similar notations Ž ᎐ .12 and Ž ᎐ . 23 will be used. 䢇
ŽP11222 . Similarly, ŽP11222 . is equivalent to
Ž  X , Y . 12 Ž  X , Y ⬘ . 13 s  X , YmY ⬘
Ž 2.4.
for all X g A; Y, Y⬘ g B. ŽP11121 . Let X, X ⬘ g A; V, V⬘ g V . The left-hand side of ŽP11121 . for Ž X, X ⬘, V, V ⬘. becomes 䢇
528
D. TAMBARA
X 䉺 Ž X ⬘ 䉺 Ž V 䉺V⬘.. s X m X ⬘ m V m A m V⬘ Ž y1 X⬘ .
X 䉺 ␣ 1121
6
6
X 䉺 ŽŽ X ⬘ 䉺V . 䉺V⬘. s X m X ⬘ m V m A m V⬘ ␣ 1121
Ž y1 X .
6
6
Ž X 䉺 Ž X ⬘ 䉺V .. 䉺V⬘ s X m X ⬘ m V m A m V⬘ ␣ 1112 䉺V X
1
6
6
ŽŽ X 䉺 X ⬘. 䉺V .. 䉺V ⬘ sX m X ⬘ m V m A m V ⬘, while the right-hand side becomes X 䉺 Ž X ⬘ 䉺 Ž V 䉺V⬘.. s X m X ⬘ m V m A m V ⬘ ␣ 1111
1
6
6
Ž X 䉺 X ⬘. 䉺 Ž V 䉺V⬘. s X m X ⬘ m V m A m V ⬘ ␣ 1121
Ž y1 Xm X ⬘ .
6
6
ŽŽ X 䉺 X ⬘. 䉺V .. 䉺V⬘ sX m X ⬘ m V m A m V⬘. So it is enough to show X m X⬘ m A
X m X⬘ m A s Ž X ⬘ . 23
X m X⬘ m A Xm X ⬘
6
6
Ž X .13
X m X ⬘ m A.
Ž2.5.
6
X m X⬘ m A
䢇
ŽP12111 . Similarly, ŽP12111 . is equivalent to
Xm X ⬘ s Ž X . 12 Ž X ⬘ . 13
Ž 2.6.
for all X, X ⬘ g A. ŽP11221 . Let X g A; Y g B; V, V⬘ g V . The left-hand side of ŽP11221 . for Ž X, V, Y, V⬘. becomes 䢇
529
DUALITY FOR MODULES
X 䉺 Ž V 䉺 Ž Y 䉺V⬘.. s X m V m A m Y m V ⬘ X 䉺 ␣ 1221
Ž␥ Y .
6
6
X 䉺 ŽŽ V 䉺Y . 䉺V⬘. s X m V m Y m A m V ⬘ ␣ 1121
Ž y1 X .
6
6
Ž X 䉺 Ž V 䉺Y .. 䉺V⬘ s X m V m Y m A m V ⬘ X, Y
. Ž X, Y .
6
␣ 1122 䉺V ⬘
6
ŽŽ X 䉺V . 䉺Y .. 䉺V⬘ sX m V m Y m A m V⬘, while the right-hand side becomes X 䉺 Ž V 䉺 Ž Y 䉺V⬘.. s X m V m A m Y m V ⬘ ␣ 1121
Ž y1 X .
6
6
Ž X 䉺V . 䉺 Ž Y 䉺V⬘. s X m V m A m Y m V ⬘ ␣ 1221
Ž␥ Y .
6
6
ŽŽ X 䉺V . 䉺Y .. 䉺V ⬘ sX m V m Y m A m V⬘. So it is enough to show that XmAmY
Ž y1 X .13
6
XmYmA Ž  X, Y .12
Ž y1 X .12
sXmAmY Ž␥ Y . 23
6
6
XmYmA
XmAmY 6
Ž␥ Y . 23
X m Y m A.
Ž2.7.
6
XmYmA
䢇
ŽP12211 . Similarly, ŽP12211 . is equivalent to
Ž ␥ Y . 12 Ž X . 13 Ž  Y , X . 23 s Ž X . 23 Ž ␥ Y . 12 for all X g A, Y g B.
Ž 2.8.
530
D. TAMBARA
ŽP12221 . Let Y, Y⬘ g B; V, V ⬘ g V . The left-hand side of ŽP12221 . for Ž V, Y, Y ⬘, V⬘. becomes 䢇
V 䉺 Ž Y 䉺 Ž Y ⬘ 䉺V⬘.. s V m A m Y m Y ⬘ m V ⬘ V 䉺 ␣ 2221
1
6
6
V 䉺 ŽŽ Y 䉺Y ⬘. 䉺V ⬘. s V m A m Y m Y⬘ m V ⬘ ␣ 1221
Ž ␥ YmY ⬘ .
6
6
Ž V 䉺 Ž Y 䉺Y⬘.. 䉺V⬘ s V m Y m Y⬘ m A m V ⬘ ␣ 1222 䉺V ⬘
1
6
6
ŽŽ V 䉺Y . 䉺Y⬘.. 䉺V ⬘ sV m Y m Y ⬘ m A m V⬘, while the right-hand side becomes V 䉺 Ž Y 䉺 Ž Y⬘ 䉺V⬘.. s V m A m Y m Y⬘ m V ⬘ ␣ 1221
Ž␥ Y .
6
6
Ž V 䉺Y . 䉺 Ž Y ⬘ 䉺V ⬘. s V m Y m A m Y⬘ m V ⬘ ␣ 1221
Ž␥ Y ⬘.
6
6
ŽŽ V 䉺Y . 䉺Y⬘.. 䉺V ⬘ sV m Y m Y ⬘ m A m V⬘. So it is enough to show that A m Y m Y⬘
6
Y m Y⬘ m A
sY m A m Y⬘ Ž␥ Y ⬘ . 23
Ž2.9.
6
␥ YmY ⬘
Ž␥ Y .12
6
A m Y m Y⬘
Y m Y ⬘ m A. ŽP11211 . Let X, X ⬘ g A; V, V⬘ g V . The left-hand side of ŽP11211 . for Ž X, V, V⬘, X ⬘. becomes 䢇
X 䉺 Ž V 䉺 Ž V⬘ 䉺 X ⬘.. s X m V m A m V⬘ m X ⬘ Ž X ⬘.
6
X 䉺 ␣ 1211
6
X 䉺 ŽŽ V 䉺V⬘. 䉺 X ⬘. s X m V m A m V⬘ m X ⬘ ␣ 1111
1
6
6
Ž X 䉺 Ž V 䉺V ⬘.. 䉺 X ⬘ s X m V m A m V⬘ m X ⬘ Ž y1 X .
6
␣ 1121 䉺 X ⬘
6
ŽŽ X 䉺V . 䉺V⬘.. 䉺 X ⬘ sX m V m A m V⬘ m X ⬘,
531
DUALITY FOR MODULES
while the right-hand side becomes X 䉺 Ž V 䉺 Ž V⬘ 䉺 X ⬘.. s X m V m A m V⬘ m X ⬘ ␣ 1121
Ž y1 X .
6
6
Ž X 䉺V . 䉺 Ž V⬘ 䉺 X ⬘. s X m V m A m V⬘ m X ⬘ ␣ 1211
Ž X ⬘.
6
6
ŽŽ X 䉺V . 䉺V ⬘.. 䉺 X ⬘ sX m V m A m V⬘ m X ⬘. So it is enough to show that
Ž X ⬘ . 23
X m A m X⬘ Ž y1 X .12
6
X m A m X⬘ 6
X m A m X⬘ s X m A m X⬘ 6
X m A m X⬘
Ž X ⬘ . 23
Ž2.10.
6
Ž y1 X .12
X m A m X ⬘.
ŽP11212 . Let X g A; V, V⬘, V⬙ g V . The left-hand side of ŽP11212 . for Ž X, V, V⬘, V⬙ . becomes 䢇
X 䉺 Ž V 䉺 Ž V⬘ 䉺V⬙ .. s X m V m V⬘ m B m V ⬙ X 䉺 ␣ 1212
Ž .
6
6
X 䉺 ŽŽ V 䉺V⬘. 䉺V ⬙ . s X m V m A m V ⬘ m V⬙ ␣ 1112
1
6
6
Ž X 䉺 Ž V 䉺V⬘.. 䉺V⬙ s X m V m A m V ⬘ m V⬙ Ž y1 X .
6
␣ 1121 䉺V ⬙
6
ŽŽ X 䉺V . 䉺V⬘.. 䉺V⬙ sX m V m A m V ⬘ m V⬙, while the right-hand side becomes X 䉺 Ž V 䉺 Ž V⬘ 䉺V⬙ .. s X m V m V⬘ m B m V ⬙ Ž  X , V ⬘mBmV ⬙ .
6
␣ 1122
6
Ž X 䉺V . 䉺 Ž V⬘ 䉺V ⬙ . s X m V m V⬘ m B m V ⬙ Ž .
6
␣ 1212
6
ŽŽ X 䉺V . 䉺V⬘.. 䉺V⬙ sX m V m A m V ⬘ m V⬙.
532
D. TAMBARA
So it is enough to show that
Xm
XmB  X, B
6
XmB 6
XmA s XmB 6
XmA
䢇
Xm
Ž2.11.
6
y1 X
X m A.
ŽP12122 . Similarly, ŽP12122 . is equivalent to
Ž m Y . Y s A , Y Ž m Y .
Ž 2.12.
for all Y g B. ŽP12212 . Let Y g B; V, V⬘, V⬙ g V . The left-hand side of ŽP12212 . for Ž V, Y, V⬘, V⬙ . becomes 䢇
V 䉺 Ž Y 䉺 Ž V⬘ 䉺V⬙ .. s V m Y m V ⬘ m B m V⬙ Ž y1 Y .
6
V 䉺 ␣ 2212
6
V 䉺 ŽŽ Y 䉺V⬘. 䉺V ⬙ . s V m Y m V⬘ m B m V⬙ Ž .
6
␣ 1212
6
Ž V 䉺 Ž Y 䉺V⬘.. 䉺V⬙ s V m A m Y m V⬘ m V ⬙ Ž␥ Y .
6
␣ 1221 䉺V ⬙
6
ŽŽ V 䉺Y . 䉺V ⬘. 䉺V⬙ sV m Y m A m V⬘ m V ⬙, while the right-hand side becomes V 䉺 Ž Y 䉺 Ž V⬘ 䉺V⬙ .. s V m Y m V⬘ m B m V⬙ 1
6
␣ 1222
6
Ž V 䉺Y . 䉺 Ž V⬘ 䉺V ⬙ . s V m Y m V⬘ m B m V⬙ Ž .
6
␣ 1212
6
ŽŽ V 䉺Y . 䉺V⬘.. 䉺V⬙ sV m Y m A m V⬘ m V ⬙.
533
DUALITY FOR MODULES
So it is enough to show that YmB Y
6
YmB T Ž Ym .
6
A m Y sY m B 6
YmA
Ym
Ž2.13.
6
␥Y
Y m A,
where T : Y m A ª A m Y is the map y m a ¬ a m y. ŽP12112 . Similarly, ŽP12112 . is equivalent to 䢇
T Ž X m . ␥ X s y1 X Ž m X .
Ž 2.14.
for all X g A. ŽP12121 . Let V, V⬘, V⬙, V g V . The left-hand side of ŽP12121 . for Ž V, V⬘, V⬙, V . becomes 䢇
V 䉺 Ž V⬘ 䉺 Ž V⬙ 䉺V .. s V m A m V ⬘ m V⬙ m A m V V 䉺 ␣ 2121
Ž .
6
6
V 䉺 ŽŽ V⬘ 䉺V⬙ . 䉺V . s V m A m V⬘ m B m V⬙ m V ␣ 1221
Ž ␥ V ⬘m BmV ⬙ .
6
6
Ž V 䉺 Ž V⬘ 䉺V⬙ .. 䉺V s V m V ⬘ m B m V⬙ m A m V ␣ 1212 䉺V
Ž .
6
6
ŽŽ V 䉺V ⬘. 䉺V⬙ .. 䉺V sV m A m V⬘ m V⬙ m A m V , while the right-hand side becomes V 䉺 Ž V⬘ 䉺 Ž V⬙ 䉺V .. s V m A m V⬘ m V⬙ m A m V Ž V ⬙m AmV .
6
␣ 1211
6
Ž V 䉺V⬘. 䉺 Ž V⬙ 䉺V . s V m A m V⬘ m V⬙ m A m V Ž y1 Vm AmV ⬘ .
6
␣ 1121
6
ŽŽ V 䉺V ⬘. 䉺V⬙ . 䉺V sV m A m V⬘ m V⬙ m A m V .
