Invariants and semi-direct products for nite group actions on tensor categories

Transcription

1 J. Math. Soc. Japan Vol. 53, No. 2, 2001 Invariants and semi-direct products for nite group actions on tensor categories B Daisuke Tambara (Received Nov. 15, 1999) Abstract. Suppose a group G acts on a tensor categor C over a eld k. Then we have the tensor categor C G of G-invariant objects in C, and the semi-direct product tensor categor C GŠ. We show that if G is nite and k GŠ is semi-simple, there exists a one-to-one correspondence between categories with action of C G and categories with action of C GŠ. Introduction. If a group G acts on a ring S, we have the ring of G-invariants S G and the skew group ring S GŠ. This paper deals with analogous constructions for a tensor categor in place of a ring. Suppose that G acts on a tensor categor C over a eld k. This means that for each s A G, a tensor functor s : C! C is given and for each s; t A G, a tensor isomorphism s t G st is given in a coherent wa. The tensor categor C G consists of objects C of C equipped with isomorphisms s C G C satisfing certain coherence conditions. The tensor categor C GŠ is just the product 0 s A G C as a categor, whose objects are expressed as 0 s A G C s ; s with C s A C, and the tensor product in C GŠ is de ned b C; s n D; t ˆ C n s D; st. For a tensor categor A, ana-module means a categor with associative action of A. We assume here categories have direct sums and direct summands. Our result is that if G is nite and k GŠ is semi-simple, then C G -modules and C G Š-modules are in one-to-one correspondence. It is given b assigning to a C GŠ-module X the C G -module X G of G-invariant objects of X. We notice that the rings S G and S GŠ are not generall Morita equivalent. The are equivalent through the functor assigning to an S G Š-module X the S G -module X G onl when S is a G-Galois extension over S G. The above result is a simple consequence of the one-to-one correspondence between modules over the tensor categor of k G Š-modules and modules over the tensor categor of k GŠ -modules given in [T], where k GŠ is the dual of the 2000 Mathematics Subject Classi cation. 18D10, 18D05, 20J06. Ke Words and Phrases. tensor categor.

2 430 D. Tambara group algebra. This correspondence ma be thought of as a version of the dualit for cross products in [BM] and [NT]. As an application, we describe C G -modules whose underling categories are the categories of k n -modules for n V 0 when C is the tensor categor of k AŠ - modules with twisted associativit given b a 3-coccle of a group A. In this case C GŠ-modules have a simple description and so we know about C G -modules through our correspondence. In Section 1 some de nitions and constructions for modules over tensor categories are reviewed. In Section 2 group actions on tensor categories are considered and the de nitions of C G and C GŠ are given. In Section 3 we restate the dualit theorem of [T] in the case of a group algebra. In Section 4 we deduce the correspondence between C G -modules and C GŠ-modules. In Section 5 we show that the semi-simplicit of categories are preserved under the correspondence. In Section 6 we consider a tensor categor of k AŠ -modules with 3-coccle twist, called a group tensor categor, and describe modules over it. In Section 7 we appl our correspondence to a group tensor categor with a group action. In Section 8 the veri cation of the pentagon axiom for C GŠ is given. 1. Modules over tensor categories. We reproduce here some of de nitions about tensor categor modules from [T]. See [T] for details. We make a little use of terms in 2-categor theor, whose meanings are explained in our context. See [B] for generalities on 2- categories. We x a eld k throughout. A k-linear categor is a categor in which the hom-sets are k-vector spaces, the compositions are k-bilinear and nite direct sums exist. A k-linear functor between k-linear categories is a functor which is linear on all hom-spaces. Let Hom X; Y denote the categor of k-linear functors X! Y. A tensor categor is a k-linear monoidal categor. Notations for monoidal structures of a general tensor categor are as follows: A; B 7!A:B denotes the tensor product operation, I the unit object, a A; B; C : A:B :C! A: B:C the associativit isomorphism, l A : I:A! A the left unit isomorphism, r A : A:I! A the right unit isomorphism. For a tensor categor A, A n op stands for the tensor categor whose underling categor is the same as A and tensor product operation is opposite to A. For a tensor categor A, a left A-module is a k-categor X equipped with a bilinear functor A X! X : A; X 7!A:X and isomorphisms of associativit a A; B; X : A:B :X! A: B:X and unitalit l X : I:X! X for A; B A C, X A X satisfing the conditions of naturalit and coherence similar to the ones for a monoidal categor.

3 Finite group actions on tensor categories 431 For tensor categories A and B, an A; B -bimodule is a k-categor X equipped with bilinear functors A X! X and X B! X and isomorphisms a A; A 0 ; X : A:A 0 :X! A: A 0 :X, a X; B; B 0 : X:B :B 0! X: B:B 0, a A; X; B : A:X :B! A: X:B, l X : I:X! X, r X : X:I! X for A; A 0 A A, B; B 0 A B, X A X satisfing the conditions of naturalit and coherence. For left A-modules X and Y, ana-linear functor X! Y is a pair L; b of a k-linear functor L : X! Y and a famil b of isomorphisms b A; X : L A:X! A:L X for A A A, X A X which are natural in A, X and commute with the isomorphisms of associativit and unitalit for X and Y. For A-linear functors L; b ; L 0 ; b 0 : X! Y, an A-linear transformation L; b! L 0 ; b 0 is a natural transformation s : L! L 0 commuting with b and b 0. Thus we have the categor Hom A X; Y whose objects are A-linear functors X! Y and morphisms are A-linear transformations. For A-linear functors L; b : X! Y and M; g : Y! Z, their composite M; g L; b is de ned to be the A-linear functor M L; d : X! Z, where d A; X ˆ g A; L X M b A; X : Thus we have the composition functors m X; Y; Z : Hom A Y; Z Hom A X; Y!Hom A X; Z ; which are (strictl) associative. Also we have the identit A-linear functors Id X in Hom A X; X, which are (strictl) unital for composition. The collection of the categories Hom A X; Y for all A-modules X; Y together with the compositions m X; Y; Z and the identities Id X is referred as the 2-categor of A-modules and denoted b A-Mod. A-modules, A-linear functors, A-linear transformations are also called 0-cells, 1-cells, 2-cells of the 2-categor A-Mod, respectivel. The composition of 1-cells and of 2-cells given b the composition functor m X; Y; Z is called the horizontal composition, while the composition of 2-cells inside the hom-categor Hom A X; Y is called the vertical composition. An A-linear functor L : X! Y is called an equivalence of A-modules if there exist an A-linear functor L 0 : Y! X and invertible A-linear transformations L L 0! Id X 0 and L 0 L! Id X. It can be shown that this amounts to requiring L to be plainl an equivalence of categories. Let M be a B; A -bimodule. If Y is a left B-module, the categor Hom B M; Y becomes a left A-module. The action is de ned b for A A A, M A M, L A Hom B M; Y. A:L M ˆL M:A

4 432 D. Tambara Moreover we have a functor F Y; Y 0 : Hom B Y; Y 0!Hom A Hom B M; Y ; Hom B M; Y 0 F 7! L 7! F L for B-modules Y; Y 0. The functors F Y; Y 0 preserve horizontal compositions and unit 1-cells. The 2-functor Hom B M; : B-Mod! A-Mod consists of the assignment B-module Y 7! A-module Hom B M; Y and the collection of the functors F Y; Y 0 for all B-modules Y; Y 0. For a right A-module X and a left A-module Y, the tensor product categor X n A Y is de ned. Its objects are nite direct sums of smbols X; Y Š for X A X, Y A Y, and morphisms are sums of compositions of smbols for f : X! X 0 and g : Y! Y 0, and f ; gš : X; Y Š! X 0 ; Y 0 Š a X; A; Y : X:A; Y Š! X; A:Y Š for X A X, A A A, Y A Y and the formal inverse of a X; A; Y. These generating morphisms are subject to the relations of functorialit, naturalit and coherence. The precise de nition is in [T], but will not be needed in the sequel. Let M be a B; A -bimodule. If X is a left A-module, M n A X becomes a left B-module. The action is de ned b B: M; X Šˆ B:M; X Š for B A B, M A M, X A X. Moreover we have a functor C X; X 0 : Hom A X; X 0!Hom B M n A X; M n A X 0 G 7! M; X Š7! M; G X Š for A-modules X; X 0. The functors C X; X 0 preserve horizontal compositions and unit 1-cells. The 2-functor M n A : A-Mod! B-Mod consists of the assignment A-module X 7! B-module M n A X and the collection of the functors C X; X 0 for all A-modules X; X 0. We have also an A-linear functor X! Hom B M; M n A X X 7! M 7! M; X Š

5 Finite group actions on tensor categories 433 for an A-module X, and a B-linear functor M n A Hom B M; Y!Y M; LŠ 7!L M for a B-module Y. These are natural in X and Y, respectivel. Furthermore we have an equivalence Hom B M n A X; Y!Hom A X; Hom B M; Y F 7! X 7! M 7! F M; X Š : The categor of nite dimensional k-vector spaces is denoted b V. This is a tensor categor with the usual tensor product. We regard the natural isomorphisms X n Y n Z G X n Y n Z and k n X G X G X n k for vector spaces X; Y; Z as the identities. Ever k-categor X ma be viewed as a V-module. The action of the n- dimensional space k n on X is given b the functor X 7! X n. We use the smbol n rather than `dot' for this action. Thus k n n X ˆ X n for X A X. A k-categor X is said to have direct summands if ever idempotent endomorphism e : X! X in X is the projection to a direct summand of X. Given a k-categor X we can embed X to a k-categor X having direct summands such that ever object of X is a direct summand of an object of X. For an k- categor Y with direct summands, the induced functor Hom X; Y!Hom X; Y is an equivalence. For a construction of X see [GV, p. 413]. If X is an A-module, then X becomes naturall an A-module. For a right A-module X and a left A-module Y, X n A Y is written as X n A Y. A-Modk denotes the 2-categor consisting of the categories Hom A X; Y for left A- modules X, Y with direct summands. The 2-functors A-Mod M n A ƒƒƒƒƒ ƒƒƒƒƒ ƒ! ƒ B-Mod Hom B M; de ned above ield the 2-functors M n A A-Modk ƒƒƒƒƒ ƒƒƒƒƒ ƒ! ƒ B-Modk: Hom B M; 2. Group actions on tensor categories. An action of a group G on a k-categor X consists of data

6 434 D. Tambara ± functors s : X! X for all s A G ± isomorphisms f s; t : st! s t for all s; t A G ± an isomorphism n : 1! Id X which make the following diagrams commutative for all s; t; r A G and X A X. str X f s; tr X f st; r X ƒƒƒƒƒƒ! st r X f s; t r X 1 s tr X ƒƒƒƒƒƒ! s f t; r X s t r X 1 X f 1; 1 X ƒƒƒ! 1 1 X 2 1 n X 1 X f 1; 1 X ƒƒƒ! 1 1 X 3 n 1X Here commutativit of the last two diagrams means that the opposite arrows are inverse to each other. Let X, Y be categories with G-action. A G-linear functor X! Y consists of ± a k-linear functor L : X! Y ± isomorphisms h s : L s! s L for all s A G making the following diagram commutative for all s; t A G and X A X. L st X L f s; t X h st X ƒƒƒ! st L X f s; t L X 4 L s t X ƒƒƒ! s L t X ƒƒƒ! h s tx s h t X s t L X Let X be a categor with G-action. The categor of G-invariants in X, denoted b X G,isak-categor de ned as follows. An object of X G is a pair X; f, where X is an object of X and f is a famil of isomorphisms f s : s X! X for all s A G making the following diagram commutative for all s; t A G. st X f s; t X ƒƒƒ! f st X x f s 5 s t X ƒƒƒ! s f t s X

