On the Rosenberg-Zelinsky sequence in abelian monoidal categories

K CL-MTH-07 -18 ZMP-HH/07-1 3 Hamburger Beitr¨ age zur Mathematik Nr. 2 94 ON THE R OSENBER G-ZELINSKY SEQUENCE IN ABELIAN MONOID AL CA TEGORIES Till Barmeier a,b , J ¨ urgen F uchs c , Ingo Runk el b , Christoph Sc h w eigert a ∗ a Or ganisationsei n heit Mathematik, Universit¨ at Hambur g Schwerpunkt A lgebr a und Zahle nthe orie Bundesstr aße 55, D – 20 146 Hambur g b Dep artment of Mathematics, King’s Col le ge L ondon Str and, L ondon WC2R 2LS , Unite d Ki n gdom c T e or etisk fysik, Karlstads Universitet Universitetsgatan 5 , S – 651 88 Karlstad Decem b er 2007 Abstract W e consider F r ob enius alge bras and t heir bimo dules in certain ab elian monoidal catego ries. In p articular we study the Picard group of the category of b imo dules o ver a F rob enius alge- bra, i.e. the group of isomorphism classes of inv ertible bimo d ules. The Rosen b erg-Zelinsky sequence describes a homomo rph ism from the group of algebra automorph isms to the Picard group, whic h ho wev er is t y p ically not su rjectiv e. W e inv estigate und er whic h conditions there exists a Morita equiv alen t F rob enius algebra for whic h the corresp onding homomorphism is surjectiv e. One motiv ation for our considerations is the orbifold constr u ction in conformal field theory . ∗ Email addresses: barmeier@math.uni-hamburg.de, jfuch s@fuchs.tekn.k au.se, ingo.runkel@k cl.ac.uk, sch weige rt@math.uni-hamburg.de 1 In tr o d u ction In the study of associative a lg ebras it is often a dv a ntageous to collect alg ebras in to a category whose morphisms are not algebra homomorphisms , but instead bimodules. One motiv ation f o r this is pro vided b y t he following observ ation. Let k b e a field and consider finite-dimensional unital asso ciativ e k -a lgebras. The condition on a k -linear map to b e a n alg ebra morphism is ob viously not linear. As a consequence the category of algebras and algebra homomorphisms ha s the unpleasan t feature of not b eing additiv e. On the other hand, instead of an algebra homomorphism ϕ : A → B one can equiv alen tly consider the B - A -bimo dule B ϕ whic h as a k -ve ctor space coincides with B and whose left action is given b y the m ultiplication of B while the right action is applicatio n o f ϕ comp o sed with m ultiplication in B . This is consisten t with comp osition in the sense tha t giv en another alg ebra homomorphism ψ : B → C there is an isomorphism C ψ ⊗ B B ϕ ∼ = C ψ ◦ ϕ of C - A - bimo dules. It is then natural not to restrict o ne’s a tten tion to such sp ecial bimo dules, but to allow all B - A -bimo dules as morphisms from A to B [Be, sect. 5.7]. Of course, a s bimo dules come with their own morphisms, one then actually deals with the structure of a bicategory . The adv an tage is that the 1-mor phism category A → B , i.e. the category of B - A - bimo dules, is additiv e and eve n ab elian. T aking bimo dules as morphisms has further in teresting consequences . First of all, the concept of isomor phy o f t w o algebras A and B is no w replaced b y Morita equiv alence, which requires the existence of an inv ertible A - B -bimo dule. Indeed, in a pplicatio ns in v olving asso ciative algebras one often finds that not only isomorphic but also Morita eq uiv alen t algebras can b e used for a giv en purp ose. The classical example is the equiv alence of the category of left (or right) mo dules o v er Morita equiv alen t algebras. Another illustration is the Morita equiv alence b etw een in v arian t subalgebras and crossed pro ducts, see e.g. [Ri]. Examples in the realm o f mathematical ph ysics include the observ ations that matrix theories on Morita equiv a lent noncomm utative tori are phy s- ically equiv alen t [Sc], and that Morita equiv alen t symmetric sp ecial F rob enius algebras in mo dular tensor categories describ e equiv alen t rational conformal field theories [FFRS1, FFRS3]. As a second consequence, instead of the a utomorphism group Aut( A ) one now deals with the in v ertible A -bimo dules. The isomorphism classes of these particular bimo dules form the Picard group Pic( A -Bimo d) of A -bimo dules. While Morita equiv alen t alg ebras ma y hav e differen t automorphism g roups, the corresp onding Picard groups are isomorphic. One finds tha t for an y algebra A the groups Aut( A ) and Pic( A -Bimo d) are related b y the exact sequence 0 − → Inn( A ) − → Aut( A ) Ψ A − → Pic( A -Bimo d) , (1.1) whic h is a v arian t of t he Rosen b erg-Zelinsky [RZ, K O] sequence. He re Inn( A ) denotes the inner automorphisms of A , and the g roup homomorphism Ψ A is g iven b y assigning to an automorphism ω of A t he bimo dule A ω obtained from A by t wisting the right action o f A on itself b y ω . In other w o rds, Pic( A -Bimo d) is t he home f or t he o bstruction to a Sk olem-No ether theorem. It should b e noticed that the gro up homomorphism Ψ A in (1.1) is not necessarily a surjection. But for practical purp oses in concrete applications it can b e o f interes t to ha v e an explicit realisa- 2 tion of the Picard group in terms of auto morphisms of the algebra av ailable. This leads naturally to t he f ollo wing questions: • Do es there exist ano t her algebra A ′ , Morita equiv alen t to A , suc h that the g roup homomorphism Ψ A ′ : Aut( A ′ ) → Pic ( A ′ -Bimo d) in (1.1) is surjectiv e? • And, o nce suc h an algebra A ′ has b een constructed: Do es this surjection admit a section, i.e. can the g roup Pic( A -Bimo d) b e iden tified with a subgroup of the automorphism gro up of t he Morita equiv alen t alg ebra A ′ ? W e will in v estigate these questions in a mo r e general setting, namely we consider alg ebras in k - linear monoidal categories more general than the one of k -v ector spaces. Lik e many other results v alid for v ector spaces, also the sequence (1.1) con tin ues to hold in this setting, see [VZ, prop. 3.14 ] and [F RS3, prop. 7]. W e start in section 2 b y collecting some asp ects of algebras and Morita equiv alence in monoidal categories and review the definition of inv ertible ob jects and of the Picard category . Section 3 collects informa t ion ab out fixed algebras under some subgroup of algebra automorphisms. In section 4 w e answ er the questions raised a b o v e for the sp ecial case that the algebra A is the t ensor unit of the monoidal category D under consideration. As recalled in section 2, the categorical dimension pro vides a c haracter on the Picard gro up with v alues in k × . The main result of section 4, Prop osition 4.3, supplies, for any finite subgroup H of the Picard gro up on whic h this character is trivial, an algebra A ′ that is Morit a equiv alen t to the tensor unit suc h that the elemen ts of H can b e iden tified with automorphisms of A . Theorem 4.12, in turn, giv es a c haracterisation of group homomorphisms H → Aut( A ) in terms of co chains on H . I n this case t he subgroup H is not only required to ha v e trivial c haracter, but in a dditio n a three-co cycle on Pic( A -Bimo d) m ust b e trivial when restricted to H . The relev an t three-co cycle is obtained from the asso ciativit y constrain t of D , see eq. (4.23 ) b elo w. W e also compute the fixed algebra under the corresp onding subgroup of automorphisms. In section 5 these results are generalised t o alg ebras not necessarily Morita equiv alen t to the tensor unit, prov iding an a ffirmativ e answ er to the ab ov e questions also in the general case. How ev er, similar to the A = 1 case, one needs to restrict oneself to a finite subgroup H of Pic( A -Bimo d) suc h that t he corresp onding inv ertible bimodules hav e categorical dimension equal to 1 in A -Bimo d and for whic h the associat ivity constrain t of A -Bimo d is trivial. This is stated in Theorem 5.6, whic h is the ma in result of this pap er. Let us a lso briefly men tion a motiv ation of our considerations whic