534
D. TAMBARA
So it is enough to show that AmA AmA A
6
AmB
6
␥B
sAmA
6
mA
Ž2.15.
y1 A
BmA
6
Am
A m A.
6
AmA By the symmetry of A and B, we need not consider the remaining ŽPi jk l m .. Thus we are reduced to showing Ž2.3. ᎐ Ž2.15.. The verifications of Ž2.3. ᎐ Ž2.10. are easy and left to the reader. Equations Ž2.11. ᎐ Ž2.15. are consequences of the identities of Proposition 2.1 as seen below. For Ž2.11., by the naturality in X, we may take X s A and it is enough to show that
y1 A Ž A m . Ž 1 m b . s Ž A m . A , B Ž 1 m b . for all b g B. Let : B ª B m A denote the right A-comodule structure corresponding to the left B-module structure on B and let Ž b . s Ýbi m a i . Let ⌬ Ž b . s ÝaX1 m aX2 . Then
Ý S Ž aX1 . m aX2 , Ž A m . A , B Ž 1 m b . s Ý ai m Ž bi . . y1 A Ž A m . Ž1 m b. s
But for any b⬘ g B, we have
ݲ b⬘, S Ž aX1 .: aX2 s ݲ S Ž b⬘. , aX1: aX2 s bX
Ž b. ,
Ý ² b⬘, ai : Ž bi . s Ž Ý ² b⬘, ai : bi . s Ž b⬘b . . Thus Ž2.11. reduces to the identity b⬘
Ž b . s Ž b⬘b .
of Proposition 2.1. Equation Ž2.12. similarly reduces to
Ž a⬘ © b . s a⬘ Ž b . .
DUALITY FOR MODULES
535
For Ž2.13., we may assume Y s B and it is enough to show that
␥B T Ž B m . Ž 1 m b . s Ž B m . B Ž 1 m b . for all b g B. Let : B ª B m A be as above and let Ž1. s Ýbi m a i , ⌬Ž b . s Ýb1 m b 2 for b g B. Then
␥ B T Ž B m . Ž 1 m b . s ␥ B Ž Ž b . m 1. s
Ý bi m Ž b . ai ,
Ž B m . B Ž 1 m b . s Ž B m . Ž Ý b1 m b 2 . s Ý b1 m Ž b 2 . . For any a⬘ g A, we have
Ý ² a⬘, bi : Ž b . ai s Ž b . a⬘, Ý ² a⬘, b1 : Ž b2 . s Ž b £ a⬘. . Thus Ž2.13. reduces to
Ž b . a⬘ s Ž b £ a⬘ . . Equation Ž2.14. similarly reduces to
Ž bb⬘ . s Ž b .
b⬘.
Finally, both sides of Ž2.15. are maps A m A ª A m A in A, where the A-module structure on the source A m A comes from the first factor. So it suffices to show that both sides coincide on elements 1 m a for all a g A. Let : B ª B m A be as above. Then
Ž m A . ␥ B Ž A m . Ž 1 m a. s Ž m A . ␥ B Ž 1 m Ž a. . s Ž m A . Ž Ž a. . , y1 y1 A A Ž 1 m a . s A Ž 1 m a . s
Ý S Ž a1 . m a2 .
So it is enough to show that
Ž m A . s Ž S m A . ⌬ . The left B-linearity b⬘
Ž b . s Ž b⬘b .
of is rephrased as the right A-colinearity
Ž m A. s T Ž S m A. ⌬ ,
536
D. TAMBARA
where T : A m A ª A m A is the map a m a⬘ ¬ a⬘ m a. Using this, we have
Ž S m A . Ž m A . s Ž S m A . T Ž S m A . ⌬ s T Ž S m S . ⌬ s ⌬ S . Thus Ž2.15. follows from the identity S s 1 of Proposition 2.1. This concludes the proof.
3. THE CORRESPONDENCE BETWEEN A-MODULES AND B-MODULES We keep the notation and the assumptions of the preceding section. Let A denote the 2-category of right A-modules with direct summands Modk-A ŽSection 1Že... We will construct a 2-equivalence between the 2-categories A and Modk-B B. Modk-A Since E is a bicategory, E12 naturally becomes an Ž E11 , E22 .-bimodule ŽSection 1Žd... That is, V becomes an Ž A, B .-bimodule. And similarly, V becomes a Ž B, A .-bimodule. The composition 䉺121: E12 = E21 ª E11 yields an Ž E11 , E11 .-linear functor E12 mE 22 E21 ª E11 , that is, an Ž A, A .linear functor P : V mB V ª A . Similarly, we obtain a Ž B, B .-linear functor Q: V mA V ª B . Since A has direct summands, by the universal property of the envelope ŽSection 1Že.. the functor P extends to an Ž A, A .-linear functor P : V mB V ª A . Similarly, Q extends to a Ž B, B .-linear functor Q: V mA V ª B . As V is an Ž A, B .-bimodule, if M is a right A-module then M mA V becomes a right B-module, and its envelope M mA V becomes a right B-module with direct summands. Similarly, if N is a right B-module, then N mB V becomes a right A-module with direct summands.
537
DUALITY FOR MODULES
For a right A-module M with direct summands, the functor P induces an A-linear functor PM> :
Ž M mA
V
. mB
V , M mA Ž V mB V
. ª M mA
A , M,
and for a right B-module N with direct summands, Q similarly induces a B-linear functor Q>N : Ž N mB V
. mA
V ª N.
THEOREM 3.1. For any right A-module M with direct summands, PM> is an equi¨ alence of A-modules. And Q>N is an equivalence of B-modules as well. In short, the 2-functors 6
ymA V
6
A Modk-A
ymB V
B Modk-B
are quasi-inverse to each other. The theorem will follow from PROPOSITION 3.2. The functor P: V mB V ª A is an equi¨ alence of Ž A, A .-bimodules. Proof. As noted in Section 1Ža., it is enough to show that P is simply an equivalence of categories. For this purpose, by the universal property of the envelope ŽSection 1Že.. it is enough to show that for any k-linear category C with direct summands the induced functor Hom Ž P , C . : Hom Ž A , C . ª Hom Ž V mB V , C . is an equivalence. Ž1. Let CMod- A denote the category of right A-module objects in C ; an object of CMod- A is a pair Ž C, . of an object C g C and a morphism : C m A ª C in C satisfying the associativity and the unital property as in usual modules. Here the tensor product C m V for any C g C and V g V is defined as in Section 1Ža.. Since A is a semisimple category with generator A and C has direct summands, any functor G: A ª C is determined by the object G Ž A. g C together with the algebra map Aop s End A A ª End GŽ A. or, equivalently, by the right action : G Ž A. m A ª GŽ A. wA, Proposition 2.7Žii.x. So we have an equivalence Hom Ž A , C . ª CMod- A G ¬ Ž G Ž A. , . .
538
D. TAMBARA
Ž2. Let D be the category defined as follows. An object of D is a pair Ž M, . of an object M of C and a family of isomorphisms Y : M m Y ª Y m M in C for all Y g B which are natural in Y and which satisfy the identities
YmY ⬘ s Ž Y m Y ⬘ . Ž Y m Y ⬘ . ,
k s 1 M
for all Y, Y ⬘ g B. Morphisms of D are obvious ones. For an object Ž M, . g D, we have a bilinear functor FM : V = V ª C
Ž V , V ⬘ . ¬ V m M m V⬘ and isomorphisms 6
V, Y , V ⬘
6
FM Ž V , Y m V⬘ . 5 V m M m Y m V⬘
Vm YmV ⬘
FM Ž V m Y , V ⬘ . 5 V m Y m M m V⬘
for V, V ⬘ g V ; Y g B. Then we obtain an object Ž FM , . of BiHom B Ž V , V ; C .. Since any bilinear functor V = V ª C is of the form FM Žup to isomorphism., we obtain an equivalence D , BiHom B Ž V , V ; C .
Ž M, . ¬ Ž FM , . . Ž3. We know the equivalence BiHom B Ž V , V ; C . , HomŽ V mB V , C . ŽSection 1Žb... Ž4. Let CB -Com denote the category of left B-comodule objects in C ; an object of this category is a pair Ž M, . of an object M of C and a morphism : M ª B m M in C satisfying the coassociativity and the counit condition. We have an isomorphism of categories CB -Com ( D
Ž M, . l Ž M, . in which and are related as MmB
B
BmM
6
Mm
6
: M and
BmMmY(BmYmM
YmM
Y m M.
6
mY
6
Y : M m Y
539
DUALITY FOR MODULES
Here : k ª B is the unit, Y : B m Y ª Y is the B-module structure on Y, and the middle isomorphism is a permutation. This is well-known and easily verified. Ž5. We have a usual isomorphism CB -Com ( CMod- A . The combination of the equivalences in Ž1. ᎐ Ž5. yields an equivalence Hom Ž A , C . ª Hom Ž V mB V , C . . Let us check that this is isomorphic to the functor HomŽ P, C . in question. Then the proof will be completed. Let ⌽ : Hom Ž A , C . ª CMod- A ( CB -Com ( D ª BiHom B Ž V , V ; C . G ¬ Ž M, . ¬ Ž M, . ¬ Ž M, . ¬ Ž FM , . be the composition of Ž1., Ž5., Ž4., and Ž2.. Recall that P: V mB V ª A arose from the B-bilinear functor
Ž 䉺121 , ␣ . : V = V ª A , where V 䉺121 V⬘ s V m A m V⬘, ␣ V, Y , V ⬘
Ž V 䉺Y . 䉺V⬘ 5 V m Y m A m V⬘,
6 6
V 䉺 ŽY 䉺 V . 5 V m A m Y m V⬘
Vm ␥ YmV ⬘
and ␥ Y is as in Section 2. So Hom Ž P , C .
Hom Ž V mB V , C . , BiHom B Ž V , V ; C .
6
⌿ : Hom Ž A , C .
is given by the composition with Ž 䉺121 , ␣ . G ¬ G( Ž 䉺121 , ␣ . . We have G Ž V 䉺121V ⬘ . s G Ž V m A m V⬘ . s V m M m V⬘, G Ž ␣ V , Y , V ⬘ . s G Ž V m ␥ Y m V⬘ . s V m ␥ M , Y m V ⬘, where ␥ M , Y is defined by MmYmA(YmMmA
Ym
YmM
6
Mm Y
6
␥M , Y : M m Y
540
D. TAMBARA
with Y : Y ª Y m A being the comodule structure and the middle isomorphism a permutation. But we see easily that ␥M , Y s Y . Hence G( Ž 䉺121 , ␣ . s Ž FM , . . Thus ⌽ s ⌿, as required. For any right A-module M with direct summands, the A-linear functor PM> :
Ž M mA
V
. mB
VªM
corresponds to a B-linear functor TM : M mA V ª Hom A Ž V , M . by the hom-tensor adjoint in Section 1Žc.. A formal consequence of Proposition 3.2 is PROPOSITION 3.3. The functor TM is an equi¨ alence of right B-modules. Proof. We have commutative diagrams
where the arrows labeled ‘‘can’’ are canonical functors. By Proposition 3.2, the lower arrows of the diagrams are equivalences. As the composite of the upper arrows of the first diagram, the functor M ª Hom B Ž V , Hom A Ž V , M . . is an equivalence of A-modules. Likewise, for any right B-module N with direct summands, we have an equivalence of B-modules N ª Hom A Ž V , Hom B Ž V , N . . .
DUALITY FOR MODULES
541
Thus the 2-functors 6
Hom A Ž V , ᎐ .
6
A Modk-A
Hom B Ž V , ᎐ .
B Modk-B
are quasi-inverse to each other. Now, by the first diagram, Hom B Ž V , TM . has a right quasi-inverse. Applying H om A Ž V , ᎐ . and using the equivalence Ž ᎐ . , Hom A Ž V , Hom B Ž V , ᎐ .., we know that TM has a right quasi-inverse. By the second diagram, TM mB V has a left quasi-inverse. Applying ᎐mA V and using the equivalence ŽŽ ᎐ . mB V . mA V , Ž ᎐ ., we know that TM has a left inverse as well. Hence TM is an equivalence. Finally, we give another description of Hom A Ž V , M .. Let M be a right A-module with action 䉺 : M = A ª M . The category of right A-comodule objects in M , denoted by MCom- A , is defined as follows. An object is a pair Ž M, M . of an object M in M and a morphism M : M ª M䉺 A in M satisfying the coassociativity and the counit condition as in usual comodules. Here A in M䉺 A is the regular A-module. The morphisms of MCom- A are obvious ones. The category MCom- A has a structure of a right B-module; for an object Ž M, M . in MCom- A and a B-module Y, an object Ž M, M . 䉺Y in MCom- A is defined as
where Y : Y ª Y m A is the comodule structure map, : A m A ª A is the multiplication map, and the middle isomorphism comes from the bilinearity of 䉺. ŽWe have the tensor product M m V g M for M g M and V g V . See the end of Section 1Ža... The associativity isomorphism Ž M, . 䉺 Ž Y m Y ⬘. ( ŽŽ M, . 䉺Y . 䉺Y ⬘ is just the identity on M m Y m Y⬘. PROPOSITION 3.4. For any right A-module M with direct summands, we ha¨ e an equi¨ alence of right B-modules Hom A Ž V , M . , MCom- A . Proof. We have an obvious equivalence M , Hom Ž V , M . M ¬ Ž FM : V ¬ M m V . .