7 Finite group actions on tensor categories 435 A morphism X; f! X 0 ; f 0 in X G is a morphism u : X! X 0 in X such that f 0 s s u ˆ u f s for all s A G. Example 2.1. Let G act on the categor V of vector spaces triviall. This means that all s, f s; t, n are the identities. Then V G is the categor of k G Š-modules. Let C be a tensor categor with tensor product A; B 7!A:B, unit object I, associativit isomorphisms a A; B; C : A:B :C! A: B:C, and unit isomorphisms l A : I:A! A, r A : A:I! A. An action of G on the tensor categor C means an action of G on the k-categor C preserving the tensor structure. Namel it consists of data ± tensor functors s : C! C for all s A G ± isomorphisms f s; t : st! s t of tensor functors for all s; t A G ± an isomorphism n : 1! Id C of tensor functors making the diagrams (1), (2), (3) commutative with obvious change of letters. We also use the word G-tensor categor for tensor categor with G-action. B the de nition of a tensor functor, the above s consists of ± a functor s : C! C ± natural isomorphisms c s A; B : s A:s B! s A:B for all A; B A C ± an isomorphism i s : I! s I making the following diagrams commutative for all A; B; C A C. s A:s B :s C c s A; B :s C s A:B :s C c s A:B; C a sa; sb; sc ƒƒƒƒƒƒ! s A: s B:s C s A:c s B; C s A:s B:C c s A; B:C 6 s A:B :C ƒƒƒƒƒƒ! s a A; B; C s A: B:C i s :i s I:I s I:s I ƒƒƒ! l I ƒƒƒ! s I:I ƒƒƒ! c s I; I s l I I i s s I 7

8 436 D. Tambara The requirement that f s; t is a morphism of tensor functors means that the following diagram is commutative for all A; B A C. st A: st B f s; t A :f s; t B ƒƒƒƒƒƒƒ! s t A:s t B c s ta; tb c st A; B s t A:t B st A:B ƒƒƒƒƒƒƒ! f s; t A:B s c t A; B s t A:B 8 In the presence of the commutativit of (3) and (8), n : 1! Id C is automaticall a morphism of tensor functors. Thus we could sa that a G-action on the tensor categor C consists of the data s, f s; t, n, c s, i s making the diagrams of (1), (2), (3), (6), (7), (8) commutative. Let C be a G-tensor categor. The categor C G becomes a tensor categor as follows. The tensor product is de ned b A; f : B; g ˆ A:B; h 9 where h s ˆf s :g s c s 1 A; B : 10 The unit object is I; i 1. The associativit and unit isomorphisms are inherited from C. We now construct another tensor categor C G Š from a G-tensor categor C. We set C GŠ ˆ0 s A G C as categories. So an object of C GŠ is expressed as 0 s A G A s ; s with A s A C, and a morphism from 0 s A G A s ; s to 0 s A G B s ; s is expressed as 0 s A G f s ; s with f s : A s! B s a morphism in C. The tensor product operation in C G Š is de ned b A; s : B; t ˆ A:s B; st f ; s : g; t ˆ f :s g; st for objects; for morphisms: The unit object is I; 1. The associativit is given b A; s : B; t : C; r ˆ A:s B; st : C; r ˆ A:s B : st C; str a A; s ; B; t ; C; r a A; s; B; t; C ; str A; s : B; t : C; r ˆ A; s : B:t C; tr ˆ A:s B:t C ; str 11

9 Finite group actions on tensor categories 437 where a A; s; B; t; C is the composite A:s B : st C A:s B :f s; t C A:s B :s t C a A; sb; stc A: s B:s t C A:c s B; tc A:s B:t C : 12 The left unitalit l A; s : I; 1 : A; s ˆ I:1 A; s! A; s is given b I:1 A ƒƒ! I:n A I:A ƒƒ! l A A: 13 The right unitalit r A; s : A; s : I; 1 ˆ A:s I; s! A; s is given b A:i s 1 A:s I ƒƒƒƒ! A:I ƒƒƒƒ! r A A: 14 These data satisf the axiom of a tensor categor. In [M] Magid introduced the double cross product of tensor categories, of which C G Š is regarded as a special case. [ M] does not contain the proof of the axiom of a tensor categor for the double cross product. We will verif the pentagon axiom for C G Š in Section 8. Example 2.2. With respect to the trivial action of G on V, we have the tensor categor V GŠ. Objects are of the form 0 s A G V s ; s with V s A V. The tensor product is given b V; s : W; t ˆ V n W; st : Thus V G Š is the categor of G-graded vector spaces, or the categor of k GŠ -modules when G is nite.