h comes fro m confo r ma l field theory . A consisten t rational conformal field theory (o n oriented surfaces with p ossibly non- empt y b oundary) is determined b y a mo dule category M o ver a mo dular tensor category C [FRS1]. The Picard group of the category o f mo dule endofunctors of M describ es the symmetries of this CFT [FFRS3]. The explicit construction of this CFT requires not just the abstract mo dule category , but rather a concrete realizatio n as category of mo dules ov er a F rob enius algebra A , as this pro vides a natural forg etful functor from M to C whic h en ters crucially in the construction. The mo dule endofunctors are r ealised a s the category of A - A -bimo dules. F or practical purp oses it can b e useful 3 to c ho o se the algebra A suc h that a giv en subgroup H of the symmetries Pic( A -Bimo d) of the CFT is realised as automorphisms of A . Theorem 5.6 pro vides us with conditions for when suc h a represen tative exists. Finally , the fixed algebra under this subgroup of automorphisms is related to t he CFT o btained b y ‘orbifolding ’ the original CFT b y the symmetry H . A ckno wledgements : TB is suppo rted b y the Europ ean Sup erstring Theory Net w ork (MCFH- 2004-51 2194) and thanks K ing’s College Lo ndon fo r hospitalit y . JF is par t ia lly supp o rted b y VR under pro ject no. 62 1-2006- 3343. IR is partia lly supp o rted b y the EPSR C F irst G ran t EP/E005047/1, t he PP AR C r o lling gran t PP/C507145 / 1 and the Marie Curie net w ork ‘Sup er- string Theory’ (MR TN-CT-2004-5 12194). CS is partially supp orted by the Collab ora t iv e Researc h Cen tre 676 “Particles , Strings and the Early Univers e - the Str ucture of Matter and Space-Time”. 2 Algebras in monoidal c ategories In this section w e collect information ab out a f ew basic structures that will b e needed b elo w. Let D b e an ab elian catego ry enric hed o v er the category V ect k of finite- dimensional v ector spaces ov er a field k . An ob ject X o f D is called simple iff it has no prop er sub o b jects. An endomorphism o f a simple ob ject X is either zero or an isomorphism (Sch ur’s lemma), and hence the endomorphism space Hom( X , X ) is a finite-dimensional division algebra o v er k . An ob j ect X of D is called absolutely simple iff Hom( X , X ) = k id X . If k is a lgebraically closed, then eve ry simple ob ject is absolutely simple; the conv erse holds e.g. if D is semisimple. When D is monoidal, then without loss o f generality w e a ssume it to b e strict. More sp ecifically , for the rest of this pap er w e make the f ollo wing assumption. Con v en t ion 2.1. ( D , ⊗ , 1 ) is an ab elian strict monoidal category with simple and absolutely simple tensor unit 1 , and enriche d o v er V ect k for a field k of c haracteristic zero. In particular, Hom( 1 , 1 ) = k id 1 , whic h w e iden tify with k . Definition 2.2. A right duality on D assigns to eac h ob ject X o f D an ob ject X ∨ , called the righ t dual ob ject of X , a nd morphisms b X ∈ Hom( 1 , X ⊗ X ∨ ) and d X ∈ Hom( X ∨ ⊗ X , 1 ) such that (id X ⊗ d X ) ◦ ( b X ⊗ id X ) = id X and ( d X ⊗ id X ∨ ) ◦ (id X ∨ ⊗ b X ) = id X ∨ . (2.1) A left duality on D assigns to eac h ob j ect X of D a left dual ob ject ∨ X together with morphisms ˜ b X ∈ Hom( 1 , ∨ X ⊗ X ) and ˜ d X ∈ Hom( X ⊗ ∨ X , 1 ) suc h that ( ˜ d X ⊗ id X ) ◦ (id X ⊗ ˜ b X ) = id X and (id ∨ X ⊗ ˜ d X ) ◦ ( ˜ b X ⊗ id ∨ X ) = id ∨ X . (2.2) Note that 1 ∨ ∼ = → 1 ∨ ⊗ 1 d 1 → 1 is nonzero; since by assumption 1 is simple, we th us ha v e 1 ∨ ∼ = 1 . In the same w a y one sees that ∨ 1 ∼ = 1 . F urther, giv en a righ t dualit y , the right dua l morphism t o a morphism f ∈ Hom( X , Y ) is the morphism f ∨ := ( d Y ⊗ id X ∨ ) ◦ (id Y ∨ ⊗ f ⊗ id X ∨ ) ◦ (id Y ∨ ⊗ b X ) ∈ Hom( Y ∨ , X ∨ ) . (2.3) 4 Left dual morphisms are defined analogously . Hereb y eac h duality furnishes a functor from D to D op . F urther, the ob jects ( X ⊗ Y ) ∨ and Y ∨ ⊗ X ∨ are isomorphic. Definition 2.3. A sover e ign 1 category is a monoidal category that is equipped with a left and a right duality whic h coincide as functors, i.e. X ∨ = ∨ X for ev ery o b ject X a nd f ∨ = ∨ f for ev ery morphism f . In a so v ereign category the left and right tr ac e s of an endomorphism f ∈ Hom( X , X ) a re the scalars (remem b er that we identify End( 1 ) with k ) tr l ( f ) := d X ◦ (id X ∨ ⊗ f ) ◦ ˜ b X and tr r ( f ) := ˜ d X ◦ ( f ⊗ id X ∨ ) ◦ b X , (2.4) resp ectiv ely , and the left and righ t dim e nsions of an ob ject X are the scalars dim l ( X ) := tr l (id X ) , dim r ( X ) := tr r (id X ) . (2.5) Both traces are cyclic, and dimensions are constant on isomorphism classes, multiplicativ e under the tensor pro duct and additiv e under direct sums. F urther, o ne has tr l ( f ) = tr r ( f ∨ ), and using the fact that in a sov ereign category each ob ject X is isomorphic to its double dual X ∨∨ it follows that t he r igh t dimension of the dual ob ject equals the left dimension o f the ob ject itself, dim l ( X ) = dim r ( X ∨ ) , (2.6) and vice v ersa. In particular, an y ob ject that is isomor phic to its dual, X ∼ = X ∨ , has equal left and righ t dimen sion, whic h w e then denote b y dim( X ). The tensor unit 1 is isomorphic to its dual and has dimension dim( 1 ) = 1. Next w e collect some information ab out algebra ob jects in monoida l categories. Recall tha t a (unital, a sso ciativ e) algebr a in D is a triple ( A, m, η ) consisting of an ob j ect A of D and morphisms m ∈ Hom( A ⊗ A, A ) and η ∈ Hom( 1 , A ), suc h that m ◦ (id A ⊗ m ) = m ◦ ( m ⊗ id A ) and m ◦ (id A ⊗ η ) = id A = m ◦ ( η ⊗ id A ) . (2.7) Dually , a (counital, coa sso ciativ e) c o alg e br a is a triple ( C , ∆ , ε ) with C an o b ject of D and mor- phisms ∆ ∈ Hom( C , C ⊗ C ) and ε ∈ Hom( C , 1 ), suc h that (∆ ⊗ id C ) ◦ ∆ = (id C ⊗ ∆) ◦ ∆ and (id C ⊗ ε ) ◦ ∆ = id C = ( ε ⊗ id C ) ◦ ∆ . (2.8) The follow ing concepts a r e also we ll kno wn, see e.g. [M ¨ u, FR S1]. Definition 2.4. (i) A F r ob enius algebr a in D is a quintuple ( A, m, η , ∆ , ε ), suc h that ( A, m, η ) is an algebra in D , ( A, ∆ , ε ) is a coalgebra and the compatibility relation (id A ⊗ m ) ◦ (∆ ⊗ id A ) = ∆ ◦ m = ( m ⊗ id A ) ◦ (id A ⊗ ∆) (2.9) b et w een t he a lgebra a nd coalgebra structures is satisfied. 1 What we call sovereign is sometimes referr ed to as strictly sover eign , co mpare [Bi, Br]. 5 (ii) A F rob enius algebra A is called sp e cial iff m ◦ ∆ = β A id A and ε ◦ η = β 1 id 1 with β 1 , β A ∈ k × . A is called normalise d sp e cial iff A is sp ecial with β A = 1. (iii) If D is in addition sov ereign, an algebra A in D is called s ymmetric iff the tw o morphisms Φ 1 := (( ε ◦ m ) ⊗ id A ∨ ) ◦ (id A ⊗ b A ) and Φ 2 := (id A ∨ ⊗ ( ε ◦ m )) ◦ ( ˜ b A ⊗ id A ) (2.10) in Hom( A, A ∨ ) are equal. F or ( A, m A , η A ) and ( B , m B , η B ) alg ebras in D , a mo r phism f : A → B is called a (unital) morphism of algebras iff f ◦ m A = m B ◦ ( f ⊗ f ) in Hom( A ⊗ A, B ) and f ◦ η A = η B . Similarly one defines (counita l) morphisms of coalgebras and morphisms of F rob enius algebras. An algebra S is called a sub algebr a of A iff there is a monic i : S → A tha t is a morphism of algebras. A (unital) left A - mo dule is a pair ( M , ρ ) consisting of an ob ject M in D and a morphism ρ ∈ Hom( A ⊗ M , M ), suc h that ρ ◦ (id A ⊗ ρ ) = ρ ◦ ( m ⊗ id M ) and ρ ◦ ( η ⊗ id M ) = id M . (2.11) Similarly one defines righ t A -mo dules. An A - A - bim o dule (o r A - b imo dule f or short) is a triple ( M , ρ,  ) suc h that ( M , ρ ) is a left A -mo dule, ( M ,  ) a rig h t A -mo dule, a nd the left and rig h t actions of A on M comm ute. Analogo usly , A - B -bimo dules carry a left action of the algebra A and a comm uting right action of the algebra B . F or ( M , ρ M ) and ( N , ρ N ) left A -mo dules, a morphism f ∈ Hom( M , N ) is said to b e a morphism of left A -mo dules (or briefly , a mo dule morphism) iff f ◦ ρ M = ρ N ◦ (id A ⊗ f ). Analogously one defines morphisms o f A - B -bimo dules. Thereb y one obtains a category , with ob j ects the A - B - bi- mo dules and morphisms the A - B -bimo dule morphisms. W e denote this category b y D A | B and the set of bimo dule morphisms from M to N b y Hom A | B ( M , N ). The F rob enius prop erty (2.9) means that t he copro duct ∆ is a morphism of A -bimo dules. Definition 2.5. An algebra is called (absolutely) simple iff it is (absolutely) simple as a bimo dule o v er itself. Thu s A is absolutely simple iff Hom A | A ( A, A ) = k id A . Remark 2.6. Since D is ab elian, one can define a tensor pro duct of A -bimo dules. This turns the bimo dule category D A | A in to a monoidal category . F or example, D ∼ = D 1 | 1 as monoidal categories. See the app endix for more details on this and esp ecially on tensor pro ducts ov er sp ecial F rob enius algebras. Remark 2.7. If A is a (not necessarily symme tric) F rob enius algebra in a so vere ign category , then the morphisms Φ 1 and Φ 2 in (2.10) are in v ertible, with inv erses Φ − 1 1 = ( d A ⊗ id A ) ◦ (id A ∨ ⊗ (∆ ◦ η ) ) and Φ − 1 2 = (id A ⊗ ˜ d A ) ◦ ((∆ ◦ η ) ⊗ id A ∨ ) , (2.12) resp ectiv ely . So if A is F rob enius, A and A ∨ are isomorphic, hence the left and right dimension of A are equal. Accordingly we will write dim( A ) for t he dimension of a F r o b enius algebra in the 6 sequel. F urther one can sho w (see [FRS1], section 3) that for any symmetric special F rob enius a lgebra A the relation β A β 1 = dim( A ) holds. In particular, dim( A ) 6 = 0. F urthermore, without loss of generalit y one can assume that the coproduct is normalised suc h that β 1 = dim( A ) a nd β A = 1, i.e. A is normalised sp ecial. Lemma 2.8. L et ( A, m, η ) b e an algebr a with dim k Hom( 1 , A ) = 1 . Then A is an absolutely simple algebr a. Pr o of. By Prop osition 4.7 of [FS] one has Hom( 1 , A ) ∼ = Hom A ( A, A ). The result thus follows from 1 ≤ dim k Hom A | A ( A, A ) ≤ dim k Hom A ( A, A ). Remark 2.9. Ob viously the tensor unit 1 is a symmetric sp ecial F rob enius algebra. One also easily verifies that for a n y ob ject X in a so v ereign category the ob ject X ⊗ X ∨ with structural morphisms m := id X ⊗ d X ⊗ id X ∨ , η := b X , ∆ := id X ⊗ ˜ b X ⊗ id X ∨ , ε := ˜ d X (2.13) pro vides an example of a symmetric F rob enius alg ebra. If the ob ject X has nonzero left and r ig h t dimensions, then this algebra is a lso sp ecial, with β X ⊗ X ∨ = dim l ( X ) , β 1 = dim r ( X ) . (2.14) The ob ject X is naturally a left mo dule ov er X ⊗ X ∨ , with represen tation morphism ρ = id X ⊗ d X , while the ob ject X ∨ is a rig h t mo dule ov er X ⊗ X ∨ with  = d X ⊗ id X ∨ . Next w e recall the concept of Morita equiv alence of algebras (for details see e.g. [P a , VZ]). Definition 2.10. A Morita c ontext in D is a sextuple ( A, B , P , Q, f , g ), where A and B are alge- bras in D , P ≡ A P B is an A - B -bimo dule and Q ≡ B Q A is a B - A -bimo dule, suc h that f : P ⊗ B Q ∼ = → A and g : Q ⊗ A P ∼ = → B are isomorphisms of A - and B -bimo dules, resp ectiv ely , and the tw o diagrams ( P ⊗ B Q ) ⊗ A P f ⊗ i d / / ∼ =   A ⊗ A P ∼ =   P ⊗ B ( Q ⊗ A P ) id ⊗ g   P ⊗ B B ∼ = / / P ( Q ⊗ A P ) ⊗ B Q g ⊗ i d / / ∼ =   B ⊗ B Q ∼ =   Q ⊗ A ( P ⊗ B Q ) id ⊗ f   Q ⊗ A A ∼ = / / Q (2.15) comm ute. If suc h a Morita context exists, we call the algebras A and B Morit a equiv alen t . In the sequel we will suppress the isomorphisms f and g and write a Morita contex t as A P ,Q ← → B . Lemma 2.11. L et D b e in addition sover eign and let U b e an o b je ct of D with nonzer o left and right dimension . Then the symmetric sp e cial F r ob enius a l g e br a U ⊗ U ∨ is Morita e quivalent to the tensor unit, with Morita c ontext 1 U ∨ ,U ← → U ⊗ U ∨ . 7 Pr o of. W e only need to sho w that U ∨ ⊗ U ⊗ U ∨ U ∼ = 1 . Since U ⊗ U ∨ is symmetric sp ecial F rob enius, the idemp otent P U ∨ ,U for the tensor pro duct ov er U ⊗ U ∨ , as describ ed in app endix A, is well defined. One calculates that P U ∨ ,U = (dim l ( U )) − 1 ˜ b U ◦ d U . This implies that the t ensor unit is indeed isomorphic to the image of P U ∨ ,U . Finally , comm utativity o f the diagrams (2.15) follows using the tec hniques of pro jectors as presen ted in t he a pp endix. Definition 2.12. An ob ject X in a mono idal categor y is called invertible iff there exists an ob ject X ′ suc h that X ⊗ X ′ ∼ = 1 ∼ = X ′ ⊗ X . If the category D has small sk eleton, then the set of isomorphism classes of inv ertible ob jects forms a group under the tensor pro duct. This group is called the Pic ar d gr oup Pic( D ) of D . Lemma 2.13. L et D b e in addi tion sover eign. (i) Every invertible obje ct of D i s simple. (ii) An obje ct X in D is invertible iff X ∨ is invertible. (iii) An obje ct X in D is invertible iff the m orphisms b X and ˜ b X ar e invertible. (iv) Every invertible o b j e ct of D is absolutely s i m ple. Pr o of. (i) Let X ⊗ X ′ ∼ = X ′ ⊗ X ∼ = 1 . Assume that e : U → X is monic for some ob ject U . Then id X ′ ⊗ e : X ′ ⊗ U → X ′ ⊗ X is monic. Indeed, if (id X ′ ⊗ e ) ◦ f = (id X ′ ⊗ e ) ◦ g for some morphisms f and g , then b y applying the duality morphism d X ′ w e obtain e ◦ ( d X ′ ⊗ id U ) ◦ (id X ′ ∨ ⊗ f ) = e ◦ ( d X ′ ⊗ id U ) ◦ (id X ′ ∨ ⊗ g ). As e is monic this amoun ts to ( d X ′ ⊗ id U ) ◦ (id X ′ ∨ ⊗ f ) = ( d X ′ ⊗ id U ) ◦ (id X ′ ∨ ⊗ g ), whic h by a pplying b X ′ and using the dua lity prop ert y of d X ′ and b X ′ sho ws that f = g . Thus id X ′ ⊗ e : X ′ ⊗ U → X ′ ⊗ X ∼ = 1 is monic. As 1 is required to b e simple, it is th us an isomorphism. Then id X ⊗ id X ′ ⊗ e is an isomorphism as w ell. By assumption there exists an isomorphism b : 1 → X ⊗ X ′ . With the help of b w e can write e = ( b − 1 ⊗ id X ) ◦ (id X ⊗ id X ′ ⊗ e ) ◦ ( b ⊗ id U ). Th us e is a comp o sition of isomorphisms, and hence an isomorphism. In summary , e : U → X b eing monic implies that e is an isomorphism. Hence X is simple. (ii) Note that X ∨ ⊗ X ′∨ ∼ = ( X ′ ⊗ X ) ∨ ∼ = 1 ∨ ∼ = 1 , and similarly X ′∨ ⊗ X ∨ ∼ = 1 . (iii) Since b y part (ii) X ∨ is inv ertible, so is X ⊗ X ∨ . By part ( i), X ⊗ X ∨ is therefore simple and b X : 1 → X ⊗ X ∨ is a nonzero mo r phism betw een simple ob jects. By Sch ur’s lemma it is an isomorphism. The argumen t for ˜ b X pro ceeds along the same lines, and the con ve rse statement follo ws b y definition. (iv) The duality mor phisms giv e an isomorphism Hom( X , X ) ∼ = Hom( X ⊗ X ∨ , 1 ). F rom part s (i) and (ii) w e kno w that X ⊗ X ∨ ∼ = 1 , and so Hom( X, X ) ∼ = Hom( 1 , 1 ). That X is a bsolutely simple no w follow s b ecause 1 is a bsolutely simple b y assumption. Lemma 2.13 implies that fo r an in v ertible ob ject X one has dim l ( X ) dim r ( X ) = dim l ( X ) dim l ( X ∨ ) = dim l ( X ⊗ X ∨ ) = dim l ( 1 ) = 1 . ( 2 .16) 8 With the help of this equalit y o ne c heck s that the inv erse o f b X is giv en b y dim l ( X ) ˜ d X , dim l ( X ) ˜ d X ◦ b X = dim l ( X ) dim r ( X ) id 1 = id 1 . (2.17) Analogously w e hav e dim r ( X ) d X ◦ ˜ b X = id 1 ; th us in pa r t icular the left and right dimensions of an in v ertible ob ject X are nonzero. F urther we ha v e dim l ( X ) b X ◦ ˜ d X = id X ⊗ X ∨ and dim r ( X ) ˜ b X ◦ d X = id X ∨ ⊗ X . (2.18) W e denote the ob ject represen ting an isomorphism class g in Pic( D ) by L g , i.e. [ L g ] = g ∈ Pic( D ). Then L g ⊗ L h ∼ = L g h . As the represen tat ive of the unit class 1 w e tak e the tensor unit, L 1 = 1 . Lemma 2.14. L et D b e in addi tion sover eign and H a sub gr o up o f Pic( D ) . (i) The mappin gs h 7→ dim l ( L h ) and h 7→ dim r ( L h ) ar e char a cters on H . (ii) If H is fi nite, then dim l | r ( L h ∈ H L h ) is e ither 0 or | H | . It is e qual to | H | iff dim l | r ( L h ) = 1 for al l h ∈ H . Pr o of. Claim (i) follow s directly from the m ultiplicativit y of the left a nd r ig h t dimension under the tensor pro duct a nd from the fact that t he dimension o nly dep ends on the isomorphism class of a n ob ject. Because of dim l | r ( L h ∈ H L h ) = P h ∈ H dim l | r ( L h ), part (ii) is a consequence of the orthog onalit y of c ha racters. Definition 2.15. The Pic ar d c ate gory P ic ( D ) o f D is the full sub category of D whose o b jects are direct sums of in v ertible ob jects of D . 