542
D. TAMBARA
Suppose that FM has a structure of A-linearity
V , X : FM Ž V m X . ª FM Ž V . 䉺 X for V g V , X g A, so that Ž FM , . is an object of Hom A Ž V , M .. By the naturality of and the semisimplicity of A, is determined by k, A : M m A ª M䉺 A, where A is the regular left A-module. As k, A should commute with the right action of A on A, it is determined by its restriction s k , A ( Ž 1 m . : M ª M m A ª M䉺 A, where : k ª A is the unit. Explicitly, V , X is recovered from as the composite
where X : A m X ª X is the module structure map of X and X t means X is considered just a vector space. The compatibilities of with the associativity and the unit isomorphisms are translated into the coassociativity and the counit condition for . Thus Ž M, . is an object of MCom- A . Conversely, given any comodule structure : M ª M䉺 A, the morphism >V , X is an isomorphism. Indeed, the inverse is given by
where the vertical arrows are natural isomorphisms. ŽNote that X Ž S m 1.: A m X ª X t is left A-linear.. Thus we have established an equivalence MCom- A , Hom A Ž V , M .
Ž M, . ¬ ž FM , >/ .
543
DUALITY FOR MODULES
It remains to show that this equivalence is compatible with the actions of B. For any object Ž F, . in Hom A Ž V , M . and a B-module Y, recalling the definition in Section 1Žc. we have Ž F, . 䉺Y s Ž F⬘, ⬘., where F⬘ Ž V . s F Ž Y m V . and XV , X is the composite
6
F ŽŽ Y m V . m X .
YmV , X
F⬘Ž V . 䉺 X 5 Ž F Y m V . 䉺X
6
F⬘Ž V m X . 5 Ž Ž F Y m V m X ..
with the first arrow induced by the map YmAmX
1m X
YmX
6
Ym1
6
Y , X : Y m X
of Section 2. Now let Ž M, M . g MCom- A . Then it is easy to see that Ž FM , >M . 䉺Y s Ž FMmY , Ž MmY . >., where MmY : M m Y ª Ž M m Y . 䉺 A is the comodule structure defined before Proposition 3.4. Thus Ž FM , >M . 䉺Y corresponds to Ž M, M . 䉺Y under the equivalence. 4. DUALITY FOR HOPF ALGEBRA ACTIONS In this section we relate the correspondence between category modules in Section 3 with the duality for Hopf algebra actions on algebras due to Blattner and Montgomery wBMx. In the beginning we only assume that A is a finite dimensional Hopf algebra and B is the dual of A. For a left A-module algebra R with action written as A = R ª R: Ž a, r . ¬ ad r , the smash product R噛A is the algebra with underlying space R m A and multiplication
Ž r m a. Ž r ⬘ m a⬘ . s Ý r Ž a1 d r ⬘ . m a2 a⬘, where ⌬Ž a. s Ýa1 m a2 . A left R噛A-module is thought of as a vector space M with two structures, that of an R-module and an A-module, such that the R-module structure map R m M ª M is A-linear. Here are several facts whose verifications are straightforward. Ž1. If R is a left A-module algebra, then R噛A has a structure of a
544
D. TAMBARA
left B-module algebra. The action d of B on R噛A is given by bd Ž r m a . s r m Ž b © a . . Ž2. If R is a left A-module algebra, then the category R噛A-Mod becomes a right module over A-Mod. The action 䉺 : R噛A-Mod = A-Mod ª R噛A-Mod is defined as follows. For an R噛A-module V and an A-module X, we set V䉺X s V m X on which R and A act by r Ž ¨ m x . s r¨ m x, aŽ ¨ m x . s
Ý a1¨ m a2 x
for r g R, a g A. With this action V m X becomes an R噛A-module. The associativity isomorphism V 䉺 Ž X m X ⬘. ª Ž V 䉺 X . 䉺 X ⬘ is the identity on V m X m X ⬘. Ž3. If R is a left A-module algebra, then the category R-Mod becomes a right module over B-Mod. Indeed, for an R-module V and a B-module Y, we set V 䉺Y s V m Y on which R acts by r Ž ¨ m y. s
Ý Ž a i d r . ¨ m yi .
Here Ž y . s Ý yi m a i and : Y ª Y m A is the A-comodule structure corresponding to the B-module structure on Y. The associativity isomorphism V 䉺 Ž Y m Y ⬘. ª Ž V 䉺Y . 䉺Y⬘ is the identity on V m Y m Y ⬘. Ž4. The action of A-Mod on R噛A-Mod in Ž2. and the one in Ž3. with R噛A regarded as a B-module algebra as in Ž1. coincide. Proof. Let V be an R噛A-module and X an A-module. In either action, V 䉺 X has the underlying space V m X. Relative to the action in Ž2., we have
Ž r m a. Ž ¨ m x . s Ý Ž r m a1 . ¨ m a2 x, while relative to the action in Ž3.,
Ž r m a. Ž ¨ m x . s Ý Ž bi d Ž r m a. . ¨ m x i s
Ý Ž r m Ž bi © a. . ¨ m x i s Ý Ž r m a1² bi , a2 : . ¨ m x i s Ý Ž r m a1 . ¨ m a2 x i ,
545
DUALITY FOR MODULES
where Ž x . s Ý x i m bi , : X ª X m B is the comodule structure map. Thus the two module structures coincide. Let R be a B-module algebra. Put A s A-Mod, B s B-Mod. By Ž3., R-Mod is a right A-module. Then we have the right B-module Ž RMod. Com- A , the category of right A-comodule objects in R-Mod ŽSection 3.. PROPOSITION 4.1. We ha¨ e an isomorphism of right B-modules
Ž R-Mod. Com-A ( R噛B-Mod. Proof. Let M be a left R-module with module structure map : R m M ª M. By the definition of the action of A-Mod on R-Mod in Ž3., the R-module structure of M䉺 A s M m A is given by r Ž m m a. s
Ý Ž bi d r . m m ai
for r g R, m g M, a g A, where Ýa i m bi s AŽ a. and A : A ª A m B is the right B-comodule structure of A corresponding to the left A-module structure of A. Suppose furthermore that M has an A-comodule structure : M ª M m A. Let : B m M ª M be the corresponding B-module structure. We will write Ž b m m. s bm as usual. If Ž m. s Ým i m a i , then r Ž m. s
Ý Ž bi j d r . m i m ai j ,
where AŽ a i . s Ýa i j m bi j . So
Ž rm . s r Ž m . m Ž rm . s
Ý Ž bi j d r . m i m ai j m b Ž rm . s Ý Ž bi j d r . m i ² b, a i j :
for all b g B.
But we have ݲ b, a i j : bi j s ݲ b 2 , a i : b1 , where ⌬Ž b . s Ýb1 m b 2 . Hence the right-hand side of the last equation is rewritten as
Ý Ž ² b 2 , a i : b1 d r . m i s Ž b1 d r . Ž b 2 m . . Thus Ž rm. s r Ž m. if and only if bŽ rm. s ÝŽ b1 d r .Ž b 2 m. for all b g B. Thus is R-linear if and only if : R m M ª M is B-linear. That is, Ž M, , . is an object of Ž R-Mod. Com- A if and only if Ž M, , . is an object of R噛B-Mod. This establishes the isomorphism of categories Ž RMod. Com- A ( R噛B-Mod. It remains to take care of the actions of B. Let Y be a B-module. Let M, , , be as above. Relative to the action of B on Ž R-Mod. Com- A
546
D. TAMBARA
defined in Section 3, we have Ž M, , . 䉺Y s Ž M m Y⬘, ⬘, ⬘., where ⬘ s m Y and the A-comodule structure ⬘ of M m Y is the standard structure of the tensor product of the A-comodules M and Y. Relative to the action of B on R噛B-Mod defined in Ž2., we have Ž M, , . 䉺Y s Ž M m Y, ⬘, ⬘., where ⬘ is the B-module structure of the tensor product of the B-modules M and Y. Since ⬘ corresponds to ⬘, Ž M, , . 䉺Y corresponds to Ž M, , . 䉺Y. Thus the above category isomorphism is B-linear. Assume that A is semisimple and cosemisimple. PROPOSITION 4.2. A-modules
For any B-module algebra R we ha¨ e an equi¨ alence of R-Mod , Ž R噛B . 噛A-Mod.
Proof. By Propositions 3.4 and 4.1 the 2-functor A ª Modk-B B Hom A Ž V , ᎐ . : Modk-A takes the A-module R-Mod to the B-module R噛B-Mod, and in view of Ž4. the 2-functor B ª Modk-A A Hom B Ž V , ᎐ . : Modk-B takes R噛B-Mod to Ž R噛B .噛A-Mod. We saw in the proof of Proposition 3.3 that these 2-functors are quasi-inverse to each other. Hence Ž R噛B .噛A-Mod must be equivalent to R-Mod. Tracing a series of the equivalences in Propositions 3.2, 3.3, 3.4, and 4.1, one can verify that the above equivalence is given by the functor sending an R-module M to the Ž R噛B .噛A-module M m A. Here R, B, and A act on M m A as r Ž m m a. s
Ý Ž bi d r . m m ai
b Ž m m a. s m m Ž b © a. a⬘ Ž m m a . s m m aSy1 Ž a⬘ . for a, a⬘ g A, b g B, r g R, and m g M, where Ž a. s Ýa i m bi and : A ª A m B is the comodule structure. To recover the duality isomorphism for Hopf algebra actions, take M to be the R-module R. Then the left action of Ž R噛B .噛A on R m A commutes with the right action of R given by
Ž r m a. r ⬘ s rr ⬘ m a.
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By the equivalence the two-sided action yields the ring isomorphism Ž R噛B .噛A ( End R Ž R m A. ( R m End A. REFERENCES wAx wBx wBMx wEGx wFSx wGVx wKx wLR1x wLR2x wLSx wNTx wSx
M. Auslander, Representation theory of artin algebras, I, Comm. Algebra 1 Ž1974., 177᎐268. J. Benabou, Introduction to bicategories, in ‘‘Reports of the Midwest Category ´ Seminar,’’ Lecture Notes in Mathematics, Vol. 47, pp. 1᎐78, Springer-Verlag, New York, 1967. R. J. Blattner and S. Montgomery, A duality theorem for Hopf module algebras, J. Algebra 95 Ž1985., 153᎐172. P. Etingof and S. Gelaki, On finite-dimensional semisimple and cosemisimple Hopf algebras in positive characteristic, Int. Math. Res. Notices 16 Ž1998., 851᎐864. P. J. Freyd and A. Scedrov, ‘‘Categories, Allegories,’’ North-Holland, Amsterdam, 1990. A. Grothendieck and J.-L. Verdier, Topos, in ‘‘Theorie des Topos et Cohomologie ´ Etale des Schemas,’’ Lecture Notes in Mathematics, Vol. 269, pp. 299᎐519, Springer´ Verlag, New York, 1972. C. Kassel, ‘‘Quantum Groups,’’ Springer-Verlag, New York, 1995. R. G. Larson and D. E. Radford, Semisimple cosemisimple Hopf algebras, Amer. J. Math. 109 Ž1987., 187᎐195. R. G. Larson and D. E. Radford, Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple, J. Algebra 117 Ž1988., 267᎐289. R. G. Larson and M. E. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 Ž1969., 75᎐94. Y. Nakagami and M. Takesaki, ‘‘Duality for Crossed Products of von Neumann Algebras,’’ Lecture Notes in Mathematics, Vol. 731, Springer-Verlag, New York, 1979. M. E. Sweedler, ‘‘Hopf Algebras,’’ Benjamin, New York, 1969.