10 438 D. Tambara Example 2.3. Suppose G acts on a group A. Then the action of G on the tensor categor V AŠ is induced. We have obviousl V AŠ G Š ˆ V A c G Š. Let C be a G-tensor categor. We ma view a C GŠ-module as a categor having actions of C and G in a compatible wa. To be precise, a C GŠ-module structure on a k-categor X amounts to the data ± a C-module structure on X : :; a; l ± a G-action on X : s ; f s; t ; n ± natural isomorphisms c s C; X : s C:s X! s C:X for all C A C, X A X, s A G making the following diagrams commutative: (6), (8) with appropriate change of letters, and in place of (7), the diagram I:s X l sx i s :s X ƒƒƒƒ! s I:s X c s I; X 15 s X ƒƒƒƒ s l X s I:X : Indeed, given these data, we de ne the action of C G Š on X b C; s :X ˆ C:s X and the associativit and the unitalit a C; s ; D; t ; X : C; s : D; t :X! C; s : D; t :X l X : I; 1 :X! X b the formulas similar to (11), (12), (13). Example 2.4. C itself is a C GŠ-module: C; s :C 0 ˆ C:s C 0. Example 2.5. k; s :X ˆ s X. A V G Š-module is nothing but a k-categor with G-action: Let X; Y be C G Š-modules. A structure of a C G Š-linear functor on a k-linear functor L : X! Y amounts to the data ± isomorphisms b C; X : L C:X!C:L X for C A C, X A X ± isomorphisms h s X : L s X!s L X for s A G, X A X satisfing the following conditions. ± L; b is a C-linear functor. ± L; h is a G-linear functor.

11 Finite group actions on tensor categories 439 ± the diagram L s C:s X b sc; sx s C:L s X s C:h s X s C:s L X L c s C; X ƒƒƒƒƒƒ! ƒƒƒƒƒƒ! c s C; L X L s C:X h s C:X s L C:X s b C; X s C:L X commutes for all C A C, s A G, X A X. If X is a C GŠ-module, X G becomes a C G -module b a similar action to (9), (10). 3. V G -modules and V GŠ-modules. Hereafter we assume G is a nite group and the characteristic of k does not divide jgj. We denote the categor of nite dimensional k GŠ-modules b V G, and the categor of nite dimensional k GŠ -modules b V GŠ (see Example 2.1, 2.2). In this section we review the one-to-one correspondence between V G -modules and V GŠ-modules from [T]. We make V into a V GŠ; V G -bimodule. The action of objects are given b Y:V ˆ Y n V; V:X ˆ V n X for X A V G, Y A V GŠ, V A V. The associativit of actions are the identit maps, while is the map Y:Y 0 :V! Y: Y 0 :V ; V:X :X 0! V: X:X 0 Y:V :X! Y: V:X ; t n v n x 7! ; t n v n t 1 x; where x A X, v A V, t A G, and ; t is an element in the t-component of the G- graded space Y. The dualit theorem of [T] in the case of a group algebra is as follows.

12 440 D. Tambara Theorem 3.1. The 2-functors V G -Modk V n V G ƒƒƒƒƒƒ ƒƒƒƒƒƒƒ ƒ! V G Š-Modk Hom V G Š V; are quasi-inverse to each other through the adjunction. Namel, for an V G - module X and V G Š-module Y with direct summands, the canonical functors X! Hom V G Š V; V n V G X ; V n V G Hom V G Š V; Y!Y are equivalences of V G -modules and of V GŠ-modules, respectivel. In this situation we also sa the pair V n V G ; Hom V G Š V; is a 2- equivalence. The 2-equivalence implies the following: (i) For ever V G -module X with direct summands there exist a V G Š-module Y with direct summands and an equivalence X! Hom V G Š V; Y of V G -modules. (ii) For V GŠ-modules Y; Y 0 with direct summands, the functor Hom V G Š Y; Y 0!Hom V G Hom V G Š V; Y ; Hom V G Š V; Y 0 is an equivalence. Recall that V G Š-modules are just k-categories with G-action (Example 2.5). So we ma use the notations s, f s; t of G-action for V GŠ-modules. The left V GŠ-module structure on the bimodule V is trivial: s V ˆ V. The associativit Y:V :X! Y: V:X for Y ˆ k; s takes the form Proposition 3.2. V G -modules b s : s V n X! s V n X v n x 7! v n s 1 x: For an V G Š-module X, we have an equivalence of X G F Hom V G Š V; X : Proof. This is [T, Proposition 3.4] adapted to our present situation. But we give here a direct description of the equivalence. Recall that an object of Hom V G Š V; X, ag-linear functor V! X, is represented as a pair F; h of a k-linear functor V! X and a famil h of isomorphisms for s A G, V A V. Let h s V : F s V!s F V F : X G! Hom V G Š V; X

13 Finite group actions on tensor categories 441 be the functor sending X; f to F; h, where F is given b F V ˆV n X (here n denotes the canonical action of V on X, see Section 1) and h is de ned b 1n f s 1 h s V : F s V ˆF V ˆV n X ƒƒƒƒƒ! V n s X ˆ s V n X ˆs F V : It is eas to see that F is an equivalence. Indeed, a quasi-inverse to F is the functor sending F; h to X; f with X ˆ F k and f s ˆh s 1 k. The proof will be completed once we show that F has a structure of a V G - linear functor. Let M A V G, X; f A X G. We wish to de ne an isomorphism b M; X; f : F M: X; f! M:F X; f in Hom V G Š V; X. Let F X; f ˆ F; h as above. As the left action of V G on Hom V G Š V; X comes from the right action on V, we have M: F; h ˆ H; x, where H V ˆF V:M ˆV n M n X; and x s V : H s V!s H V is the composite F b s F s V n M ƒƒƒƒ! h s VnM F s V n M ƒƒƒƒ! s F V n M ; where b s was given preceding to Proposition 3.2. This is equal to 1n s 1 M V n M n X ƒƒƒƒƒ n1 1n1n f s 1 ƒ! V n M n X ƒƒƒƒƒ ƒ! V n M n s X; that is, 1n s 1 M V n M n X ƒƒƒƒƒƒƒƒ! n f s 1 V n M n s X; where s M : M! M is the action of s on M. On the other hand, b the de nition of the action of V G on X G we have M: X; f ˆ MnX; h, where s M n f s h s : s M n X ˆM n s X ƒƒƒƒƒ! M n X: So we have F M n X; h ˆ H 0 ; x 0, where H 0 V ˆV n M n X and x 0 s V : H 0 s V!s H 0 V