3 Fixed algebras W e in tro duce the notion of a fixed a lgebra under a group of alg ebra auto mo r phisms and establish some basic results on fixed algebras. Definition 3.1. Let ( A, m, η ) b e an alg ebra in D a nd H ≤ Aut( A ) a group of (unital) automor- phisms of A . Then a fixe d alg ebr a under the action of H is a pa ir ( A H , j ), where A H is an ob ject of D and j : A H → A is a monic with α ◦ j = j for all α ∈ H , suc h tha t the fo llo wing unive rsal prop ert y is fulfilled: F or ev ery ob j ect B in D and morphism f : B → A with α ◦ f = f fo r all α ∈ H , there is a unique morphism ¯ f : B → A H suc h that j ◦ ¯ f = f . The ob ject A H defined this w ay is unique up to isomorphism . The follo wing result justifies using the term ‘fixed algebra’, ra ther than ‘fixed ob ject’. Lemma 3.2. Given A , H and ( A H , j ) as in d efinition 3.1, ther e exists a unique algebr a s tructur e on the o bje ct A H such that the inclusion j : A H → A is a morph ism o f alge b r as. 9 Pr o of. F or arbitra r y α ∈ H consider the diag r a ms A H j / / A α / / A A H j / / A α / / A A H ⊗ A H m ◦ ( j ⊗ j ) O O and 1 η O O (3.1) Since α is a morphism of a lgebras, we ha v e α ◦ m ◦ ( j ⊗ j ) = m ◦ (( α ◦ j ) ⊗ ( α ◦ j )) = m ◦ ( j ⊗ j ); as this holds for all α ∈ H , the univers al prop ert y o f the fixed algebra yields a unique pro duct morphism µ : A H ⊗ A H → A H suc h that j ◦ µ = m ◦ ( j ⊗ j ). By asso ciativity of A w e hav e j ◦ µ ◦ ( µ ⊗ id A H ) = m ◦ ( m ⊗ id A ) ◦ ( j ⊗ j ⊗ j ) = m ◦ (id A ⊗ m ) ◦ ( j ⊗ j ⊗ j ) = j ◦ µ ◦ (id A H ⊗ µ ) . (3.2) Since j is monic, this implies a sso ciativit y of the pro duct morphism µ . Similarly , applying the univ ersal prop ert y of ( A H , j ) on η giv es a morphism η ′ : 1 → A H that has the prop erties of a unit for the pro duct µ . So ( A H , µ, η ′ ) is an asso ciativ e algebra with unit. W e pro ceed to sho w that fixed algebras under finite groups of aut o morphisms alw ays exist in the situation studied here. Let H ≤ Aut( A ) b e a finite subgroup of the group o f algebra automorphisms of A . Set P = P H := 1 | H | X α ∈ H α ∈ End( A ) . (3.3) Then P ◦ P = 1 | H | 2 P α,β ∈ H α ◦ β = 1 | H | 2 P α,β ′ ∈ H β ′ = 1 | H | P β ′ ∈ H β ′ = P , i.e. P is an idemp otent. Anal- ogously one sho ws t hat α ◦ P = P for ev ery α ∈ H . F urther, since D is ab elian, w e can write P = e ◦ r with e monic and r epi. Denote the imag e of P ≡ P H b y A P , so that e : A P → A and r ◦ e = id A P . Lemma 3.3. T h e p a i r ( A P , e ) satisfies the universal pr op erty of the fixe d algebr a. Pr o of. F rom r ◦ e = id A P w e see tha t α ◦ e = α ◦ e ◦ r ◦ e = α ◦ P ◦ e = P ◦ e = e for all α ∈ H . F o r B an ob ject of D and f : B → A a morphism with α ◦ f = f for all α ∈ H , set ¯ f := r ◦ f . Then e ◦ ¯ f = e ◦ r ◦ f = P ◦ f = 1 | H | P α ∈ H α ◦ f = 1 | H | P α ∈ H f = f . F urther, if f ′ is a no ther morphism sat- isfying e ◦ f ′ = f , then e ◦ f ′ = e ◦ ¯ f and, since e is monic, ¯ f = f ′ , so ¯ f is unique. Hence the ob j ect A P satisfies the univ ersal pro p ert y of the fixed algebra A H . W e would lik e to express the structural morphisms of the fixed a lg ebra through e and r . T o this end w e in tro duce a candidate pro duct m P and candidate unit η P on A P : w e set m P := r ◦ m ◦ ( e ⊗ e ) and η P := r ◦ η . (3.4) 10 Note that e is a morphism of unital alg ebras: e ◦ m P = e ◦ r ◦ m ◦ ( e ⊗ e ) = P ◦ m ◦ ( e ⊗ e ) = 1 | H | P α ∈ H α ◦ m ◦ ( e ⊗ e ) = 1 | H | P α ∈ H m ◦ (( α ◦ e ) ⊗ ( α ◦ e )) = 1 | H | P α ∈ H m ◦ ( e ⊗ e ) = m ◦ ( e ⊗ e ) , e ◦ η P = e ◦ r ◦ η = P ◦ η = 1 | H | P α ∈ H α ◦ η = 1 | H | P α ∈ H η = η . (3.5) Lemma 3.4. The algebr a ( A P , m P , η P ) is isomorphic to the a l g e br a structur e that A P inherits a s a fixe d algebr a. Pr o of. An easy calculation sho ws that m P is asso ciative a nd η P is a unit fo r m P ; th us ( A P , m P , η P ) is an alg ebra. Moreo v er, since according to lemma 3.2 t here is a unique alg ebra structure o n A H suc h that the inclusion into A is a morphism of algebras, it f ollo ws that A P and A H are isomorphic as algebras. In the follow ing discussion the term fixed algebra will alwa ys refer t o the a lgebra A P . Lemma 3.5. With P = 1 | H | P α ∈ H α = e ◦ r as in (3.3), we have the f o l lowing e qualities of m or- phisms: r ◦ m ◦ ( e ⊗ P ) = r ◦ m ◦ ( e ⊗ id A ) , r ◦ m ◦ ( P ⊗ e ) = r ◦ m ◦ (id A ⊗ e ) and P ◦ m ◦ ( e ⊗ e ) = m ◦ ( e ⊗ e ) . (3.6) Pr o of. Indeed, making use of r ◦ α = r and α ◦ e = e for all α ∈ H , w e hav e r ◦ m ◦ ( e ⊗ P ) = 1 | H | X α ∈ H r ◦ m ◦ ( e ⊗ α ) = 1 | H | X α ∈ H r ◦ α ◦ m ◦ (( α − 1 ◦ e ) ⊗ id A ) = 1 | H | X α ∈ H r ◦ m ◦ ( e ⊗ id A ) = r ◦ m ◦ ( e ⊗ id A ) . (3.7) The other t w o equalities a re established analogously . Remark 3.6. With the help of the graphical calculus f or morphisms in strict monoidal categories (see [JS, Ka, Ma , BK], and e.g. App endix A o f [FFRS2] for the graphical represen t a tion of the structural morphisms of F rob enius alg ebras in suc h catego ries), the equalities in Lemma 3.5 can b e visualised as follows : A P A A P e P r = A P A A P e r A A P A P P e r = A A P A P e r A P A P A e e P = A P A P A e e (3.8) If A is a F rob enius alg ebra, it is understoo d that Aut( A ) consists of all algebra automorphisms of 11 A whic h are at the same time also coalgebra automorphisms. Then for a F rob enius algebra A the idemp oten t P can also b e omitted in the follow ing situations, whic h w e describ e a g ain pictorially: A P A A P r P e A A P A P P r e A P A P A r r P (3.9) Prop osition 3.7. L et A b e a F r ob en ius algeb r a i n D and H ≤ Aut( A ) a finite gr o up of automor- phisms of A . (i) A P is a F r ob enius alg e b r a, and the emb e d d ing e : A P → A is a m orphism of algebr as while the r estriction r : A → A P is a morphism of c o a l g ebr as. (ii) If the c ate gory D is sover eign and A is symm e tric, then A P is symm e tric, to o. (iii) If the c ate gory D is so ver eign, A is symmetric sp e ci a l and A P is an abs o lutely si m ple a lgebr a and has nonzer o left (e q uiva l e ntly right, cf. r emark 2.7) d imension, then A P is sp e cia l . Pr o of. (i) The algebra structure on A P has already b een defined in (3.4 ), and a ccording to (3.5) e is a morphism of algebras. D enoting the copro duct on A by ∆ and the counit b y ε , we further set ∆ P := ( r ⊗ r ) ◦ ∆ ◦ e a nd ε P := ε ◦ e . Similarly to the calculation in (3.5) one v erifies that r is a morphism of coalg ebras, and that ∆ P is coasso ciative and ε P is a counit. Regarding the F rob enius prop ert y , we give a graphical pro of of one o f the equalities that m ust b e satisfied: ∆ P ◦ m P = A P A P A P A P r r P e e = A P A P A P A P r r e e = A P A P A P A P r r e e = A P A P A P A P r r P e e = ( m P ⊗ id A P ) ◦ (id A P ⊗ ∆ P ) . (3.10) Here it is used that according to remark 3 .6 w e are allow ed to remov e and insert idemp oten ts P , and then the F rob enius prop erty of A is inv ok ed. The ot her half o f the F rob enius prop ert y is seen analogously . (ii) The follow ing chain of equalities shows that A P is symmetric: A ∨ P A P e r e e = A ∨ P A P e e ∨ = A ∨ P A P e ∨ e = A ∨ P A P e r e e (3.11) 12 Here the no tations b X = X X ∨ and ˜ b X = X ∨ X (3.12) are used for the duality morphisms b X and ˜ b X , resp ectiv ely , of an o b ject X . The morphisms d X and ˜ d X are dra wn in a similar w ay . (iii) W e ha v e ε P ◦ η P = ε ◦ e ◦ r ◦ η = ε ◦ η , whic h is nonzero b y sp ecialness of A . As A P is asso- ciativ e, m P is a morphism of A P -bimo dules. The F ro b enius