A Duality for Modules over Monoidal Categories of Representations of Semisimple Hopf Algebras D. Tambara Department of Mathematical System Science, Hirosaki Uni¨ ersity, Hirosaki 036-8561, Japan Communicated by Susan Montgomery Received January 4, 1999
For a finite dimensional semisimple cosemisimple Hopf algebra A and its dual Hopf algebra B, we set up a natural one-to-one correspondence between categories with actions of the monoidal categories of representations of A and of B. This gives a categorical interpretation of the duality for actions of Hopf algebras on algebras. 䊚 2001 Academic Press
INTRODUCTION Let A be a finite dimensional semisimple cosemisimple Hopf algebra over a field k. Let A be the category of finite dimensional A-modules. As A is a monoidal category, we have a notion of A-modules: A right A-module is a linear category M equipped with a bilinear functor M = A ª M and coherent isomorphisms of associativity and unit. Let B be the dual Hopf algebra of A and B the category of finite dimensional B-modules. The purpose of the paper is to set up a natural one-to-one correspondence between right A-modules and right B-modules with direct summands. Recall the duality theorem for Hopf algebra actions on algebras due to Blattner and Montgomery wBMx, generalizing the duality for group actions due to Nakagami and Takesaki wNTx. It asserts that if R is an algebra on which B acts, the two-step smash product algebra Ž R噛B .噛A is isomorphic to a full matrix algebra over R, without the semisimplicity of A and B assumed. In the case of semisimple Hopf algebras, this is related to the above correspondence as follows. The category R-Mod of R-modules becomes naturally a right A-module. Similarly if S is an algebra with 515 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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A-action, the category S-Mod becomes a right B-module. In our correspondence, the A-module R-Mod is mapped to the B-module R噛B-Mod, and B-module S-Mod is mapped to the A-module S噛A-Mod. The mappings are inverse to each other. So Ž R噛B .噛A-Mod should be equivalent to R-Mod, which is in accord with the duality theorem. The correspondence between A-modules and B-modules is given by categorical analogues of Hom and m functors for usual modules. Let V be the category of finite dimensional k-modules. Then V becomes an Ž A, B .-bimodule and a Ž B, A .-bimodule as well. For an A-module M with direct summands, the corresponding B-module is the category Hom A Ž V , M . of A-linear functors V ª M . This is also equivalent to the B-module M mA V , which is obtained by firstly making the tensor product M mA V and then adjoining direct summands. The duality equivalence M mA V mB V , M is induced by an equivalence of Ž A, A .-bimodules V mB V , A. In addition, the equivalences V mB V , A, V mA V , B can be taken in a coherent way so that the monoidal categories A, B, the Ž A, B .-bimodule V , and the Ž B, A .-bimodule V form a monoidal category of matrix A V ., or in other words a bicategory with two objects. form Ž V B In Section 1 we review basic facts about modules over monoidal categories, including the definition of tensor product categories. In Section 2 we construct the bicategory mentioned above. This gives the bimodule structures on V and the functors V mB V ª A, V mA V ª B. In Section 3 we prove that the induced functors V mB V ª A, V mA V ª B are equivalences, from which the main result follows. In Section 4 we describe the correspondence for categories of modules over rings with Hopf algebra actions in terms of the smash product construction. Notations. For a Hopf algebra A, ⌬ denotes the comultiplication map, and S denotes the antipode of A. For a ring R, R-Mod denotes the category of finitely generated left R-modules.
1. MODULES OVER MONOIDAL CATEGORIES This section consists mainly of the definitions of modules over monoidal categories and tensor products of those modules. Nothing will be new. For generality about monoidal categories see wKx. A k-linear category is a category in which the Hom-sets are k-vector spaces, the compositions are k-bilinear operations and finite direct sums exist. The notion of a k-linear functor C ª D, and a k-bilinear functor C = C ⬘ ª D for k-linear categories C , C ⬘, D will be obvious. Let HomŽ C , D . denote the category of k-linear functors C ª D.
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Ža. A monoidal category over k is a k-linear category A equipped with a k-bilinear functor 䉺: A = A ª A, an object I, and natural isomorphisms
␣ X , Y , Z : X 䉺 Ž Y 䉺Z . ª Ž X 䉺Y . 䉺Z, X : X ª I䉺 X ,
X : X ª X 䉺I
satisfying the identities
Ž ␣ X , Y , Z 䉺W . ␣ X , Y 䉺 Z , W Ž X 䉺 ␣ Y , Z , W . s ␣ X 䉺Y , Z , W ␣ X , Y , Z䉺W , Ž M1. ␣ X , I , Y Ž X 䉺 Y . s X 䉺Y
Ž M2.
for all objects X, Y, Z, W in A. For a monoidal category A, a left A-module is a k-linear category M equipped with a k-bilinear functor 䉺: A = M ª M and natural isomorphisms
␣ X , Y , M : X 䉺 Ž Y 䉺M . ª Ž X 䉺Y . 䉺M, M : M ª I䉺M for X, Y g A, M g M satisfying ŽM1. with Ž X, Y, Z, W . replaced with Ž X, Y, Z, M . and ŽM2. with Ž X, Y . replaced with Ž X, M . for all X, Y, Z g A, M g M . A right A-module is similarly defined. For monoidal categories A and B, an Ž A, B .-bimodule is a k-linear category M equipped with k-bilinear functors 䉺: A = M ª M , 䉺: M = B ª M , and natural isomorphisms
␣ X , Y , M : X 䉺 Ž Y 䉺M . ª Ž X 䉺Y . 䉺M, ␣ X , M , S : X 䉺 Ž M䉺S . ª Ž X 䉺M . 䉺S, ␣ M , S , T : M䉺 Ž S䉺T . ª Ž M䉺S . 䉺T , M : M ª I䉺M,
M : M ª M䉺I
for X, Y g A; M g M ; S, T g B satisfying ŽM1. with Ž X, Y, Z, W . replaced with Ž X, Y, Z, M ., Ž X, Y, M, S ., Ž X, M, S, T ., and Ž M, S, T, U . and ŽM2. with Ž X, Y . replaced with Ž X, M ., Ž M, S . for all X, Y, Z g A; M g M ; S, T, U g B. For left A-modules M and N , an A-linear functor Ž F, .: M ª N consists of a k-linear functor F: M ª N and natural isomorphisms
X , M : F Ž X 䉺M . ª X 䉺F Ž M .
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satisfying the identities
X 䉺Y , M F Ž ␣ X , Y , M . s ␣ X , Y , F Ž M . Ž X 䉺 Y , M . X , Y 䉺 M , I , M F Ž M . s F Ž M . for all X, Y g A, M g M . We write Ž F, . s F occasionally. For A-linear functors Ž F, ., Ž F⬘, ⬘.: M ª N , a morphism Ž F, . ª Ž F⬘, ⬘. is a natural transformation : F ª F⬘ satisfying
XX , M X 䉺 M s Ž X 䉺 M . X , M for all X g A, M g M . With this notion of morphisms, we have the category of A-linear functors M ª N , denoted by Hom A Ž M , N .. For A-linear functors Ž G, .: L ª M and Ž F, .: M ª N , the composition Ž F, .(Ž G, . is the A-linear functor Ž F (G, .: L ª N where
X , L s X , GŽ L. F Ž X , L . for X g A, L g L . We then have the functor Hom A Ž M , N . =Hom A Ž L , M . ª Hom A Ž L , N .
Ž Ž F , . , Ž G, . . ¬ Ž F , . ( Ž G, . . The composition is strictly associative. Thus we have a 2-category in which the objects are left A-modules, the Hom-categories are Hom A Ž M , N ., and the horizontal compositions are the above compositions. This 2-category is denoted by A-Mod. An A-linear functor F: M ª N is called an equivalence if there is an A-linear functor G: N ª M with isomorphisms F (G ( 1 in Hom A Ž N , N . and G( F ( 1 in Hom A Ž M , M .. Then G is called a quasi-inverse of F. It should be noted that F is an equivalence of A-modules if it is simply an equivalence of categories. Indeed, any quasi-inverse of F, as categories, can be made into an A-linear functor giving a quasi-inverse of F as A-modules. Let V be the monoidal category of finite dimensional vector spaces over k. Any k-linear category C becomes a left V-module by setting k n m X s X n, the n-fold direct sum. Žb. For a right A-module M , a left A-module N , and a k-linear category L , an A-bilinear functor Ž F, ␣ .: M = N ª L consists of a k-bilinear functor F: M = N ª L and natural isomorphisms
␣ M , X , N : F Ž M, X 䉺N . ª F Ž M䉺 X , N .
519
DUALITY FOR MODULES
satisfying F Ž ␣ M , X , Y , N . ␣ M , X 䉺Y , N F Ž M, ␣ X , Y , N . s ␣ M 䉺 X , Y , N ␣ M , X , Y 䉺 N ,
␣ M , I , N F Ž M, N . s F Ž M , N . for all M g M , N g N , X, Y g A. With an obvious definition of morphisms, we have the category of A-bilinear functors M = N ª L , denoted by BiHom A Ž M , N ; L .. We will construct a k-linear category M mA N and an A-bilinear functor M = N ª M mA N inducing an equivalence HomŽ M mA N , L . ª BiHom A Ž M , N ; L . for any k-linear category L . As a k-linear category, M mA N has the following presentation by generators and relations. Objects are finite direct sums of the symbols w M, N x for M g M , N g N . Generators for morphisms are the symbols
w f , g x : w M, N x ª w M⬘, N⬘ x for morphisms f : M ª M⬘ in M and g: N ª N⬘ in N , and the symbols
␣ M , X , N : w M, X 䉺N x ª w M䉺 X , N x , X ␣M , X , N : w M䉺 X , N x ª w M, X 䉺N x
for objects M g M , X g A, and N g N . Relations among them are Ži. ᎐ Žv. below. Ži. ŽLinearity .
w f q f ⬘, g x s w f , g x q w f ⬘, g x ,
w f , g q g ⬘x s w f , g x q w f , g ⬘x ,
w af , g x s aw f , g x s w f , ag x for morphisms f, f ⬘: M ª M⬘ in M ; g, g ⬘: N ª N⬘ in N ; and a g k. Žii. ŽFunctoriality.
w f ⬘ f , g ⬘g x s w f ⬘, g ⬘ xw f , g x , w 1 M , 1 N x s 1w M , N x for morphisms f : M ª M⬘, f ⬘: M⬘ ª M⬙ in N⬘ ª N⬙ in N . Žiii. ŽIsomorphism. X ␣M , X , N ␣M , X , N s 1,
M and g: N ª N⬘, g ⬘:
X ␣M , X , N ␣ M , X , N s 1.
Živ. ŽNaturality.
␣ M ⬘, X ⬘, N ⬘ w f , u䉺 g x s w f 䉺u, g x ␣ M , X , N
520
D. TAMBARA
for morphisms f : M ª M⬘ in M , u: X ª X ⬘ in A, and g: N ª N⬘ in N . Žv. ŽPentagon and triangle.
w ␣ M , X , Y , N x ␣ M , X 䉺Y , N w M, ␣ X , Y , N x s ␣ M 䉺 X , Y , N ␣ M , X , Y 䉺 N , ␣ M , I , N w M, N x s w M , N x , where wy, N x s wy, 1 N x, w M, yx s w1 M , yx. The bilinear functor T : M = N ª M mA N is defined by T Ž M, N . s w M, N x for objects and T Ž f , g . s w f , g x for morphisms. The isomorphisms ␣ M , X , N then give T a structure of an A-bilinear functor. From this construction, it will be obvious that for any k-linear category L , the functor Hom Ž M mA N , L . ª BiHom A Ž M , N ; L . G ¬ G(T is an equivalence. Žc. If M is a right A-module, N is an Ž A, B .-bimodule and L is a right B-module, then Hom B Ž N , L . has a structure of a right A-module and M mA N has a structure of a right B-module. The action of an object X g A on an object Ž F, . g Hom B Ž N , L . is given by
Ž F , . 䉺 X s Ž F⬘, ⬘ . , where F⬘ Ž N . s F Ž X 䉺N . and F⬘Ž N䉺Y . s F Ž X 䉺 Ž N䉺Y .. F Ž ␣ X, N, Y .
6
X N, Y
F ŽŽ X 䉺N . 䉺Y . X 䉺 N, Y
6
6
F⬘Ž N . 䉺Y s F Ž X 䉺N . 䉺Y for N g N , Y g B. The associativity of this action is given by the isomorphisms F Ž ␣ X , X ⬘, N . : F Ž X 䉺 Ž X ⬘ 䉺N . . ª F Ž Ž X 䉺 X ⬘ . 䉺N . for X, X ⬘ g A, N g N .