14 442 is the map D. Tambara 1n h s 1 V n M n X ƒƒƒƒƒ! V n s M n X : Hence H ˆ H 0 and x ˆ x 0. Thus the identit H! H 0 gives the required isomorphism b M; X; f. It is obvious that b M; X; f is compatible with associativit of actions. Thus F; b : X G! Hom V G Š V; X is a V G -linear functor. r 4. C G -modules and C GŠ-modules. Let C be a tensor categor with G-action. Theorem 4.1. The 2-functors are quasi-inverse to each other. C G -Modk C n C G ƒƒ ƒƒ ƒ! ƒ C GŠ-Modk G Here if X is a C GŠ-module, then X G becomes a C G -module as noted in Section 3. Also in the tensor product C n C G, C is viewed as a C GŠ; C G - bimodule in which the left action of C GŠ on C is the standard one (Example 2.4), the right action of C G on C comes from the forgetful functor C G! C, and the associativit X; s :Y : Z; f! X; s : Y: Z; f for X; s A C GŠ, Y A C, Z; f A C G is given b a X; sy; Z X:s Y :Z ƒƒƒƒƒƒƒƒ! X:c s Y; Z ƒƒƒƒƒƒƒƒ! X: s Y: f s 1 X: s Y:Z ƒƒƒƒƒƒƒƒ! X: s Y:s Z X:s Y:Z : That C is reall a bimodule is veri ed in the course of the proof of the theorem. Lemma 4.2. Let B be a tensor categor and M a left B-module. Set A ˆ End B M nop so that M becomes a B; A -bimodule. Suppose that the 2-functors M n A A-Modk ƒƒƒƒƒ ƒƒƒƒƒ ƒ! ƒ B-Modk Hom B M; are quasi-inverse to each other through the adjunction. Let j : B! E be a tensor functor. Set N ˆ E n B M, D ˆ End E N nop so that N becomes an E; D -

15 Finite group actions on tensor categories 443 bimodule and a tensor functor i : A! D is induced. Then the 2-functors N n D D-Modk ƒƒƒƒƒƒ! E-Modk Hom E N; are quasi-inverse to each other through the adjunction. Also the diagrams N n D D-Modk ƒƒƒƒƒ ƒ! E-Modk Hom E N; D-Modk ƒƒƒƒƒ ƒ E-Modk A-Modk ƒƒƒƒƒ ƒ! M n A B-Modk; A-Modk ƒƒƒƒƒ ƒ Hom B M; B-Modk are commutative up to natural isomorphisms, where the vertical arrows are the restrictions through i and j. Proof. The commutativit of the right diagram follows readil from the hom-tensor adjoint. For the left one, we have rstl D ˆ Hom E E n B M; E n B M F Hom B M; E n B M : Since M n A is quasi-inverse to Hom B M;, we have an equivalence M n A D F E n B M 16 of B-modules. Moreover, this is an equivalence of B; D -bimodules. Indeed, this factors as the composite M n A D F M n A Hom B M; E n B M F E n B M: 17 For an B-module Y, the canonical functor M n A Hom B M; Y!Y is B; End B Y nop -linear. In particular, the second equivalence of (17) is B; D -linear. Next, the equivalence D F Hom B M; E n B M is A; D -linear as seen from the diagram A Hom B M; M i Hom B M; j n M D ƒƒƒ! Hom B M; E n B M : Hence the rst equivalence of (17) is B; D -linear. well. So (16) is B; D -linear as

16 444 D. Tambara For an D-module X we now have M n A X F M n A D n D X F E n B M n D X ˆ N n D X as B-modules. This proves the commutativit of the left diagram. We next show that for an D-module X the canonical functor X! Hom E N; N n D X is an equivalence of D-modules. For this it is enough to show that this is just an equivalence of categories. B the commutativit we have shown, the functor restricted to A-Mod is isomorphic to the functor X! Hom B M; M n A X : This is an equivalence of A-modules b assumption. Similarl we have N n D Hom E N; G Id. Proof of Theorem 4.1. tensor functor V G Š!C GŠ. The trivial tensor functor V! C induces the We have an equivalence C GŠ n V G Š V ˆ C n V GŠ n V G Š V F C; in which an object X A C corresponds to the object X; 1 ; kš A C GŠ n V G Š V. This is an equivalence of C G Š-modules, because X; s : X 0 ; 1 ; kš ˆ X; s : X 0 ; 1 ; kš ˆ X:s X 0 ; s ; kš ˆ X:s X 0 ; 1 : I; s ; kš G X:s X 0 ; 1 ; k; s :kš ˆ X:s X 0 ; 1 ; kš: Let D ˆ End V G Š C nop. Appling the lemma to the 2-equivalence of Theorem 3.1 with A ˆ V G, B ˆ V GŠ, M ˆ V and E ˆ C GŠ, we have a 2-equivalence D-Modk C n D ƒƒƒƒƒƒƒ! C G Š-Modk: 18 Hom C G Š C; For an C GŠ-module X, b the equivalence C GŠ n V G Š V F C, the equivalence X G! Hom V G Š V; X of Proposition 3.2 ields an equivalence F X : X G! Hom C G Š C; X : We express an object of Hom C G Š C; X as a pair F; h of a C-linear functor F : C! X and a famil h of isomorphisms h s : F s C!s F C. Then F X maps an object X; f A X G to the object F; h A Hom C G Š C; X de ned as follows. F : C! X is given b F C ˆC:X r