prop ert y ensures that ∆ P is also a morphism o f bimo dules. Hence m P ◦ ∆ P is a morphism of bimo dules, and b y absolute simplicit y of A P it is a multiple of the iden tit y . Moreov er, m P ◦ ∆ P is not zero: w e ha v e ε P ◦ m P ◦ ∆ P ◦ η P = ε ◦ m ◦ (id A ⊗ P ) ◦ ∆ ◦ η (3.13) whic h, as A is symmetric, is equal t o tr l ( P ) = dim l ( A P ) 6 = 0. W e conclude that m P ◦ ∆ P 6 = 0. Hence A P is sp ecial. Remark 3.8. In the ab ov e discuss ion the category D is assumed to b e ab elian, but this assumption can b e relaxed. O f the prop erties of an ab elian category w e only used that the morphism sets are ab elian groups, that comp o sition is bilinear, and that the relev ant idemp oten ts factorise in a monic and an epi, i.e. that D is idemp otent complete. In addition we assumed that mor phisms sets are finite- dimensional k -vec tor spaces. F rom eq. (3.3) on w ards, and in part icular in prop osition 3.7, it is in addition used t ha t D is enric hed ov er V ect k . If this is not the case, one can no longer, in general, define an idemp otent P through 1 | H | P α ∈ H α , and t here need not exist a copro duct o n the fixed algebra A H , ev en if there is one on A . 4 Algebras in the Mo r i ta class of the tensor unit Recall that according to our con v en tion 2.1 ( D , ⊗ , 1 ) is ab elian strict monoidal, with simple and absolutely simple tensor unit a nd enric hed ov er V ect k with k of c haracteristic zero. F rom now on w e further assume that D is sk eletally small and sov ereign. W e no w asso ciate to an algebra ( A, m, η ) in D a sp ecific subgroup of its automorphism group – the inner a utomorphisms – whic h are defined as follows. The space Hom( 1 , A ) b ecomes a k -algebra b y defining the pro duct as f ∗ g := m ◦ ( f ⊗ g ) for f , g ∈ Hom( 1 , A ). The morphism η ∈ Hom( 1 , A ) is a unit f or this pro duct. W e call a morphism f in Hom( 1 , A ) in v ertible iff there exists a morphism f − ∈ Hom( 1 , A ) suc h t ha t f ∗ f − = η = f − ∗ f . Now the morphism ω f := m ◦ ( m ⊗ f − ) ◦ ( f ⊗ id A ) ∈ Hom( A, A ) (4.1) is easily seen to b e an algebra automorphism. The automorphisms of this form are called i n ner automorphisms; they fo rm a normal subgroup Inn( A ) ≤ Aut( A ) as is seen b elow . 13 Definition 4.1. F or A an algebra in D and α , β ∈ Aut( A ), t he A -bimo dule α A β = ( A, ρ α ,  β ) is the bimo dule whic h has A as underlying ob ject and left and right actions o f A g iv en b y ρ α := m ◦ ( α ⊗ id A ) and  β := m ◦ (id A ⊗ β ) , (4.2) resp ectiv ely . These left and r igh t actio ns o f A are said to b e twiste d b y α and β , resp ectiv ely , and α A β is called a twiste d bimo dule . That this indeed defines an A -bimo dule structure on the ob j ect A is easily c hec k ed with the help of the m ultiplicativit y and unitality of α and β . F urther, as shown in [VZ, FRS3], the bimo dules α A β are in v ertible. Denote the isomorphism class of a bimo dule X by [ X ]. By setting Ψ A ( α ) := [ id A α ] (4.3) one obtains an exact sequence 0 − → Inn( A ) − → Aut( A ) Ψ A − → Pic( D A | A ) (4.4) of g roups. In particular one sees that t he subgroup Inn ( A ) is in fact a normal subgroup, as it is the kernel of the homomorphism Ψ A . The pro of o f exactness of this sequence in [VZ, FRS3] is not only v alid in braided monoidal categories, but also in the presen t mor e general situation. Let no w A and B b e Morita equiv alen t algebras in D , with A P ,Q ← → B a Morita c ontex t ( P ≡ A P B , Q ≡ B Q A ). Then the mapping Π Q,P : Pic( D A | A ) ∼ = − → Pic( D B | B ) [ X ] 7− → [ Q ⊗ A X ⊗ A P ] (4.5) constitutes an isomorphism b et w een the Picard groups Pic( D A | A ) and Pic( D B | B ). In par t icular, if A is an algebra that is Morita equiv alen t t o the tensor unit 1 , then w e hav e an isomorphism Pic( D A | A ) ∼ = Pic( D ). As Morita equiv alent alg ebras need not ha v e isomorphic a utomorphism groups, the images of the group homo mo r phisms Ψ A : Aut( A ) → Pic( D A | A ) and Ψ B : Aut( B ) → Pic( D B | B ) will in general b e no n-isomorphic. In the following we will consider subgroups of the group Pic( D ). F o r a subgroup H ≤ Pic( D ) w e put Q ≡ Q ( H ) := M h ∈ H L h . (4.6) Remark 4.2. Since the ob ject Q is the direct sum ov er a whole subgroup of Pic( D ) and L ∨ g ∼ = L g − 1 , it follows that Q ∼ = Q ∨ . As a consequence, left and right dimensions o f Q ar e equal, and accordingly in the sequel w e use the not a tion dim( Q ) for b oth of them. Prop osition 4.3. L e t H ≤ Pic( D ) b e a fin i te sub gr oup such that dim( Q ) 6 = 0 for Q ≡ Q ( H ) . Then with the algebr a A ≡ A ( H ) := Q ⊗ Q ∨ (4.7) 14 and the Morita c ontext 1 Q ∨ ,Q ← → A intr o duc e d in lemma 2.11, we have H = im(Π Q ∨ ,Q ◦ Ψ A ) , i.e. the sub gr oup H is r e c ove r e d as the im age o f the c omp o s ite map Aut( A ) Ψ A − → Pic( D A | A ) Π Q ∨ ,Q − − − − → Pic ( D ) . (4.8) Pr o of. The isomorphism Π Q,Q ∨ : Pic( D ) → Pic( D A | A ) is given b y [ L g ] 7→ [ Q ⊗ L g ⊗ Q ∨ ]. F or h ∈ H w e wan t to find auto morphisms α h of A suc h that id A α h ∼ = Q ⊗ L h ⊗ Q ∨ as A -bimo dules. W e first observ e the isomorphisms Q ⊗ L h ∼ = L g ∈ H L g ⊗ L h ∼ = L g ∈ H L g h ∼ = Q . W e make a (in general non- canonical) choice of isomorphisms f h : Q ⊗ L h ∼ = → Q , with the morphism f 1 c ho sen to b e the iden tity id Q . Then for eac h h ∈ H w e define the endomorphism α h of Q ⊗ Q ∨ b y α h := Q ∨ Q ∨ Q Q f − 1 h L h f ∨ h (4.9) These a r e algebra morphisms: m ◦ ( α h ⊗ α h ) = Q Q ∨ Q Q ∨ Q Q ∨ f − 1 h f ∨ h f − 1 h L h f ∨ h = Q Q ∨ Q Q ∨ Q Q ∨ f − 1 h f ∨ h f − 1 h f ∨ h = Q Q ∨ Q Q ∨ Q Q ∨ f − 1 h L h f ∨ h f − 1 h f ∨ h = Q Q ∨ Q Q ∨ Q Q ∨ f − 1 h f ∨ h = α h ◦ m . (4.10) Here w e ha v e used that by lemma 2.14 (ii) w e hav e dim l | r ( L h ) = 1 f or h ∈ H . The third equality is then a consequence of id L ∨ h ⊗ L h = ˜ b L h ◦ d L h , see equation ( 2.18); in the f o urth equalit y f h is cancelled against f − 1 h b y using prop erties of the dualit y . (Also, for b etter reada bility , here and b elow w e refrain from lab elling some of the L h -lines.) F urther, the morphisms α h are also unital: α h ◦ η = Q ∨ Q f − 1 h f ∨ h = Q ∨ Q f − 1 h f h = L h Q ∨ Q = Q ∨ Q = η , (4.11) where again b y lemma 2.1 4 we ha v e dim r ( L h ) = 1. The in v erse o f α h is giv en b y α − 1 h = Q ∨ Q ∨ Q Q f h f −∨ h L h (4.12) 15 as is seen in the following calculations: α h ◦ α − 1 h = Q Q Q ∨ Q ∨ f − 1 h f −∨ h f h f ∨ h = Q ∨ Q ∨ Q Q L h = id Q ⊗ Q ∨ , α − 1 h ◦ α h = Q ∨ Q ∨ Q Q f − 1 h f −∨ h f h f ∨ h = Q ∨ Q ∨ Q Q f − 1 h f −∨ h f h f ∨ h = id Q ⊗ Q ∨ . (4.13) where in particular (2.1 8) and dim l | r ( L h ) = 1 is used. A bimo dule isomorphism Q ⊗ L h ⊗ Q ∨ → id A α h is no w giv en by F h := id Q ⊗ (( ˜ d L h ⊗ id Q ∨ ) ◦ (id L h ⊗ f ∨ h )) . (4.14) First w e see that F h is in v ertible with in ve rse F − 1 h = id Q ⊗ ((id L h ⊗ f −∨ h ) ◦ ( b L h ⊗ id Q ∨ )), where f −∨ h stands for the dual of the inv erse o f f h . That F − 1 h is indeed in v erse to F h is seen as follow s: F h ◦ F − 1 h = Q ∨ Q ∨ Q Q f ∨ h f −∨ h = Q ∨ Q ∨ Q Q = id Q ⊗ Q ∨ , F − 1 h ◦ F h = Q ∨ Q ∨ Q Q L h L h f −∨ h f ∨ h = Q ∨ Q ∨ Q Q L h L h f −∨ h f ∨ h = id Q ⊗ L h ⊗ Q ∨ . (4 .15) Moreo v er, F h clearly intert wines the left actions o f A on Q ⊗ L h ⊗ Q ∨ and on id A α h . That it in tert wines the right actions a s well is verifie d as follow s: Q ∨ Q ∨ Q ∨ Q Q Q L h f ∨ h f ∨ h f − 1 h = Q ∨ Q ∨ Q ∨ Q Q Q L h f ∨ h f ∨ h f − 1 h = Q ∨ Q ∨ Q ∨ Q Q Q L h f ∨ h f ∨ h f − 1 h = Q ∨ Q ∨ Q ∨ Q Q Q L h f ∨ h (4.16) Here similar steps are p erformed as in t he pro of that α h resp ects the pro duct of A . W e conclude that we ha v e [ Q ⊗ L h ⊗ Q ∨ ] ∈ im (Ψ A ) for all h ∈ H , and th us Π Q,Q ∨ ( H ) is a subgroup of im(Ψ A ). 