DUALITY FOR MODULES
521
The action of an object Y g B on an object w M, N x g M mA N is given by
w M, N x 䉺Y s w M, N䉺Y x , and the action on the morphism ␣ M , X , N is given by w M, X 䉺N x 䉺Y sw M, Ž X 䉺N . 䉺Y x w M, ␣ y1 X, N, Y x
6
␣ M, X , N 䉺Y
w M, X 䉺 Ž N䉺Y .x ␣ M, X , N 䉺Y
6
6
w M䉺 X, N x 䉺Y s w M䉺 X, N䉺Y x. The associativity of this action is given by the isomorphisms
w M, ␣ N , Y , Y ⬘ x : M, N䉺 Ž Y 䉺Y⬘ . ª M, Ž N䉺Y . 䉺Y ⬘ for Y, Y ⬘ g B. We have an equivalence of k-categories Hom B Ž M mA N , L . ª Hom A Ž M , Hom B Ž N , L . . G ¬ Ž M ¬ Ž N ¬ G Ž w M, N x . . . This follows from the equivalence at the end of Žb.. Žd. A bicategory E consists of a set J; a collection of k-linear categories Ei j for i, j g J; bilinear functors 䉺 i jk : Ei j = Ejk ª Ei k for i, j, k g I; objects Ii g Eii ; and natural isomorphisms
␣ X , Y , Z : X 䉺i jl Ž Y 䉺jk l Z . ª Ž X 䉺i jk Y . 䉺i k l Z, X : X ª Ii 䉺ii j X ,
X : X ª X 䉺i j j I j
for X g Ei j , Y g Ejk , and Z g Ek l satisfying identities analogous to ŽM1. and ŽM2.. See wBx. Each category Eii becomes a monoidal category and Ei j becomes an Ž Eii , Ej j .-bimodule. Moreover, 䉺i jk : Ei j = Ejk ª Ei k becomes an Ej j-bilinear functor and hence induces a functor Ei j mE j j Ejk ª Ei k . This in turn becomes an Ž Ei i , Ek k .-linear functor. Že. A k-linear category C is said to have direct summands if any idempotent endomorphism e: X ª X in C is a projection to a direct summand of X. The envelope C of C is the category defined as follows: An object of C is a pair Ž X, e . of an object X g C and an idempotent
522
D. TAMBARA
e g End X. A morphism Ž X, e . ª Ž X ⬘, e⬘. is a morphism f : X ª X ⬘ in C such that fe s f s e⬘ f. Then C has direct summands, and the functor C ª C : X ¬ Ž X, 1. has the following universality: For any k-linear category D with direct summands, the induced functor Hom Ž C , D . ª Hom Ž C , D . is an equivalence. Indeed, an idempotent f g EndŽ X, e . gives rise to the decomposition Ž X, e . ( Ž X, f . [ Ž X, e y f . so that f is isomorphic to the projection to the summand Ž X, f .. Any functor F : C ª D extends to F: C ª D which takes Ž X, e . to the direct summand of F Ž X . corresponding to the idempotent F Ž e . and takes a morphism f : Ž X, e . ª Ž X ⬘, e⬘. to the component of F Ž f .. See wGV, p. 413x or wFS, p. 15x. For a monoidal category A, A-Modk denotes the 2-category whose objects are left A-modules with direct summands and whose Hom-categories are categories of A-linear functors as in A-Mod. If M is an A-module, then M becomes an A-module by setting X 䉺 Ž M, e . s Ž X 䉺M, X 䉺e .. For a right A-module M and a left A-module N , the envelope of M mA N is denoted by M mA N . 2. THE BICATEGORY ASSOCIATED WITH THE DUAL PAIR Ž A, B . Let A be a finite dimensional semisimple cosemisimple Hopf algebra. Actually, semisimplicity implies cosemisimplicity in characteristic zero wLR2x. We need the result that the square of the antipode S of A is the identity, due to Larson and Radford wLR1x in characteristic zero and to Etingof and Gelaki wEGx in positive characteristic. Let B s A*, the dual Hopf algebra. Put A s A-Mod, B s B-Mod, and V s k-Mod. In this section we construct a bicategory E , in the sense of Section 1Žd., with index set J s 1, 24 such that
ž
E11 E21
E12 A s E22 V
/
ž
V . B
/
The canonical pairing between A and B is denoted by ² ᎐, ᎐ :. After Sweedler’s book wSx, the left action © and the right action £ of A on B are defined by a©bs
Ý b1² a, b2 : , b £ a s Ý ² a, b1 : b 2
523
DUALITY FOR MODULES
for a g A, b g B with ⌬Ž b . s Ýb1 m b 2 , so that ² a⬘, a © b : s ² a⬘a, b : , ² a⬘, b £ a: s ² aa⬘, b : . Then the left action
and the right action a
b s b £ S Ž a. ,
b
a s S Ž a . © b.
of A on B are defined by
We need to choose linear isomorphisms A ª B, B ª A in a special way. PROPOSITION 2.1. There exist linear isomorphisms : A ª B, : B ª A such that
Ž a⬘a . s a⬘
Ž a. ,
Ž b⬘ © a . s b⬘ Ž a . , Ž b⬘b . s b⬘
Ž aa⬘ . s Ž a . Ž a £ b⬘ . s Ž a . b⬘,
Ž b. ,
Ž a⬘ © b . s a⬘ Ž b . ,
a⬘,
Ž bb⬘ . s Ž b .
b⬘,
Ž b £ a⬘ . s Ž b . a⬘
for all a, a⬘ g A, b, b⬘ g B, and that S s 1,
S s 1.
Such a pair Ž , . is unique modulo the relation Ž , . ; Ž , y1 . for scalars / 0. Proof. Take integrals ⌳ g A, ⌫ g B. Since A, B are unimodular, left integrals and right integrals are the same and SŽ ⌳ . s ⌳, SŽ ⌫ . s ⌫ wLS, Corollary to Proposition 4 and the first corollary to Proposition 8x. Since A is involutory, we have ² aa⬘, ⌫ : s ² a⬘a, ⌫ : wLS, the second corollary to Proposition 8x. It follows that a ⌫s⌫ a. Define : A ª B by
Ž a. s a
⌫s⌫
a.
Then is an isomorphism of two-sided Hopf A-modules, that is, the first four identities in the statement hold for all a, a⬘ g A, b⬘ g B wS, p. 96x. Conversely, any map A ª B satisfying those identities comes from an integral in B in this way, hence it is unique up to scalar. Similarly, define : B ª A by
Ž b. s b
⌳s⌳
Then satisfies the next four identities.
b.
524
D. TAMBARA
Using an identity b © S Ž a. s S Ž a £ S Ž b . . and the fact that SŽ ⌫ . s ⌫, S 2 s 1, one has S Ž a . s S Ž a
⌫. s SŽ ⌫.
Sy1 Ž a . s ⌫
S Ž a. s S Ž a. .
Thus S s S. And S s S as well. Now the computations
² S Ž a. , b: s ² S Ž a.
⌳ , b: s ² ⌳ £ Ž a . , b:
s ² ⌳ , Ž a . b: s ² ⌳ , Ž a £ b .: s² ⌳ , Ž a £ b.
⌫: s ² S Ž a £ b . ⌳ , ⌫:
s ² ⑀ S Ž a £ b . ⌳ , ⌫: s ²² a, b: ⌳ , ⌫: s ² a, b :² ⌳ , ⌫ : yield S s ² ⌳, ⌫ :1. We can take ⌳, ⌫ so that ² ⌳, ⌫ : s 1. Then S s 1. Thus S s 1, S s 1. We fix such a choice of : A ª B, : B ª A throughout the paper. For X g A, Y g B we have the maps
X : X m A ª X m A xma¬
Ý a1 x m a2 ,
X : A m X ª A m X amx¬
Ý a1 m a2 x,
X , Y : X m Y ª X m Y xmy¬
Ý ai x m yi s Ý x j m bj y,
␥Y : A m Y ª Y m A amy¬
Ý yi m aai ,
where ⌬ Ž a. s
Ý a1 m a2 ,
Ž y. s
Ý yi m a i ,
Ž x. s
Ý x j m bj ,
and : Y ª Y m A, : X ª X m B are the right comodule structures coming from the left module structures.
DUALITY FOR MODULES
525
These are all bijections with inverses given by
y1 X : xma¬
Ý Sy1 Ž a1 . x m a2 , y1 X : a m x ¬ Ý a1 m S Ž a 2 . x, y1 X , Y : x m y ¬ Ý S Ž a i . x m yi s Ý x j m S Ž bj . y, y1 ␥y1 Ž a i . m yi . Y : y m a ¬ Ý aS Replacing the roles of A and B, we have the similar maps Y , Y ,  Y , X , ␥ X for X g A, Y g B. Now we define the bicategory E as follows. The index set is 1, 24 . Let E11 s A , E12 s V , E21 s V ,
E22 s B .
The composition functors 䉺 i jk : Ei j = Ejk ª Ei k for i, j, k s 1, 2 are given by X 䉺111 X ⬘ s X m X ⬘,
Y 䉺222 Y⬘ s Y m Y ⬘,
X 䉺112 V s X m V ,
Y 䉺221 V s Y m V ,
V 䉺211 X s V m X ,
V 䉺122 Y s V m Y ,
V 䉺121 V⬘ s V m A m V⬘,
V 䉺212 V⬘ s V m B m V⬘
for X, X ⬘ g A; Y, Y⬘ g B; V, V ⬘ g V . Here the module structures of X m X ⬘, Y m Y ⬘ are the usual ones. In X m V, V m X, Y m V, V m Y the module structures of X, Y are forgotten. In V m A m V⬘, V m B m V ⬘, we regard A, B as the left regular modules. The units I g E11 , I g E22 are the trivial modules k. Next we define the natural transformations of associativity
␣ i jk l : 䉺i jl ( Ž 1 E i j = 䉺jk l . ª 䉺i k l ( Ž 䉺 i jk = 1 E k l . . ␣ 1111 , ␣ 2222 , ␣ 1112 , ␣ 2111 , ␣ 2221 , ␣ 1222 are the identity. ␣ 1122 , ␣ 1121 , ␣ 1211 , ␣ 1221 , ␣ 1212 are given by
Ž X 䉺V . 䉺Y
6
5 XmVmY
Ž X, Y .
Ž X 䉺V . 䉺V⬘
6
␣ 1121
6
X 䉺 Ž V 䉺V⬘ . 5 X m V m A m V⬘
␣ 1122
6
X 䉺 Ž V 䉺Y . 5 X 䉺V 䉺Y
5 X m V m A m V⬘
Ž y1 X .
526
D. TAMBARA
Ž .
Ž V 䉺V⬘ . 䉺V⬙
6
␣ 1212
5 V m Y m A m V⬘
6
Ž␥ Y .
Ž V 䉺Y . 䉺V⬘
6
␣ 1221
5 V m A m V⬘ m X
6
V 䉺 Ž V⬘ 䉺V ⬙ . 5 V m V⬘ m B m V⬙
Ž X .
Ž V 䉺V ⬘ . 䉺 X
6
V 䉺 Ž Y 䉺V . 5 V m A m Y m V⬘
␣ 1211
6
V 䉺 Ž V⬘ 䉺 X . 5 V m A m V⬘ m X
5 V m A m V ⬘ m V⬙
for X g A; Y g B; V, V⬘, V ⬙ g V . Here Ž  X , Y ., Ž␥ X ., . . . stand for the maps induced by  X , Y , ␥ X , . . . in an obvious way. The remaining ␣ 2211 , ␣ 2212 , ␣ 2122 , ␣ 2112 , ␣ 2121 are defined by interchanging A and B, and and . Finally, the natural isomorphisms for unit
i j : 1 E i j ª Ii 䉺ii j Ž ᎐ . ,
i j : 1 E i j ª Ž ᎐ . 䉺i j j I j
are given by the maps x ¬ 1 m x, x ¬ x m 1. THEOREM 2.2. The data Ei j , 䉺i jk , Ii , ␣ i jk l , i j , i j constitute a bicategory E. Proof. The triangle identities for i j , i j are obvious. We prove that ␣ i jk l satisfy the pentagon identity X 䉺 Ž Y 䉺 Ž Z䉺W .. X 䉺 Ž Y 䉺 Ž Z䉺W .. ␣ i jkm
6 X 䉺 ŽŽ Y 䉺Z . 䉺W .
6
␣ i jlm
s Ž X 䉺Y . 䉺 Ž Z䉺W . ␣ iklm
6
Ž X 䉺 Ž Y 䉺Z .. 䉺W ␣ i jkl 䉺W
6
X 䉺 ␣ jklm
ŽŽ X 䉺Y . 䉺Z .. 䉺W
6
ŽŽ X 䉺Y . 䉺Z .. 䉺W for X g Ei j , Y g Ejk , Z g Ek l , and W g El m .