17 Finite group actions on tensor categories 445 with the obvious C-linear structure, and h is given b : f s 1 c s h s C : F s C ˆs C:X ƒƒƒƒ! s C:s X ƒƒƒƒ! s C:X ˆs F C : In particular, taking X ˆ C, we have an equivalence F C : C G! Hom C G Š C; C ˆD: Recall that if X is a C-module, then X G becomes a C G -module. And D ˆ End C G Š C nop acts on Hom C G Š C; X through C. We claim that F X and F C are compatible with respect to these actions. In particular F C : C G! D is an equivalence of tensor categories. The theorem then follows from (18). To prove the claim, let C; g A C G, X; f A X G. Let F X; f ˆ F; h as above and F C; g ˆ D; d, that is, d s C 0 : D s C 0 ˆs C 0 :g s 1 :C ƒƒƒƒ! D C 0 ˆC 0 :C; s C 0 c s :s C ƒƒƒƒ! s C 0 :C ˆs D C 0 : Then, relative to the action of End C G Š C on Hom C G Š C; X we have D; d : F; h ˆ F; h D; d ˆ H; x, where H ˆ F D, so and is the composite H C 0 ˆF D C 0 ˆ C 0 :C :X; x s C 0 : H s C 0!s H C 0 FD s C 0 F d s C 0 ƒƒƒƒ! F s D C 0 h s D C 0 ƒƒƒƒ! s FD C 0 ; which is expanded as s C 0 :C :X s C 0 :s C :X c s : :g s 1 : s C 0 :C :X : f s 1 s C 0 :C :s X c s s C 0 :C :X :

18 446 D. Tambara This is equal to the left vertical composite of the diagram s C 0 :C :X G s C 0 : C:X :g s 1 : f s 1 : g s 1 : f s 1 s C 0 :s C :s X G s C 0 : s C:s X c s : s C 0 :C :s X c s :c s s C 0 :s C:X c s s C 0 :C :X G s C 0 : C:X which commutes b the naturalit of a and the commutativit in (6). other hand, relative to the action of C G on X G we have 19 On the C; g : X; f ˆ C:X; h with c s 1 h s : s C:X ƒƒƒƒ! g s : f s s C:s X ƒƒƒƒ! C:X: Then F C:X; h ˆ H 0 ; x 0, where and is the composite s C 0 :h s 1 : C:X ƒƒƒƒ! H 0 C 0 ˆC 0 : C:X x 0 s C 0 : H 0 s C 0!s H 0 C 0 s C 0 c s :s C:X ƒƒƒƒ! s C 0 : C:X : Also we have an iso- This is equal to the right vertical composite of (19). morphism p : H! H 0 given b the associativit The diagram (19) now becomes p C 0 : H C 0 ˆ C 0 :C :X G C 0 : C:X ˆH 0 C 0 : H s C 0 x s C 0 s H C 0 p sc 0 ƒƒƒ! ƒƒƒ! s p C 0 H 0 s C 0 x 0 s C 0 s H 0 C 0 :

19 Finite group actions on tensor categories 447 Thus p gives the isomorphism H; x G H 0 ; x 0 ; that is, p : F C; g :F X; F C; g : X; f in Hom C G Š C; X. Finall we have to check that p is compatible with the associativit of the actions. But this readil follows from the pentagon axiom for the action of C on X. r 5. Semi-simple modules. A k-linear categor is said to be nite semi-simple if it is equivalent to the categor of nite dimensional modules over a nite dimensional semi-simple algebra. We aim to show Proposition 5.1. Suppose a C G -module X corresponds to a C GŠ-module Y in the 2-equivalence of Theorem 4.1. Then X is nite semi-simple if and onl if Y is nite semi-simple. Clearl this will follow from Proposition 5.2. Let X be a categor with G-action. Then X is nite semisimple if and onl if X G is nite semi-simple. An object M of a categor X is called an additive generator if ever object X of X is a direct summand of M n for some n > 0. For a k-linear categor X with direct summands, X is nite semi-simple if and onl if X has an additive generator M such that End M is a nite dimensional semi-simple algebra. Let X be a categor with G-action. Let U : X G! X be the forgetful functor X; f 7!X. A left and right adjoint functor T of U is de ned b T X ˆ 0 t X; g t A G! ; where g s for s A G is given b the commutative diagram s 0 t X ƒƒƒ! g s 0 t X t t s p t p st s t X ƒƒƒ! f s; t 1 X st X

20 448 D. Tambara with p t, p st the projections. We have triviall U T ˆ 0 t : t A G For X; f A X G, let h : 0 t A G t X! X be the sum of f t : t X! X. diagram s 0 t X ƒƒƒ! s h s X t g s 0 t t X ƒƒƒ! h f s X The is commutative, because it is composed of the commutative diagrams f s; t 1 X s t X st X s f t ƒƒƒ! ƒƒƒ! f st s X f s X for t A G. So h de nes a morphism TU X; f! X; f. Similarl, let l : X! 0 t A G t X be the sum of f t 1. Then l de nes a morphism X; f!tu X; f in X G. The composite Id! T U! Id equals jgj1. As we are assuming jgj is invertible in k, Id! T U is a split injection. The following is well-known and the proof is omitted. Lemma 5.3. Let A be a nite dimensional algebra with G-action. Then the skew group algebra A G Š is semi-simple if and onl if A is semi-simple. Proof of Proposition 5.2. End X b If X; f A X G, then G acts on the algebra s:u ˆ f s s u f s 1 for u A End X. The isomorphisms End T X G Hom X; UT X G Hom X; 0 t X G End X GŠ give the algebra isomorphism of End T X to the skew group algebra End X GŠ. B Lemma 5.3, End T X is semi-simple if and onl if End X is semi-simple. Hence for an object Y A X G, End TU Y is semi-simple if and onl if End U Y is semi-simple.