16 On the other hand, for g 6∈ H , Q ⊗ L g ⊗ Q ∨ ∼ = L h,h ′ ∈ H L hg h ′ is not isomorphic to Q ⊗ Q ∨ , not ev en as an o b ject, so that Π Q,Q ∨ ( g ) 6∈ im(Ψ A ). T ogether it follows that im(Π Q,Q ∨ ◦ Ψ A ) = H . Remark 4.4. Similarly to the calculation that the morphisms α h in (4.9) a re morphisms of algebras, one sho ws that t hey also resp ect the copro duct and the counit of A ( H ). So in fact w e ha v e found a uto morphisms of F rob enius algebras. W e denote the inclusion morphisms L h → Q = L g ∈ H L g b y e h and the pro jections Q → L h b y r h , suc h t ha t r g ◦ e h = 0 f or g 6 = h a nd r g ◦ e g = id L g . Then e g ◦ r g = P g is a nonzero idemp o ten t in End( Q ), and w e hav e P h ∈ H P h = id Q . Lemma 4.5. L e t H ≤ Pic ( D ) b e a finite sub gr oup such that dim ( Q ) 6 = 0 for Q ≡ Q ( H ) . Given g ∈ H and an automorphism α o f A = Q ⊗ Q ∨ such that Ψ A ( α ) = [ Q ⊗ L g ⊗ Q ∨ ] , ther e exists a unique isomorphism f g ∈ Hom( Q ⊗ L g , Q ) such that r g ◦ f g ◦ ( e 1 ⊗ id L g ) = id L g and α = α g with α g as in (4.9). Pr o of. W e start b y proving existence . Let ϕ g : Q ⊗ L g ⊗ Q ∨ ∼ = − → id A α b e an isomorphism of bimo d- ules. As a first step w e sho w tha t ϕ g = id Q ⊗ h for some mor phism h : L g ⊗ Q ∨ ∼ = → Q ∨ : Q ∨ Q ∨ Q Q L g ϕ g = 1 dim( Q ) Q ∨ Q ∨ Q Q L g ϕ g = 1 dim( Q ) Q ∨ Q ∨ Q Q L g ϕ g (4.17) Here in the first step w e just inserted the dimension of Q , using that it is nonzero, and the second step is the statemen t that ϕ g in tert wines the left action of A o n Q ⊗ L g ⊗ Q ∨ and on id A α . Note that up on setting f g := 1 dim( Q ) Q Q L g ϕ g (4.18) this amoun ts to ϕ g = id Q ⊗ (( ˜ d L g ⊗ id Q ∨ ) ◦ (id L g ⊗ f ∨ g )), a s in prop osition 4.3. Similarly the condition that ϕ g in tert wines the right action of A on Q ⊗ L g ⊗ Q ∨ and id A α means that Q ∨ Q ∨ Q ∨ Q Q Q L g f ∨ g α = Q ∨ Q ∨ Q ∨ Q Q Q L g f ∨ g (4.19) and this is equiv alen t to equalit y o f the same pictures with the identit y morphisms at the left sides 17 remo v ed. Applying duality mo r phisms to b oth sides of the resulting equalit y w e obtain Q ∨ Q ∨ Q Q L ∨ g f g α = Q ∨ Q ∨ Q Q L ∨ g f ∨ g (4.20) Next w e apply the morphism (id Q ⊗ ˜ d L g ⊗ id Q ∨ ) ◦ ( f − 1 g ⊗ id L ∨ g ⊗ id Q ∨ ), leading to Q ∨ Q ∨ Q Q α = Q ∨ Q ∨ Q Q f − 1 g f ∨ g (4.21) Using also that b y lemma 2.14 (ii) we ha v e dim r ( L g ) = 1, this sho ws that α = α g with α g as in (4.9). Since L g is absolutely simple (see lemma 2 .1 3 (iv)), w e hav e r g ◦ f g ◦ ( e 1 ⊗ id L g ) = ξ id L g for some ξ ∈ k . As r g is injectiv e and f g is an isomorphism, r g ◦ f g : Q ⊗ L g → L g is nonzero. But this morphism can only b e non v anishing on the direct summand 1 of Q , a nd hence also r g ◦ f g ◦ ( e 1 ⊗ id L g ) 6 = 0, i.e. ξ ∈ k × . Finally note that α do es not change if w e replace f g b y a nonze- ro m ultiple of f g ; hence after suitable rescaling f g ob eys b oth α = α g and r g ◦ f g ◦ ( e 1 ⊗ id L g ) = id L g , th us provin g existence. T o show uniquenes s, supp ose that there is anot her isomorphism f ′ g : Q ⊗ L g → Q suc h that α = α ′ g (where α ′ g is giv en by (4.9) with f ′ g instead of f g ) and r g ◦ f ′ g ◦ ( e 1 ⊗ id L g ) = id L g . Comp osing b oth sides of the equality α g = α ′ g with f g ⊗ id Q ∨ from the right and ta king a pa r t ial trace ov er Q of the resulting morphism Q ⊗ L g ⊗ Q ∨ → Q ⊗ Q ∨ giv es ( d Q ⊗ id Q ∨ ) ◦ (id Q ∨ ⊗ α g ) ◦ (id Q ∨ ⊗ f g ⊗ id Q ∨ ) ◦ ( ˜ b Q ⊗ id L g ⊗ id Q ∨ ) = ( d Q ⊗ id Q ∨ ) ◦ (id Q ∨ ⊗ α ′ g ) ◦ (id Q ∨ ⊗ f g ⊗ id Q ∨ ) ◦ ( ˜ b Q ⊗ id L g ⊗ id Q ∨ ) (4.22) Substituting the explicit form of α g and α ′ g , and using t ha t dim ( Q ) 6 = 0 a nd that L g is absolutely simple, one finds that f g = λ f ′ g for some λ ∈ k . The normalisation conditions r g ◦ f g ◦ ( e 1 ⊗ id L g ) = id L g and r g ◦ f ′ g ◦ ( e 1 ⊗ id L g ) = id L g then force λ = 1, pro ving uniqueness. The construction of the automorphisms α h presen ted in prop osition 4.3 and lemma 4.5 still dep ends on the c hoice of isomorphisms f h or ϕ h . As eac h suc h automorphism gets mapp ed to [ Q ⊗ L h ⊗ Q ∨ ], due to exactness of the sequence (4.4) differen t c hoices of f h lead to automor phisms whic h differ only by inner automorphisms. On the other hand, the mapping h 7→ α h need not b e a homo mo r phism of groups for any c hoice of the isomor phisms f h . In the following we will formulate necessary and sufficien t conditions that H ≤ Pic( D ) m ust satisfy for t he assignmen t h 7→ α h to yield a gro up homomorphism fro m H to Aut( A ). Recall that L g ⊗ L h ∼ = L g h . Th us b y lemma 2.13 (iv) the spaces Ho m( L g ⊗ L h , L g h ) are one-dimensional (but there is no canonical c hoice of an isomor phism to the g round field k ). F or each pair 18 g , h ∈ Pic( D ) w e select a basis isomorphism g b h ∈ Hom( L g ⊗ L h , L g h ). W e denote their in v erses by g b h ∈ Hom( L g h , L g ⊗ L h ), i.e. g b h ◦ g b h = id L g ⊗ L h and g b h ◦ g b h = id L gh . F or g = 1 w e tak e 1 b g and g b 1 to b e the iden tit y , which is p ossible b y the assumed strictness of D . F or an y triple g 1 , g 2 , g 3 ∈ Pic( D ) the collection { g b h } of morphisms provide s us with tw o bases of the one-dimensional space Hom( L g 1 ⊗ L g 2 ⊗ L g 3 , L g 1 g 2 g 3 ), namely with g 1 g 2 b g 3 ◦ ( g 1 b g 2 ⊗ id L g 3 ) as w ell as g 1 b g 2 g 3 ◦ (id L g 1 ⊗ g 2 b g 3 ). These differ b y a nonzero scalar ψ ( g 1 , g 2 , g 3 ) ∈ k : g 1 g 2 b g 3 ◦ ( g 1 b g 2 ⊗ id L g 3 ) = ψ ( g 1 , g 2 , g 3 ) g 1 b g 2 g 3 ◦ (id L g 1 ⊗ g 2 b g 3 ) . (4.23) The p entagon a xiom for the asso ciativity constraints of D implies that ψ is a three-co cycle o n the group Pic( D ) with v alues in k × (see e.g. app endix E of [MS], chapter 7.5 of [FK], or [Y a]). An y other c hoice of bases leads to a cohomolog o us t hree-cocycle. Observ e that b y taking 1 b h and h b 1 to b e the identit y on L h the co cycle ψ is normalised, i.e. satisfies ψ ( g 1 , g 2 , g 3 ) = 1 as so on as one of t he g i equals 1. Lemma 4.6. F or g , h, k ∈ Pic( D ) , the b ases intr o duc e d ab ove ob ey the r elation L k L h L k − 1 g L h − 1 g k b k − 1 g h b h − 1 g = 1 ψ ( h, h − 1 k , k − 1 g ) L k L h L k − 1 g L h − 1 g h b h − 1 g h − 1 k b k − 1 g (4.24) Pr o of. In pictures: L k − 1 g L h − 1 g L k L h h b h − 1 g k b k − 1 g = L k − 1 g L h − 1 g L k L h h − 1 k b k − 1 g k b k − 1 g h − 1 k b k − 1 g h b h − 1 g = 1 ψ L k − 1 g L h − 1 g L k L h h b h − 1 k h − 1 k b k − 1 g k b k − 1 g k b k − 1 g = 1 ψ L k − 1 g L h − 1 g L k L h h b h − 1 k h − 1 k b k − 1 g (4.25) where in the second step we inserted the definition of ψ and abbreviated ψ ≡ ψ ( h, h − 1 k , k − 1 g ). Definition 4.7. Give n the normalised co cycle ψ on Pic( D ), a no rmalised t w o -co c hain ω on H with v alues in k × is called a trivialisation of ψ on H iff it satisfies d ω = ψ   H . Prop osition 4.8. Given a finite sub gr oup H of Pic( D ) and a function ω : H × H → k × , defi n e Q := L h ∈ H L h and m ≡ m ( H , ω ) := P g ,h ∈ H ω ( g , h ) e g h ◦ g b h ◦ ( r g ⊗ r h ) ∈ Hom( Q ⊗ Q, Q ) , η ≡ η ( H , ω ) := e 1 ∈ Hom( 1 , Q ) , ∆ ≡ ∆( H , ω ) := | H | − 1 P g ,h ∈ H ω ( g , h ) − 1 ( e g ⊗ e h ) ◦ g b h ◦ r g h ∈ Hom( Q, Q ⊗ Q ) , ε ≡ ε ( H , ω ) := | H | r 1 ∈ Hom( Q, 1 ) . (4.26) Then the fol lowing statements ar e e q uiva lent: (i) ω is a trivialisation of ψ on H . 