ŽPi jk l m .
DUALITY FOR MODULES
527
ŽPi jk l m . is clear for Ž ijklm. s Ž11111., Ž11112., Ž12222., and for these with 1 and 2 interchanged, because all ␣ involved are the identity maps. ŽP11122 . Let X, X ⬘ g A; V g V ; Y g B. The left-hand side of ŽP11122 . for Ž X, X ⬘, V, Y . becomes 䢇
䢇
X 䉺 Ž X ⬘ 䉺 Ž V 䉺Y .. s X m X ⬘ m V m Y X 䉺 ␣ 1122
Ž  X ⬘, Y .
6
6
X 䉺 ŽŽ X ⬘ 䉺V . 䉺Y . s X m X ⬘ m V m Y ␣ 1122
Ž X, Y .
6
6
Ž X 䉺 Ž X ⬘ 䉺V .. 䉺Y s X m X ⬘ m V m Y ␣ 1112 䉺Y
1
6
6
ŽŽ X 䉺 X ⬘. 䉺V . 䉺Y sX m X ⬘ m V m Y, while the right-hand side becomes X 䉺 Ž X ⬘ 䉺 Ž V 䉺Y .. s X m X ⬘ m V m Y ␣ 1112
1
6
6
Ž X 䉺 X ⬘. 䉺 Ž V 䉺Y . s X m X ⬘ m V m Y ␣ 1122
Ž  Xm X ⬘, Y .
6
6
ŽŽ X 䉺 X ⬘. 䉺V . 䉺Y sX m X ⬘ m V m Y. So we are reduced to showing the identity X m X⬘ m Y Ž  X ⬘, Y . 23
Ž  X, Y .13
s
 Xm X ⬘, Y
6
6
X m X⬘ m Y
X m X⬘ m Y
X m X ⬘ m Y.
Ž2.3.
6
X m X⬘ m Y Here Ž  X , Y .13 is the map operating as  X , Y on the Ž1, 3. position of the tensor product X m X ⬘ m Y. Similar notations Ž ᎐ .12 and Ž ᎐ . 23 will be used. 䢇
ŽP11222 . Similarly, ŽP11222 . is equivalent to
Ž  X , Y . 12 Ž  X , Y ⬘ . 13 s  X , YmY ⬘
Ž 2.4.
for all X g A; Y, Y⬘ g B. ŽP11121 . Let X, X ⬘ g A; V, V⬘ g V . The left-hand side of ŽP11121 . for Ž X, X ⬘, V, V ⬘. becomes 䢇
528
D. TAMBARA
X 䉺 Ž X ⬘ 䉺 Ž V 䉺V⬘.. s X m X ⬘ m V m A m V⬘ Ž y1 X⬘ .
X 䉺 ␣ 1121
6
6
X 䉺 ŽŽ X ⬘ 䉺V . 䉺V⬘. s X m X ⬘ m V m A m V⬘ ␣ 1121
Ž y1 X .
6
6
Ž X 䉺 Ž X ⬘ 䉺V .. 䉺V⬘ s X m X ⬘ m V m A m V⬘ ␣ 1112 䉺V X
1
6
6
ŽŽ X 䉺 X ⬘. 䉺V .. 䉺V ⬘ sX m X ⬘ m V m A m V ⬘, while the right-hand side becomes X 䉺 Ž X ⬘ 䉺 Ž V 䉺V⬘.. s X m X ⬘ m V m A m V ⬘ ␣ 1111
1
6
6
Ž X 䉺 X ⬘. 䉺 Ž V 䉺V⬘. s X m X ⬘ m V m A m V ⬘ ␣ 1121
Ž y1 Xm X ⬘ .
6
6
ŽŽ X 䉺 X ⬘. 䉺V .. 䉺V⬘ sX m X ⬘ m V m A m V⬘. So it is enough to show X m X⬘ m A
X m X⬘ m A s Ž X ⬘ . 23
X m X⬘ m A Xm X ⬘
6
6
Ž X .13
X m X ⬘ m A.
Ž2.5.
6
X m X⬘ m A
䢇
ŽP12111 . Similarly, ŽP12111 . is equivalent to
Xm X ⬘ s Ž X . 12 Ž X ⬘ . 13
Ž 2.6.
for all X, X ⬘ g A. ŽP11221 . Let X g A; Y g B; V, V⬘ g V . The left-hand side of ŽP11221 . for Ž X, V, Y, V⬘. becomes 䢇
529
DUALITY FOR MODULES
X 䉺 Ž V 䉺 Ž Y 䉺V⬘.. s X m V m A m Y m V ⬘ X 䉺 ␣ 1221
Ž␥ Y .
6
6
X 䉺 ŽŽ V 䉺Y . 䉺V⬘. s X m V m Y m A m V ⬘ ␣ 1121
Ž y1 X .
6
6
Ž X 䉺 Ž V 䉺Y .. 䉺V⬘ s X m V m Y m A m V ⬘ X, Y
. Ž X, Y .
6
␣ 1122 䉺V ⬘
6
ŽŽ X 䉺V . 䉺Y .. 䉺V⬘ sX m V m Y m A m V⬘, while the right-hand side becomes X 䉺 Ž V 䉺 Ž Y 䉺V⬘.. s X m V m A m Y m V ⬘ ␣ 1121
Ž y1 X .
6
6
Ž X 䉺V . 䉺 Ž Y 䉺V⬘. s X m V m A m Y m V ⬘ ␣ 1221
Ž␥ Y .
6
6
ŽŽ X 䉺V . 䉺Y .. 䉺V ⬘ sX m V m Y m A m V⬘. So it is enough to show that XmAmY
Ž y1 X .13
6
XmYmA Ž  X, Y .12
Ž y1 X .12
sXmAmY Ž␥ Y . 23
6
6
XmYmA
XmAmY 6
Ž␥ Y . 23
X m Y m A.
Ž2.7.
6
XmYmA
䢇
ŽP12211 . Similarly, ŽP12211 . is equivalent to
Ž ␥ Y . 12 Ž X . 13 Ž  Y , X . 23 s Ž X . 23 Ž ␥ Y . 12 for all X g A, Y g B.
Ž 2.8.
530
D. TAMBARA
ŽP12221 . Let Y, Y⬘ g B; V, V ⬘ g V . The left-hand side of ŽP12221 . for Ž V, Y, Y ⬘, V⬘. becomes 䢇
V 䉺 Ž Y 䉺 Ž Y ⬘ 䉺V⬘.. s V m A m Y m Y ⬘ m V ⬘ V 䉺 ␣ 2221
1
6
6
V 䉺 ŽŽ Y 䉺Y ⬘. 䉺V ⬘. s V m A m Y m Y⬘ m V ⬘ ␣ 1221
Ž ␥ YmY ⬘ .
6
6
Ž V 䉺 Ž Y 䉺Y⬘.. 䉺V⬘ s V m Y m Y⬘ m A m V ⬘ ␣ 1222 䉺V ⬘
1
6
6
ŽŽ V 䉺Y . 䉺Y⬘.. 䉺V ⬘ sV m Y m Y ⬘ m A m V⬘, while the right-hand side becomes V 䉺 Ž Y 䉺 Ž Y⬘ 䉺V⬘.. s V m A m Y m Y⬘ m V ⬘ ␣ 1221
Ž␥ Y .
6
6
Ž V 䉺Y . 䉺 Ž Y ⬘ 䉺V ⬘. s V m Y m A m Y⬘ m V ⬘ ␣ 1221
Ž␥ Y ⬘.
6
6
ŽŽ V 䉺Y . 䉺Y⬘.. 䉺V ⬘ sV m Y m Y ⬘ m A m V⬘. So it is enough to show that A m Y m Y⬘
6
Y m Y⬘ m A
sY m A m Y⬘ Ž␥ Y ⬘ . 23
Ž2.9.
6
␥ YmY ⬘
Ž␥ Y .12
6
A m Y m Y⬘
Y m Y ⬘ m A. ŽP11211 . Let X, X ⬘ g A; V, V⬘ g V . The left-hand side of ŽP11211 . for Ž X, V, V⬘, X ⬘. becomes 䢇
X 䉺 Ž V 䉺 Ž V⬘ 䉺 X ⬘.. s X m V m A m V⬘ m X ⬘ Ž X ⬘.
6
X 䉺 ␣ 1211
6
X 䉺 ŽŽ V 䉺V⬘. 䉺 X ⬘. s X m V m A m V⬘ m X ⬘ ␣ 1111
1
6
6
Ž X 䉺 Ž V 䉺V ⬘.. 䉺 X ⬘ s X m V m A m V⬘ m X ⬘ Ž y1 X .
6
␣ 1121 䉺 X ⬘
6
ŽŽ X 䉺V . 䉺V⬘.. 䉺 X ⬘ sX m V m A m V⬘ m X ⬘,
531
DUALITY FOR MODULES
while the right-hand side becomes X 䉺 Ž V 䉺 Ž V⬘ 䉺 X ⬘.. s X m V m A m V⬘ m X ⬘ ␣ 1121
Ž y1 X .
6
6
Ž X 䉺V . 䉺 Ž V⬘ 䉺 X ⬘. s X m V m A m V⬘ m X ⬘ ␣ 1211
Ž X ⬘.
6
6
ŽŽ X 䉺V . 䉺V ⬘.. 䉺 X ⬘ sX m V m A m V⬘ m X ⬘. So it is enough to show that
Ž X ⬘ . 23
X m A m X⬘ Ž y1 X .12
6
X m A m X⬘ 6
X m A m X⬘ s X m A m X⬘ 6
X m A m X⬘
Ž X ⬘ . 23
Ž2.10.
6
Ž y1 X .12
X m A m X ⬘.
ŽP11212 . Let X g A; V, V⬘, V⬙ g V . The left-hand side of ŽP11212 . for Ž X, V, V⬘, V⬙ . becomes 䢇
X 䉺 Ž V 䉺 Ž V⬘ 䉺V⬙ .. s X m V m V⬘ m B m V ⬙ X 䉺 ␣ 1212
Ž .
6
6
X 䉺 ŽŽ V 䉺V⬘. 䉺V ⬙ . s X m V m A m V ⬘ m V⬙ ␣ 1112
1
6
6
Ž X 䉺 Ž V 䉺V⬘.. 䉺V⬙ s X m V m A m V ⬘ m V⬙ Ž y1 X .
6
␣ 1121 䉺V ⬙
6
ŽŽ X 䉺V . 䉺V⬘.. 䉺V⬙ sX m V m A m V ⬘ m V⬙, while the right-hand side becomes X 䉺 Ž V 䉺 Ž V⬘ 䉺V⬙ .. s X m V m V⬘ m B m V ⬙ Ž  X , V ⬘mBmV ⬙ .
6
␣ 1122
6
Ž X 䉺V . 䉺 Ž V⬘ 䉺V ⬙ . s X m V m V⬘ m B m V ⬙ Ž .
6
␣ 1212
6
ŽŽ X 䉺V . 䉺V⬘.. 䉺V⬙ sX m V m A m V ⬘ m V⬙.
532
D. TAMBARA
So it is enough to show that
Xm
XmB  X, B
6
XmB 6
XmA s XmB 6
XmA
䢇
Xm
Ž2.11.
6
y1 X
X m A.
ŽP12122 . Similarly, ŽP12122 . is equivalent to
Ž m Y . Y s A , Y Ž m Y .
Ž 2.12.
for all Y g B. ŽP12212 . Let Y g B; V, V⬘, V⬙ g V . The left-hand side of ŽP12212 . for Ž V, Y, V⬘, V⬙ . becomes 䢇
V 䉺 Ž Y 䉺 Ž V⬘ 䉺V⬙ .. s V m Y m V ⬘ m B m V⬙ Ž y1 Y .
6
V 䉺 ␣ 2212
6
V 䉺 ŽŽ Y 䉺V⬘. 䉺V ⬙ . s V m Y m V⬘ m B m V⬙ Ž .
6
␣ 1212
6
Ž V 䉺 Ž Y 䉺V⬘.. 䉺V⬙ s V m A m Y m V⬘ m V ⬙ Ž␥ Y .
6
␣ 1221 䉺V ⬙
6
ŽŽ V 䉺Y . 䉺V ⬘. 䉺V⬙ sV m Y m A m V⬘ m V ⬙, while the right-hand side becomes V 䉺 Ž Y 䉺 Ž V⬘ 䉺V⬙ .. s V m Y m V⬘ m B m V⬙ 1
6
␣ 1222
6
Ž V 䉺Y . 䉺 Ž V⬘ 䉺V ⬙ . s V m Y m V⬘ m B m V⬙ Ž .