21 Finite group actions on tensor categories 449 Suppose rst that X is nite semi-simple. For an Y A X G, End U Y is semi-simple. B the above observation, End TU Y is semi-simple. Since Y is a direct summand of TU Y, it follows that End Y is semi-simple. Take an additive generator M A X. Then U Y is a summand of M n for some n. Then TU Y is a summand of T M n, and so is Y. Thus T M is an additive generator for X G. This proves that X G is nite semi-simple. Suppose next that X G is nite semi-simple. For an X A X, End TUT X is semi-simple. Hence b the above observation End UT X is semi-simple. Since X is a summand of UT X, End X is semi-simple. Let N be an additive generator in X G. Then T X is a summand of N n for some n. Then UT X is a summand of U N n, and so is X. Thus U N is an additive generator for X. This proves that X is nite semi-simple. r 6. Modules over group tensor categories. In this section we describe modules over a 3-coccle deformation of V G Š. Most of statements here are simple translations of de nitions. For s A G we write the object k; s of V GŠ simpl as s. Let w : G 3! k be a 3-coccle. We have the tensor categor V G; wš whose underling k- categor, tensor product and unit object are the same as those of V G Š, and whose associativit and unit isomorphisms are given b a s; t; r ˆ w s; t; r 1 str l s ˆ w 1; 1; s 1 1 s r s ˆ w s; 1; 1 1 s for s; t; r A G. We call V G; wš the group tensor categor of the pair G; w. Analogousl to the identi cation of a V G Š-module with a categor with G- action, a V G; wš-module is thought of as a k-categor equipped with s, f s; t, n satisfing the commutativit of the diagrams w s; t; r 1 s tr X ƒƒƒƒƒ f s; tr X (instead of (1)), (2) and (3). s tr X ƒƒƒƒƒ! s f t; r X st r X f st; r X st r X f s; t r X s t r X 20

22 450 D. Tambara A k-categor that is equivalent to a nite direct sum of V is called a 2- vector space. All V G; wš-modules that are 2-vector spaces as categories can be obtained as follows. Let X be a nite G-set and v : G G X! k a map satisfing w s; t; r ˆv st; r; x v s; t; rx v t; r; x v s; tr; x for s; t; r A G, x A X. equations read as If v is viewed as a map G G! Map X; k, the i w ˆqv 1 in Map G 3 ; Map X; k, where q is the coboundar operator for the group G and i is the map induced b the embedding i : k! Map X; k. Let V X Š denote 0 x A X V, the categor of X-graded vector spaces. We ma regard an element x A X as a simple object of V X Š. The action of V G; wš on V X Š is then de ned b s x ˆ sx f s; t x ˆ v s; t; x 1 stx n x ˆ 1 v 1; 1; x 1 x for s; t A G, x A X. We denote b V X; vš the V G; wš-module obtained in this wa. Given two pairs X; v, X 0 ; v 0 as above, the V G; wš-modules V X; vš and V X 0 ; v 0 Š are equivalent if and onl if there exists an isomorphism f : X! X 0 of G-sets such that f v 0 and v are cohomologue in the group Map G 2 ; Map X; k. Thus the equivalence class of a V G; wš-module which is a 2-vector space bijectivel corresponds to the isomorphism class of a pair X; vš of a nite G-set X and an element vš in the quotient set fv A Map G 2 ; Map X; k j qv ˆ i w 1 g fqt j t A Map G; Map X; k : g Here the group in the denominator acts on the set in the numerator b translation. Note that the quotient is either an empt set or a regular H 2 G; Map X; k -set. Let w ˆ 1. Then V G; wš-modules are just k-categories with G-action. So we know that the equivalence class of a 2-vector space X with G-action bijectivel corresponds to the isomorphism class of a pair X; vš of a nite G-set X and a cohomolog class vš in H 2 G; Map X; k.

23 The categor V X; vš G can be described as follows. An object of V X; vš G is a pair V; f, where V is a famil of vector spaces V x for x A X and f is a famil of linear maps f s; x : V x! V sx for s A G, x A X satisfing f st; x ˆf s; tx f t; x v s; t; x for all s; t A G, x A X. Suppose X is a transitive G-set and let K be the stabilizer of an element x 0 A X. The map v 0 : K 2! k de ned b v 0 s; t ˆv s; t; x 0 is a 2-coccle on K. And we have Shapiro's isomorphism H 2 G; Map X; k G H 2 K; k in which vš corresponds to v 0 Š. The pair V; f above is determined b the pair V x0 ; f 0, where f 0 : K! End V x0 is de ned b f 0 s ˆf s; x 0. Such a pair V x0 ; f 0 is just a module over the skew group algebra k K; v 0 Š relative to the 2-coccle v 0. Thus V X; vš G is equivalent to the categor of k K; v 0 Š- modules. Also V G is the categor of k GŠ-modules. The action of V G on V X; vš G is given b the tensor product through the restriction to the subgroup K. 7. Group actions on group tensor categories. In this section we appl the 2-equivalence of Theorem 4.1 to a group tensor categor with G-action. An G-action on a group tensor categor is obtained in the following wa. Let A be a group with G-action denoted b s; a 7! s a. Let be maps satisfing Finite group actions on tensor categories 451 t : A A A! k u : G A A! k v : G G A! k 1 ˆ t b; c; d t a; bc; d t a; b; c t ab; c; d t a; b; cd t a; b; c u s; b; c u s; a; bc t s a; s b; s ˆ c u s; ab; c u s; a; b u s; t a; t b u t; a; b u st; a; b ˆ v st; r; a v s; t; r a v t; r; a v s; tr; a ˆ 1 v s; t; ab v s; t; a v s; t; b