19 (ii) ( Q, m, η ) is an asso cia tive unital algeb r a. (iii) ( Q, ∆ , ε ) is a c o asso ciative c ounital c o algebr a. Mor e over, if any of these e quivalent c ond i tion s hol d s, then Q ( H , ω ) ≡ ( Q, m, ∆ , η , ε ) is a sp e c i a l F r ob enius algebr a with m ◦ ∆ = id Q and ε ◦ η = | H | id 1 . Pr o of. The equiv a lence of conditions (i) – (iii) follows by direct computation using only the defi- nitions; w e refrain from g iving the details. The F rob enius prop ert y then follow s with the help of lemma 4.6. Lemma 4.9. L et ω b e a trivialisation of ψ on H ≤ Pic( D ) and let Q ≡ Q ( H , ω ) b e the sp e cial F r ob enius algebr a define d in pr op o s i tion 4.8. Then Q is symmetric iff dim( Q ) 6 = 0 . Pr o of. Recall t he morphisms Φ 1 and Φ 2 from (2.10). Since Q = L h ∈ H L h , the condition Φ 1 = Φ 2 is equiv alen t to Φ 1 ◦ e g = Φ 2 ◦ e g for a ll g ∈ H . By the definition of the m ultiplication on Q , this amoun ts to ω ( g , g − 1 ) L ∨ g − 1 1 L g g b g − 1 = ω ( g − 1 , g ) L ∨ g − 1 1 L g g − 1 b g (4.27) whic h in turn, by a pplying duality morphisms and comp osing with the morphisms g b g − 1 , is equiv- alen t to ω ( g , g − 1 ) 1 1 g b g − 1 g b g − 1 = ω ( g − 1 , g ) 1 g − 1 b g g b g − 1 (4.28) The righ t hand side of (4.2 8) is ev aluat ed t o ω ( g − 1 , g ) 1 g − 1 b g g b g − 1 = ω ( g − 1 , g ) ψ ( g − 1 , g , g − 1 ) 1 g − 1 b 1 1 b g − 1 = ω ( g − 1 , g ) ψ ( g − 1 , g , g − 1 ) L g = ω ( g − 1 , g ) ψ ( g − 1 , g , g − 1 ) dim l ( L g ) id 1 . (4.29) The first step is an application of lemma 4.6, the second is due to our con v en tion that the mo r - phisms 1 b g are chose n t o b e iden tit y morphisms. So the condition that Q is a symmetric F rob enius algebra is equiv a len t to t he condition ω ( g , g − 1 ) = ω ( g − 1 , g ) ψ ( g − 1 , g , g − 1 ) dim l ( L g ) for a ll g ∈ H . As ω is a trivialisation of ψ and d ω ( g − 1 , g , g − 1 ) = ω ( g , g − 1 ) /ω ( g − 1 , g ), this is equiv alen t to dim l ( L g ) = 1 for all g ∈ H . By lemma 2.1 4 the la tter condition holds iff dim( Q ) 6 = 0. Definition 4.10. An a dmissible subgroup of Pic( D ) is a finite subgroup H ≤ Pic( D ) such that dim l | r ( L h ) = 1 for all h ∈ H and suc h that there exists a t rivialisation ω o f ψ on H . 20 Remark 4.11. W e see tha t Q ( H, ω ) is a symmetric special F rob enius algebra if a nd only if H is an admissible subgroup of Pic( D ) and ω is a trivialisation o f ψ on H . One can sho w that ev ery structure o f a sp ecial F rob enius algebra on the ob ject Q = L h ∈ H L h is of the t yp e Q ( H , ω ) described in prop o sition 4.8 for a suitable trivialisation ω of ψ (see [FR S2], prop osition 3.14) . So giving pro duct and copro duct mo r phisms on Q is equiv alen t to giving a trivialisation ω of ψ . Also observ e that multiply ing a trivialisation ω of ψ with a tw o-co cycle γ on H g iv es another triv- ialisation ω ′ of ψ . One can sho w that the F rob enius algebras Q ( H , ω ) and Q ( H, ω ′ ) ar e isomorphic as F ro b enius algebras if and only if ω and ω ′ differ by multiplication with an exact t w o-co cycle. Accordingly , in the sequel we call tw o trivialisations ω and ω ′ for ψ equiv alen t iff ω /ω ′ = d η for some one-co chain η . Th us if ψ is trivialisable on H , then the equiv alence classes o f trivialisations form a torsor o v er H 2 ( H , k × ). Theorem 4.12. L et H b e an admissible sub gr oup of Pic( D ) , put Q = Q ( H ) as in (4.6), a nd let A = A ( H ) = Q ⊗ Q ∨ b e the algebr a defi n e d in (4.7 ). Then ther e is a bije ction b etwe en • trivialisations ω of ψ on H and • gr oup homo m orphisms α : H → Aut( A ) with Π Q ∨ ,Q ◦ Ψ A ◦ α = id H . Pr o of. Denote b y T the set of all trivialisations ω of ψ o n H , and b y H the set of all group homomorphisms α : H → Aut ( A ) satisfying Π Q ∨ ,Q ◦ Ψ A ◦ α = id H . The pro o f that T ∼ = H a s sets is or g anised in three steps: defining maps F : T → H and G : H → T , and sho wing t hat they are eac h other’s inv erse. (i) Let ω ∈ T . F or eac h h ∈ H define f h := X g ∈ H ω ( g , h ) Q L h Q g b h e gh r g (4.30) Since ω t a k es v alues in k × , these are in fact isomorphisms, with inv erse g iven by f − 1 h = X g ∈ H ω ( g , h ) − 1 ( e g ⊗ id L h ) ◦ g b h ◦ r g h . (4.31) Define the function F ( ω ) fro m H to Aut ( A ) b y F ( ω ): h 7→ α h for α h giv en b y (4.9) with f h as in (4.30). W e pro ceed to sho w t hat F ( ω ) ∈ H . Abbreviate α ≡ F ( ω ). That Π Q ∨ ,Q ◦ Ψ A ◦ α = id H follo ws from the pro of of prop osition 4.3 . T o see 21 that α ( g ) ◦ α ( h ) = α ( g h ), first rewrite α ( g ) ◦ α ( h ) using ( 4.30) and ( 4 .31): α ( g ) ◦ α ( h ) = X k ,l,m,n ω ( k , g ) ω ( l , h ) ω ( m, g ) ω ( n, h ) Q ∨ Q ∨ Q Q m b g n b h k b g ∨ l b h ∨ r k ∨ r l ∨ r mg r nh e m e n e kg ∨ e lh ∨ L g L h = X k ,m ω ( k , g ) ω ( k g , h ) ω ( m, g ) ω ( mg , h ) Q ∨ Q ∨ Q Q m b g mg b h k b g ∨ kg b h ∨ r k ∨ r mgh e m e kg h ∨ L g L h = X k ,m ξ k m Q ∨ Q ∨ Q Q g b h m b gh g b h ∨ k b gh ∨ r k ∨ r mgh e m e kg h ∨ = X k ,m ξ k m Q ∨ Q ∨ Q Q m b gh k b gh ∨ r k ∨ r mgh e m e kg h ∨ L gh (4.32) with ξ k m = ω ( k , g ) ω ( k g , h ) ψ ( k , g , h ) ω ( m, g ) ω ( mg , h ) ψ ( m, g , h ) . (4.33) Here the second step uses that there a re no nonzero morphisms L n → L mg unless n = mg ; b y the same argument w e conclude that l = k g . In the third step one applies relation (4.23). No w the condition α ( g ) ◦ α ( h ) = α ( g h ) is equiv alen t to ω ( k , g ) ω ( k g , h ) ψ ( k , g , h ) ω ( m, g ) ω ( mg , h ) ψ ( m, g , h ) = ω ( k , g h ) ω ( m, g h ) for all m, k ∈ H , (4.34) whic h in turn can b e rewritten a s d ω ( m, g , h ) d ω ( k , g , h ) = ψ ( m, g , h ) ψ ( k , g , h ) for all m, k ∈ H . (4.35) The last conditio n is satisfied b ecause b y assumption d ω = ψ | H . So indeed we hav e F ( ω ) ∈ H . (ii) Giv en α ∈ H , for eac h h ∈ H the automorphism α ( h ) satisfies the conditions of lemma 4.5. As a consequenc e we obtain a unique isomorphism f h : Q ⊗ L h → Q suc h that α ( h ) = α h and r h ◦ f h ◦ ( e 1 ⊗ id L h ) = id L h . Define a function ω : H × H → k via ω ( g , h ) L gh L h L g g b h = L gh L h L g e g f h r gh (4.36) Then define the map G from H to functions H × H → k b y G ( α ) := ω , with ω obtained as in (4.36). W e will sho w tha t G ( α ) ∈ T . 22 Giv en α ∈ H , a bbreviate ω ≡ G ( α ). First note that ω take s v a lues in k × , as f h is an isomorphism. Next, the normalisation condition r h ◦ f h ◦ ( e 1 ⊗ id L h ) = id L h implies ω (1 , h ) = 1 for all h ∈ H . Since α is a group homomorphism we ha v e α (1) = id Q ⊗ Q ∨ . By the uniqueness result of lemma 4.5 this implies that f 1 = id Q , a nd so ω ( g , 1) = 1 for all g ∈ H . Altogether it fo llo ws that ω is a normalised tw o-co c hain with v alues in k × . By follo wing ag a in the steps (4.32) t o (4.35) o ne sho ws that, since α is a group ho mo mo r phism, ω mus t satisfy (4.35 ). Setting k = 1 and using tha t ω and ψ a r e normalised finally demonstrates that d ω = ψ   H . Th us indeed G ( α ) ∈ T . (iii) That F ( G ( α )) = α is immediate by construction, and that G ( F ( ω )) = ω f ollo ws from the uniqueness result of lemma 4.5 . W e ha v e seen that if H is an admissible subgroup of Pic ( D ) and ω a trivialisation o f ψ , t hen Q ( H , ω ) is a symmetric sp ecial F r ob enius algebra. Using the pro duct and copro duct morphisms of Q ( H , ω ), the automorphisms α h induced b y ω as describ ed in theorem 4 .1 2 can b e written as α h = | H | Q ∨ Q ∨ Q Q P h (4.37) Note that in this picture the circle on the left stands for the copro duct of Q , while the circle on the righ t stands for the dual of the pro duct. Giv en a trivialisation ω of ψ , theorem 4.12 allows us to realise H as a subgro up of Aut( A ). In particular the fixed algebra under the action o f H is w ell defined; w e denote it b y A H . W e a lready kno w fro m prop osition 3.7 that A H is a symmetric F rob enius algebra. I t will turn out that it is isomorphic to Q ( H , ω ). In particular, by remark 4.11 an y equiv alen t c hoice o f a t r ivialisatio n ω ′ of ψ will give an isomorphic fixed alg ebra. Theorem 4.13. L et H b e an admi ssible sub gr oup of Pic ( D ) with trivialisation ω , put A = A ( H ) as in (4. 