6
␣ 1212
6
ŽŽ V 䉺Y . 䉺V⬘.. 䉺V⬙ sV m Y m A m V⬘ m V ⬙.
533
DUALITY FOR MODULES
So it is enough to show that YmB Y
6
YmB T Ž Ym .
6
A m Y sY m B 6
YmA
Ym
Ž2.13.
6
␥Y
Y m A,
where T : Y m A ª A m Y is the map y m a ¬ a m y. ŽP12112 . Similarly, ŽP12112 . is equivalent to 䢇
T Ž X m . ␥ X s y1 X Ž m X .
Ž 2.14.
for all X g A. ŽP12121 . Let V, V⬘, V⬙, V g V . The left-hand side of ŽP12121 . for Ž V, V⬘, V⬙, V . becomes 䢇
V 䉺 Ž V⬘ 䉺 Ž V⬙ 䉺V .. s V m A m V ⬘ m V⬙ m A m V V 䉺 ␣ 2121
Ž .
6
6
V 䉺 ŽŽ V⬘ 䉺V⬙ . 䉺V . s V m A m V⬘ m B m V⬙ m V ␣ 1221
Ž ␥ V ⬘m BmV ⬙ .
6
6
Ž V 䉺 Ž V⬘ 䉺V⬙ .. 䉺V s V m V ⬘ m B m V⬙ m A m V ␣ 1212 䉺V
Ž .
6
6
ŽŽ V 䉺V ⬘. 䉺V⬙ .. 䉺V sV m A m V⬘ m V⬙ m A m V , while the right-hand side becomes V 䉺 Ž V⬘ 䉺 Ž V⬙ 䉺V .. s V m A m V⬘ m V⬙ m A m V Ž V ⬙m AmV .
6
␣ 1211
6
Ž V 䉺V⬘. 䉺 Ž V⬙ 䉺V . s V m A m V⬘ m V⬙ m A m V Ž y1 Vm AmV ⬘ .
6
␣ 1121
6
ŽŽ V 䉺V ⬘. 䉺V⬙ . 䉺V sV m A m V⬘ m V⬙ m A m V .
534
D. TAMBARA
So it is enough to show that AmA AmA A
6
AmB
6
␥B
sAmA
6
mA
Ž2.15.
y1 A
BmA
6
Am
A m A.
6
AmA By the symmetry of A and B, we need not consider the remaining ŽPi jk l m .. Thus we are reduced to showing Ž2.3. ᎐ Ž2.15.. The verifications of Ž2.3. ᎐ Ž2.10. are easy and left to the reader. Equations Ž2.11. ᎐ Ž2.15. are consequences of the identities of Proposition 2.1 as seen below. For Ž2.11., by the naturality in X, we may take X s A and it is enough to show that
y1 A Ž A m . Ž 1 m b . s Ž A m . A , B Ž 1 m b . for all b g B. Let : B ª B m A denote the right A-comodule structure corresponding to the left B-module structure on B and let Ž b . s Ýbi m a i . Let ⌬ Ž b . s ÝaX1 m aX2 . Then
Ý S Ž aX1 . m aX2 , Ž A m . A , B Ž 1 m b . s Ý ai m Ž bi . . y1 A Ž A m . Ž1 m b. s
But for any b⬘ g B, we have
ݲ b⬘, S Ž aX1 .: aX2 s ݲ S Ž b⬘. , aX1: aX2 s bX
Ž b. ,
Ý ² b⬘, ai : Ž bi . s Ž Ý ² b⬘, ai : bi . s Ž b⬘b . . Thus Ž2.11. reduces to the identity b⬘
Ž b . s Ž b⬘b .
of Proposition 2.1. Equation Ž2.12. similarly reduces to
Ž a⬘ © b . s a⬘ Ž b . .
DUALITY FOR MODULES
535
For Ž2.13., we may assume Y s B and it is enough to show that
␥B T Ž B m . Ž 1 m b . s Ž B m . B Ž 1 m b . for all b g B. Let : B ª B m A be as above and let Ž1. s Ýbi m a i , ⌬Ž b . s Ýb1 m b 2 for b g B. Then
␥ B T Ž B m . Ž 1 m b . s ␥ B Ž Ž b . m 1. s
Ý bi m Ž b . ai ,
Ž B m . B Ž 1 m b . s Ž B m . Ž Ý b1 m b 2 . s Ý b1 m Ž b 2 . . For any a⬘ g A, we have
Ý ² a⬘, bi : Ž b . ai s Ž b . a⬘, Ý ² a⬘, b1 : Ž b2 . s Ž b £ a⬘. . Thus Ž2.13. reduces to
Ž b . a⬘ s Ž b £ a⬘ . . Equation Ž2.14. similarly reduces to
Ž bb⬘ . s Ž b .
b⬘.
Finally, both sides of Ž2.15. are maps A m A ª A m A in A, where the A-module structure on the source A m A comes from the first factor. So it suffices to show that both sides coincide on elements 1 m a for all a g A. Let : B ª B m A be as above. Then
Ž m A . ␥ B Ž A m . Ž 1 m a. s Ž m A . ␥ B Ž 1 m Ž a. . s Ž m A . Ž Ž a. . , y1 y1 A A Ž 1 m a . s A Ž 1 m a . s
Ý S Ž a1 . m a2 .
So it is enough to show that
Ž m A . s Ž S m A . ⌬ . The left B-linearity b⬘
Ž b . s Ž b⬘b .
of is rephrased as the right A-colinearity
Ž m A. s T Ž S m A. ⌬ ,
536
D. TAMBARA
where T : A m A ª A m A is the map a m a⬘ ¬ a⬘ m a. Using this, we have
Ž S m A . Ž m A . s Ž S m A . T Ž S m A . ⌬ s T Ž S m S . ⌬ s ⌬ S . Thus Ž2.15. follows from the identity S s 1 of Proposition 2.1. This concludes the proof.
3. THE CORRESPONDENCE BETWEEN A-MODULES AND B-MODULES We keep the notation and the assumptions of the preceding section. Let A denote the 2-category of right A-modules with direct summands Modk-A ŽSection 1Že... We will construct a 2-equivalence between the 2-categories A and Modk-B B. Modk-A Since E is a bicategory, E12 naturally becomes an Ž E11 , E22 .-bimodule ŽSection 1Žd... That is, V becomes an Ž A, B .-bimodule. And similarly, V becomes a Ž B, A .-bimodule. The composition 䉺121: E12 = E21 ª E11 yields an Ž E11 , E11 .-linear functor E12 mE 22 E21 ª E11 , that is, an Ž A, A .linear functor P : V mB V ª A . Similarly, we obtain a Ž B, B .-linear functor Q: V mA V ª B . Since A has direct summands, by the universal property of the envelope ŽSection 1Že.. the functor P extends to an Ž A, A .-linear functor P : V mB V ª A . Similarly, Q extends to a Ž B, B .-linear functor Q: V mA V ª B . As V is an Ž A, B .-bimodule, if M is a right A-module then M mA V becomes a right B-module, and its envelope M mA V becomes a right B-module with direct summands. Similarly, if N is a right B-module, then N mB V becomes a right A-module with direct summands.
537
DUALITY FOR MODULES
For a right A-module M with direct summands, the functor P induces an A-linear functor PM> :
Ž M mA
V
. mB
V , M mA Ž V mB V
. ª M mA
A , M,
and for a right B-module N with direct summands, Q similarly induces a B-linear functor Q>N : Ž N mB V
. mA
V ª N.
THEOREM 3.1. For any right A-module M with direct summands, PM> is an equi¨ alence of A-modules. And Q>N is an equivalence of B-modules as well. In short, the 2-functors 6
ymA V
6
A Modk-A
ymB V
B Modk-B
are quasi-inverse to each other. The theorem will follow from PROPOSITION 3.2. The functor P: V mB V ª A is an equi¨ alence of Ž A, A .-bimodules. Proof. As noted in Section 1Ža., it is enough to show that P is simply an equivalence of categories. For this purpose, by the universal property of the envelope ŽSection 1Že.. it is enough to show that for any k-linear category C with direct summands the induced functor Hom Ž P , C . : Hom Ž A , C . ª Hom Ž V mB V , C . is an equivalence. Ž1. Let CMod- A denote the category of right A-module objects in C ; an object of CMod- A is a pair Ž C, . of an object C g C and a morphism : C m A ª C in C satisfying the associativity and the unital property as in usual modules. Here the tensor product C m V for any C g C and V g V is defined as in Section 1Ža.. Since A is a semisimple category with generator A and C has direct summands, any functor G: A ª C is determined by the object G Ž A. g C together with the algebra map Aop s End A A ª End GŽ A. or, equivalently, by the right action : G Ž A. m A ª GŽ A. wA, Proposition 2.7Žii.x. So we have an equivalence Hom Ž A , C . ª CMod- A G ¬ Ž G Ž A. , . .
538
D. TAMBARA
Ž2. Let D be the category defined as follows. An object of D is a pair Ž M, . of an object M of C and a family of isomorphisms Y : M m Y ª Y m M in C for all Y g B which are natural in Y and which satisfy the identities
YmY ⬘ s Ž Y m Y ⬘ . Ž Y m Y ⬘ . ,
k s 1 M
for all Y, Y ⬘ g B. Morphisms of D are obvious ones. For an object Ž M, . g D, we have a bilinear functor FM : V = V ª C
Ž V , V ⬘ . ¬ V m M m V⬘ and isomorphisms 6
V, Y , V ⬘
6
FM Ž V , Y m V⬘ . 5 V m M m Y m V⬘
Vm YmV ⬘
FM Ž V m Y , V ⬘ . 5 V m Y m M m V⬘
for V, V ⬘ g V ; Y g B. Then we obtain an object Ž FM , . of BiHom B Ž V , V ; C .. Since any bilinear functor V = V ª C is of the form FM Žup to isomorphism., we obtain an equivalence D , BiHom B Ž V , V ; C .
Ž M, . ¬ Ž FM , . . Ž3. We know the equivalence BiHom B Ž V , V ; C . , HomŽ V mB V , C . ŽSection 1Žb... Ž4. Let CB -Com denote the category of left B-comodule objects in C ; an object of this category is a pair Ž M, . of an object M of C and a morphism : M ª B m M in C satisfying the coassociativity and the counit condition. We have an isomorphism of categories CB -Com ( D
Ž M, . l Ž M, . in which and are related as MmB
B
BmM
6
Mm
6
: M and
BmMmY(BmYmM
YmM
Y m M.
6
mY
6
Y : M m Y
539
DUALITY FOR MODULES
Here : k ª B is the unit, Y : B m Y ª Y is the B-module structure on Y, and the middle isomorphism is a permutation. This is well-known and easily verified. Ž5. We have a usual isomorphism CB -Com ( CMod- A . The combination of the equivalences in Ž1. ᎐ Ž5. yields an equivalence Hom Ž A , C . ª Hom Ž V mB V , C . . Let us check that this is isomorphic to the functor HomŽ P, C . in question. Then the proof will be completed. Let ⌽ : Hom Ž A , C . ª CMod- A ( CB -Com ( D ª BiHom B Ž V , V ; C . G ¬ Ž M, . ¬ Ž M, . ¬ Ž M, . ¬ Ž FM , . be the composition of Ž1., Ž5., Ž4., and Ž2.. Recall that P: V mB V ª A arose from the B-bilinear functor
Ž 䉺121 , ␣ . : V = V ª A , where V 䉺121 V⬘ s V m A m V⬘, ␣ V, Y , V ⬘
Ž V 䉺Y . 䉺V⬘ 5 V m Y m A m V⬘,
6 6
V 䉺 ŽY 䉺 V . 5 V m A m Y m V⬘
Vm ␥ YmV ⬘
and ␥ Y is as in Section 2. So Hom Ž P , C .
Hom Ž V mB V , C . , BiHom B Ž V , V ; C .
6
⌿ : Hom Ž A , C .
is given by the composition with Ž 䉺121 , ␣ . G ¬ G( Ž 䉺121 , ␣ . . We have G Ž V 䉺121V ⬘ . s G Ž V m A m V⬘ . s V m M m V⬘, G Ž ␣ V , Y , V ⬘ . s G Ž V m ␥ Y m V⬘ . s V m ␥ M , Y m V ⬘, where ␥ M , Y is defined by MmYmA(YmMmA
Ym
YmM
6
Mm Y
6
␥M , Y : M m Y
540
D. TAMBARA
with Y : Y ª Y m A being the comodule structure and the middle isomorphism a permutation. But we see easily that ␥M , Y s Y . Hence G( Ž 䉺121 , ␣ . s Ž FM , . . Thus ⌽ s ⌿, as required. For any right A-module M with direct summands, the A-linear functor PM> :
Ž M mA
V
. mB
VªM
corresponds to a B-linear functor TM : M mA V ª Hom A Ž V , M . by the hom-tensor adjoint in Section 1Žc.. A formal consequence of Proposition 3.2 is PROPOSITION 3.3. The functor TM is an equi¨ alence of right B-modules. Proof. We have commutative diagrams
where the arrows labeled ‘‘can’’ are canonical functors. By Proposition 3.2, the lower arrows of the diagrams are equivalences. As the composite of the upper arrows of the first diagram, the functor M ª Hom B Ž V , Hom A Ž V , M . . is an equivalence of A-modules. Likewise, for any right B-module N with direct summands, we have an equivalence of B-modules N ª Hom A Ž V , Hom B Ž V , N . . .