24 452 D. Tambara for all s; t; r A G, a; b; c; d A A. (21) sas t is a 3-coccle of A, so we have the group tensor categor V A; tš of Section 6. A G-action on this tensor categor is de ned b s a ˆsa f s; t a ˆ v s; t; a 1st a n a ˆ 1 v 1; 1; a 1 a c s a; b ˆ u s; a; b 1 s ab i s ˆ 1 u s; 1; for s; t A G, a; b A A. Indeed, commutativit of (6), (8), (1) is assured b (22), (23), (24), while that of (2), (3), (7) b the de nition of n, i, respectivel. B the de nition of C G Š in Section 2, we have V A; tš G Š ˆ V A c G; sš, where s is a 3-coccle on the semi-direct product A c G given b s a; s ; b; t ; c; r ˆ t a; s b; st c u s; b; t c v s; t; c : Theorem 4.1 applied to the G-tensor categor V A; tš sas that the 2-functor V A; tš G -Modk ƒƒ G V A c G; sš-modk is a 2-equivalence. Assume k is algebraicall closed. Then nite semi-simple categories are just 2-vector spaces. B Proposition 5.1 the propert of being a 2-vector space is preserved under the above 2-equivalence. We saw in Section 6 that an V A c G; sš-module which is a 2-vector space is of the form V X; rš for a nite A c G-set X and a map r : A c G 2 X! k satisfing i s ˆqr 1. Hence an V A; tš G -module which is a 2-vector space is of the form V X; rš G. 8. Pentagon identit for C G Š. We will show here that the associativit isomorphisms a A; s ; B; t ; C; r for C G Š de ned in Section 2 satisf the pentagon axiom of a monoidal categor: A; s :a B; t ; C; r ; D; p a A; s ; B; t : C; r ; D; p a A; s ; B; t ; C; r : D; p ˆ a A; s ; B; t ; C; r : D; p a A; s : B; t ; C; r ; D; p for s; t; r; p A G, A; B; C; D A C.

25 Finite group actions on tensor categories 453 Figure 8.1 A pentagon diagram for C is a A; B; C :D A: B:C :D ƒƒƒƒ a A; B:C; D A: B:C :D ƒƒƒƒ! A:a B; C; D A:B :C :D a A:B; C; D A:B : C:D a A; B; C:D A: B: C:D : We refer this diagram as I 0 A; B; C; D, and also the diagram (6) as I 1 s; A; B; C, (8) as I 2 s; t; A; B, (1) as I 3 s; t; r; X. The assumption is that I 0 A; B; C; D, I 1 s; A; B; C, I 2 s; t; A; B, I 3 s; t; r; A are commutative for all A; B; C; D A C, s; t; r A G. To save space we write here and below s A instead of s A. B (11), we have the equalities in Figure 8.1 for morphisms in C GŠ. Hence what we have to prove is the equalit

26 454 D. Tambara Figure 8.2 A: s a B; t; C; r; D a A; s; B: t C; tr; D a A; s; B; t; C : str D ˆ a A; s; B; t; C: r D a A: s B; st; C; r; D 25 in C. B the de nition of a ; ; ; ; in (12), the both sides of (25) are expanded as in Figure 8.2, respectivel. Arrows are labeled onl b their tpes. Now we have the commutative diagrams in Figures 8.3 and 8.4. There are four faces induced from the diagrams of tpe I 0 ; I 1 ; I 2 ; I 3 which are commutative. Faces labeled b aš, c s Š are commutative b the naturalit of a, c s, and faces labeled b nš are commutative b the functorialit of the tensor product. Hence in either diagram, the composite along the leftmost path and the composite along the rightmost path from A: s B : st C : str D to A: s B: t C: r D are equal. Moreover the rightmost paths in the both diagrams are literall identical. The leftmost path in Figure 8.3 and the left path in Figure 8.2 are identical, and the leftmost path in Figure 8.4 and the right path in Figure 8.2 are identical. Hence the both sides of (21) are equal.

27 Finite group actions on tensor categories 455 Figure 8.3 Figure 8.4 References [B] F. Borceux, Handbook of Categorical Albebra 1, Cambridge Universit Press, Cambridge, [BM] R. J. Blattner and S. Montgomer, A dualit theorem for Hopf module algebras, J. Algebra, 95 (1985), 153±172. [GV] A. Grothendieck and J.-L. Verdier, Topos, TheÂorie des Topos et Cohomologie Etale des ScheÂmas, Lecture Notes in Mathematics, vol. 269, Springer-Verlag, New York, 1972, 299±519.

28 456 D. Tambara [M] S. Magid, Representations, duals and quantum doubles of monoidal categories, Suppl. Rend. Circ. Math. Palermo, Series II, 26 (1991), 197±206. [NT] Y. Nakagami and M. Takesaki, Dualit for crossed products of von Neumann algebras, Lecture Notes in Mathematics, vol. 731, Springer-Verlag, New York, [T] D. Tambara, A dualit for modules over monoidal categories of representations of semisimple Hopf algebras, to appear. Daisuke Tambara Department of Mathematical Sstem Science Hirosaki Universit , Japan tambara@cc.hirosaki-u.ac.jp