DUALITY FOR MODULES
541
Thus the 2-functors 6
Hom A Ž V , ᎐ .
6
A Modk-A
Hom B Ž V , ᎐ .
B Modk-B
are quasi-inverse to each other. Now, by the first diagram, Hom B Ž V , TM . has a right quasi-inverse. Applying H om A Ž V , ᎐ . and using the equivalence Ž ᎐ . , Hom A Ž V , Hom B Ž V , ᎐ .., we know that TM has a right quasi-inverse. By the second diagram, TM mB V has a left quasi-inverse. Applying ᎐mA V and using the equivalence ŽŽ ᎐ . mB V . mA V , Ž ᎐ ., we know that TM has a left inverse as well. Hence TM is an equivalence. Finally, we give another description of Hom A Ž V , M .. Let M be a right A-module with action 䉺 : M = A ª M . The category of right A-comodule objects in M , denoted by MCom- A , is defined as follows. An object is a pair Ž M, M . of an object M in M and a morphism M : M ª M䉺 A in M satisfying the coassociativity and the counit condition as in usual comodules. Here A in M䉺 A is the regular A-module. The morphisms of MCom- A are obvious ones. The category MCom- A has a structure of a right B-module; for an object Ž M, M . in MCom- A and a B-module Y, an object Ž M, M . 䉺Y in MCom- A is defined as
where Y : Y ª Y m A is the comodule structure map, : A m A ª A is the multiplication map, and the middle isomorphism comes from the bilinearity of 䉺. ŽWe have the tensor product M m V g M for M g M and V g V . See the end of Section 1Ža... The associativity isomorphism Ž M, . 䉺 Ž Y m Y ⬘. ( ŽŽ M, . 䉺Y . 䉺Y ⬘ is just the identity on M m Y m Y⬘. PROPOSITION 3.4. For any right A-module M with direct summands, we ha¨ e an equi¨ alence of right B-modules Hom A Ž V , M . , MCom- A . Proof. We have an obvious equivalence M , Hom Ž V , M . M ¬ Ž FM : V ¬ M m V . .
542
D. TAMBARA
Suppose that FM has a structure of A-linearity
V , X : FM Ž V m X . ª FM Ž V . 䉺 X for V g V , X g A, so that Ž FM , . is an object of Hom A Ž V , M .. By the naturality of and the semisimplicity of A, is determined by k, A : M m A ª M䉺 A, where A is the regular left A-module. As k, A should commute with the right action of A on A, it is determined by its restriction s k , A ( Ž 1 m . : M ª M m A ª M䉺 A, where : k ª A is the unit. Explicitly, V , X is recovered from as the composite
where X : A m X ª X is the module structure map of X and X t means X is considered just a vector space. The compatibilities of with the associativity and the unit isomorphisms are translated into the coassociativity and the counit condition for . Thus Ž M, . is an object of MCom- A . Conversely, given any comodule structure : M ª M䉺 A, the morphism >V , X is an isomorphism. Indeed, the inverse is given by
where the vertical arrows are natural isomorphisms. ŽNote that X Ž S m 1.: A m X ª X t is left A-linear.. Thus we have established an equivalence MCom- A , Hom A Ž V , M .
Ž M, . ¬ ž FM , >/ .
543
DUALITY FOR MODULES
It remains to show that this equivalence is compatible with the actions of B. For any object Ž F, . in Hom A Ž V , M . and a B-module Y, recalling the definition in Section 1Žc. we have Ž F, . 䉺Y s Ž F⬘, ⬘., where F⬘ Ž V . s F Ž Y m V . and XV , X is the composite
6
F ŽŽ Y m V . m X .
YmV , X
F⬘Ž V . 䉺 X 5 Ž F Y m V . 䉺X
6
F⬘Ž V m X . 5 Ž Ž F Y m V m X ..
with the first arrow induced by the map YmAmX
1m X
YmX
6
Ym1
6
Y , X : Y m X
of Section 2. Now let Ž M, M . g MCom- A . Then it is easy to see that Ž FM , >M . 䉺Y s Ž FMmY , Ž MmY . >., where MmY : M m Y ª Ž M m Y . 䉺 A is the comodule structure defined before Proposition 3.4. Thus Ž FM , >M . 䉺Y corresponds to Ž M, M . 䉺Y under the equivalence. 4. DUALITY FOR HOPF ALGEBRA ACTIONS In this section we relate the correspondence between category modules in Section 3 with the duality for Hopf algebra actions on algebras due to Blattner and Montgomery wBMx. In the beginning we only assume that A is a finite dimensional Hopf algebra and B is the dual of A. For a left A-module algebra R with action written as A = R ª R: Ž a, r . ¬ ad r , the smash product R噛A is the algebra with underlying space R m A and multiplication
Ž r m a. Ž r ⬘ m a⬘ . s Ý r Ž a1 d r ⬘ . m a2 a⬘, where ⌬Ž a. s Ýa1 m a2 . A left R噛A-module is thought of as a vector space M with two structures, that of an R-module and an A-module, such that the R-module structure map R m M ª M is A-linear. Here are several facts whose verifications are straightforward. Ž1. If R is a left A-module algebra, then R噛A has a structure of a
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D. TAMBARA
left B-module algebra. The action d of B on R噛A is given by bd Ž r m a . s r m Ž b © a . . Ž2. If R is a left A-module algebra, then the category R噛A-Mod becomes a right module over A-Mod. The action 䉺 : R噛A-Mod = A-Mod ª R噛A-Mod is defined as follows. For an R噛A-module V and an A-module X, we set V䉺X s V m X on which R and A act by r Ž ¨ m x . s r¨ m x, aŽ ¨ m x . s
Ý a1¨ m a2 x
for r g R, a g A. With this action V m X becomes an R噛A-module. The associativity isomorphism V 䉺 Ž X m X ⬘. ª Ž V 䉺 X . 䉺 X ⬘ is the identity on V m X m X ⬘. Ž3. If R is a left A-module algebra, then the category R-Mod becomes a right module over B-Mod. Indeed, for an R-module V and a B-module Y, we set V 䉺Y s V m Y on which R acts by r Ž ¨ m y. s
Ý Ž a i d r . ¨ m yi .
Here Ž y . s Ý yi m a i and : Y ª Y m A is the A-comodule structure corresponding to the B-module structure on Y. The associativity isomorphism V 䉺 Ž Y m Y ⬘. ª Ž V 䉺Y . 䉺Y⬘ is the identity on V m Y m Y ⬘. Ž4. The action of A-Mod on R噛A-Mod in Ž2. and the one in Ž3. with R噛A regarded as a B-module algebra as in Ž1. coincide. Proof. Let V be an R噛A-module and X an A-module. In either action, V 䉺 X has the underlying space V m X. Relative to the action in Ž2., we have
Ž r m a. Ž ¨ m x . s Ý Ž r m a1 . ¨ m a2 x, while relative to the action in Ž3.,
Ž r m a. Ž ¨ m x . s Ý Ž bi d Ž r m a. . ¨ m x i s
Ý Ž r m Ž bi © a. . ¨ m x i s Ý Ž r m a1² bi , a2 : . ¨ m x i s Ý Ž r m a1 . ¨ m a2 x i ,
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DUALITY FOR MODULES
where Ž x . s Ý x i m bi , : X ª X m B is the comodule structure map. Thus the two module structures coincide. Let R be a B-module algebra. Put A s A-Mod, B s B-Mod. By Ž3., R-Mod is a right A-module. Then we have the right B-module Ž RMod. Com- A , the category of right A-comodule objects in R-Mod ŽSection 3.. PROPOSITION 4.1. We ha¨ e an isomorphism of right B-modules
Ž R-Mod. Com-A ( R噛B-Mod. Proof. Let M be a left R-module with module structure map : R m M ª M. By the definition of the action of A-Mod on R-Mod in Ž3., the R-module structure of M䉺 A s M m A is given by r Ž m m a. s
Ý Ž bi d r . m m ai
for r g R, m g M, a g A, where Ýa i m bi s AŽ a. and A : A ª A m B is the right B-comodule structure of A corresponding to the left A-module structure of A. Suppose furthermore that M has an A-comodule structure : M ª M m A. Let : B m M ª M be the corresponding B-module structure. We will write Ž b m m. s bm as usual. If Ž m. s Ým i m a i , then r Ž m. s
Ý Ž bi j d r . m i m ai j ,
where AŽ a i . s Ýa i j m bi j . So
Ž rm . s r Ž m . m Ž rm . s
Ý Ž bi j d r . m i m ai j m b Ž rm . s Ý Ž bi j d r . m i ² b, a i j :
for all b g B.
But we have ݲ b, a i j : bi j s ݲ b 2 , a i : b1 , where ⌬Ž b . s Ýb1 m b 2 . Hence the right-hand side of the last equation is rewritten as
Ý Ž ² b 2 , a i : b1 d r . m i s Ž b1 d r . Ž b 2 m . . Thus Ž rm. s r Ž m. if and only if bŽ rm. s ÝŽ b1 d r .Ž b 2 m. for all b g B. Thus is R-linear if and only if : R m M ª M is B-linear. That is, Ž M, , . is an object of Ž R-Mod. Com- A if and only if Ž M, , . is an object of R噛B-Mod. This establishes the isomorphism of categories Ž RMod. Com- A ( R噛B-Mod. It remains to take care of the actions of B. Let Y be a B-module. Let M, , , be as above. Relative to the action of B on Ž R-Mod. Com- A
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D. TAMBARA
defined in Section 3, we have Ž M, , . 䉺Y s Ž M m Y⬘, ⬘, ⬘., where ⬘ s m Y and the A-comodule structure ⬘ of M m Y is the standard structure of the tensor product of the A-comodules M and Y. Relative to the action of B on R噛B-Mod defined in Ž2., we have Ž M, , . 䉺Y s Ž M m Y, ⬘, ⬘., where ⬘ is the B-module structure of the tensor product of the B-modules M and Y. Since ⬘ corresponds to ⬘, Ž M, , . 䉺Y corresponds to Ž M, , . 䉺Y. Thus the above category isomorphism is B-linear. Assume that A is semisimple and cosemisimple. PROPOSITION 4.2. A-modules
For any B-module algebra R we ha¨ e an equi¨ alence of R-Mod , Ž R噛B . 噛A-Mod.
Proof. By Propositions 3.4 and 4.1 the 2-functor A ª Modk-B B Hom A Ž V , ᎐ . : Modk-A takes the A-module R-Mod to the B-module R噛B-Mod, and in view of Ž4. the 2-functor B ª Modk-A A Hom B Ž V , ᎐ . : Modk-B takes R噛B-Mod to Ž R噛B .噛A-Mod. We saw in the proof of Proposition 3.3 that these 2-functors are quasi-inverse to each other. Hence Ž R噛B .噛A-Mod must be equivalent to R-Mod. Tracing a series of the equivalences in Propositions 3.2, 3.3, 3.4, and 4.1, one can verify that the above equivalence is given by the functor sending an R-module M to the Ž R噛B .噛A-module M m A. Here R, B, and A act on M m A as r Ž m m a. s
Ý Ž bi d r . m m ai
b Ž m m a. s m m Ž b © a. a⬘ Ž m m a . s m m aSy1 Ž a⬘ . for a, a⬘ g A, b g B, r g R, and m g M, where Ž a. s Ýa i m bi and : A ª A m B is the comodule structure. To recover the duality isomorphism for Hopf algebra actions, take M to be the R-module R. Then the left action of Ž R噛B .噛A on R m A commutes with the right action of R given by
Ž r m a. r ⬘ s rr ⬘ m a.
DUALITY FOR MODULES
547
By the equivalence the two-sided action yields the ring isomorphism Ž R噛B .噛A ( End R Ž R m A. ( R m End A. REFERENCES wAx wBx wBMx wEGx wFSx wGVx wKx wLR1x wLR2x wLSx wNTx wSx
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