7), and e mb e d H → Aut( A ) as in the or em 4.12. Then the fixe d algeb r a A H is wel l define d and i t is isomorphic to Q ( H , ω ) . Pr o of. Let Q ≡ Q ( H, ω ). By lemma 2.1 4 w e ha v e dim( Q ) = | H | . Iden tify H with its image in Aut( A ) via the embedding H → Aut( A ) determined b y ω as in theorem 4.12 . Now define mor- phisms i : Q → A and s : A → Q b y i := ( m ⊗ id Q ∨ ) ◦ (id Q ⊗ b Q ) , s := (id Q ⊗ ˜ d Q ) ◦ (∆ ⊗ id Q ∨ ) . (4.38) W e hav e s ◦ i = id Q , implying that i is monic and Q is a retract of A . W e claim that i is the inclusion morphism of the fixed alg ebra. Recall from (3.3 ) the definition P = | H | − 1 P h ∈ H α h of the idempotent corresponding to t he fixed algebra ob ject. In the case under consideration, P 23 tak es the form P = Q ∨ Q ∨ Q Q (4.39) as follow s from (4.37) together with id Q = P h ∈ H P h . W e calculate that i ◦ s = P : i ◦ s = Q ∨ Q ∨ Q Q (2) = Q ∨ Q ∨ Q Q (3) = Q ∨ Q ∨ Q Q (4) = Q ∨ Q ∨ Q Q (5) = Q ∨ Q ∨ Q Q (6) = Q ∨ Q ∨ Q Q (7) = Q ∨ Q ∨ Q Q = P . (4.40) Here in the second step a counit morphism is introduced and in the third step the F rob enius prop ert y is applied. The next step uses coasso ciativit y , while the fifth step follows b ecause Q ( H , ω ) is symmetric. Then one uses the F rob enius prop erty and duality . So Q satisfies t he univ ersal prop ert y of t he image of P and hence the univ ersal prop erty of the fixed algebra. W e now calculate the pro duct morphism tha t the fixed algebra inherits from Q ⊗ Q ∨ , starting from (3.4): Q Q Q = Q Q Q = Q Q Q = Q Q Q (4.41) 24 The first step uses dualit y , the second one holds by asso ciativit y of m . In the last step o ne uses that s ◦ i = id Q . The inherited unit morphism is g iv en b y Q (4.42) whic h due to dim k Hom( 1 , Q ) = 1 is equal to ζ η f or some ζ ∈ k . Applying ε to b oth these morphisms and using that dim( Q ) = | H | , w e see t ha t ζ = 1. Similarly one shows that the copro duct and counit morphisms that Q inherits as a fixed a lgebra equal those defined in prop osition 4.8. So Q ( H , ω ) is isomorphic to the fixed algebra A H as a F rob enius algebra. Remark 4.14. The algebra structure on the ob ject Q ( H, ω ) is a kind of tw isted group algebra of t he group H whic h is not t wisted b y a closed tw o-co c hain, but rather b y a trivialisation of the asso ciator of D . Algebras of this t yp e ha v e app eared in applications in conformal field theory [FRS2]. 5 Algebras in general Morita class e s In this section w e solv e the pro blem discussed in the previous section fo r algebras that a re not Morita equiv alen t to the tensor unit. Throughout this section w e will a ssume t he f ollo wing. Con v en t ion 5.1. ( C , ⊗ , 1 ) has the prop erties listed in con v en tion 2.1 and is in addition sk eletally small and so vereign. ( A, m, η , ∆ , ε ) is a simple and absolutely simple symmetric normalised sp ecial F rob enius algebra in C , and H ≤ Pic( C A | A ) is a finite subgroup. Recall from remark 2.7 that the conditions ab ov e imply dim( A ) 6 = 0. In the sequel w e will find a symmetric sp ecial F rob enius alg ebra A ′ = A ′ ( H ) and a Morita con text A P ,P ′ ← → A ′ in C , suc h that H is a subgroup of im(Π P ,P ′ ◦ Ψ A ′ ), where Π P ,P ′ is the isomorphism in t r o duced in (4.5). This generalises the results of prop osition 4 .3. W e will apply some of the results of the previous section t o the strictification D of the categor y C A | A . Note that D has the pro p erties stated in conv en tion 2.1 and is in addition sov ereign, as can b e seen by straigh tforward calculatio ns whic h ar e pa rallel to those of [FS] fo r the category C A of left A -mo dules. By applying the inv erse equiv alence f unctor D ≃ − → C A | A this will then yield a sym- metric sp ecial F rob enius algebra in C A | A that ha s the desired pro p erties. No t e that the gr a phical represen tations of morphisms used b elow are mean t t o represen t morphisms in C . P ertinent f acts ab out the structure o f the category C A | A are collected in app endix A; in the sequel w e will f reely use the terminology presen ted there. It is w orth emphasising that fo r establishing v a rious of the results b elo w, it is essen tial tha t A is not just an algebra in C , but ev en a simple and absolutely simple symmetric sp ecial F r ob enius algebra. 25 As a first step w e study ho w concepts like algebras and mo dules ov er algebras can b e tra ns- p orted fro m C A | A to C . If X is an ob ject of C A | A , i.e. an A -bimo dule in C , we denote the corre- sp onding ob ject of C b y ˙ X . Prop osition 5.2. (i) L et ( B , m B , η B ) b e an algebr a in C A | A . Th e n ( ˙ B , m B ◦ r B ,B , η B ◦ η ) is an algebr a in C . (ii) If ( C , ∆ C , ε C ) is a c o algebr a in C A | A , then ( ˙ C , e C,C ◦ ∆ C , ε ◦ ε C ) is a c o algebr a in C . (iii) A morphism γ : B → B ′ of algebr as in C A | A is also a morphism of algebr as in C . (iv) L et ( B , m B , η B ) b e an algeb r a in C A | A and ( M , ρ ) b e a left B -mo dule in C A | A . The n ( ˙ M , ρ ◦ r B ,M ) is a left ˙ B -mo dule in C . Similarly right B -mo dules and B -bimo dules in C A | A c an b e tr ansp orte d to C . F urther, if f : ( M , ρ M ) → ( N , ρ N ) is a morphism of left B -mo d ules in C A | A , then f is also a mo rp hism of left ˙ B -mo dules in C , and an a n alo gous statement h o lds fo r morphisms of right- and bimo dules. (v) If ( B , m B , ∆ B , η B , ε B ) is a F r ob enius algebr a in C A | A , then ( ˙ B , m ◦ r B ,B , e B ,B ◦ ∆ , η B ◦ η , ε ◦ ε B ) is a F r ob enius algebr a in C . If B is sp e cial in C A | A , then ˙ B is sp e cial in C . If B is symm etric in C A | A , then ˙ B is symmetric in C . Pr o of. (i) That m B is an asso ciativ e pro duct fo r B in C A | A means that m B ◦ (id B ⊗ A m B ) ◦ α B ,B ,B = m B ◦ ( m B ⊗ A id B ) , (5.1) where α B ,B ,B is the a sso ciator defined in (A.4) . After inserting the definitions of the tensor pro duct of mo r phisms and the asso ciator α B ,B ,B this reads ( B ⊗ A B ) ⊗ A B B m B m B r B,B r B,B ⊗ A B r B,B e B,B e B,B ⊗ A B e B ⊗ A B ,B = ( B ⊗ A B ) ⊗ A B B m B m B e B ⊗ A B ,B r B,B (5.2) After comp osing b oth sides of this equalit y with the morphism r B ⊗ A B ,B ◦ ( r B ,B ⊗ id B ) t he resulting idemp oten ts P B ,B ⊗ A B , P B ⊗ A B ,B and P B ,B can b e dropp ed from the left hand side, and P B ⊗ A B ,B from the righ t ha nd side. W e see that m B ◦ r B ,B is indeed an asso ciative pro duct for ˙ B in C . Next 26 consider the morphism η B ; we hav e B B m B η B r B,B = B B m B r B,B η B = B B m B r B,B η B = B B m B r B,B η B e B,A r B,A = m B ◦ (id B ⊗ A η B ) ◦ ρ A ( B ) − 1 = id B , (5.3) with ρ A the unit constraint as given b y (A.7) in the app endix. Here in the first step the idemp otent P B ,B is intro duced and then mo v ed down w ards, and lik ewise in the third step. The fift h step is the unit prop erty of η B in C A | A . So w e see that η B ◦ η is indeed a right unit for ˙ B in C , similarly one sho ws that it is also a left unit. (ii) is prov ed analo gously to t he preceding statemen t. (iii) Let m B and m B ′ denote the pro ducts of B and B ′ in C A | A . Then γ ◦ m B ◦ r B ,B = m B ′ ◦ ( γ ⊗ A γ ) ◦ r B ,B = m B ′ ◦ r B ′ ,B ′ ◦ ( γ ⊗ γ ) ◦ P B ,B = m B ′ ◦ r B ′ ,B ′ ◦ P B ′ ,B ′ ◦ ( γ ⊗ γ ) = m B ′ ◦ r B ′ ,B ′ ◦ ( γ ⊗ γ ) , (5.4) where the third equalit y uses that γ is a morphism in C A | A . F urther we hav e γ ◦ η B ◦ η = η B ′ ◦ η , as γ resp ects the unit η B of B in C A | A . So γ is also a morphism of alg ebras in C . (iv) The statemen t that M is a left B -mo dule in C A | A reads ρ ◦ (id B ⊗ A ρ ) ◦ α B ,B ,M = ρ ◦ ( m B ⊗ A id M ) , (5.5) whic h is an equality in Hom A | A (( B ⊗ A B ) ⊗ A M , M ). Similarly to the pro o f in i), one shows tha t this indeed implies that ( M , ρ ◦ r B ,M ) is a left ˙ B -mo dule in C . No w for an y morphism f : ( M , ρ M ) → ( N , ρ N ) we hav e f ◦ ρ M ◦ r B ,M = ρ N ◦ (id B ⊗ A f ) ◦ r B ,M = ρ N ◦ r B ,N ◦ (id B ⊗ f ) ◦ P B ,M = ρ N ◦ r B ,N ◦ P B ,N ◦ (id B ⊗ f ) = ρ N ◦ r B ,N ◦ (id B ⊗ f ) , (5.6) sho wing that f is a mor phism of left ˙ B -mo dules in C . Similarly o ne v erifies the conditions for righ t- and bimo dules. (v) T o see that the pro duct and copro duct morphisms for ˙ B satisfy the F rob enius prop erty in C 27 consider the following calculation: B B B B m B ∆ B e B,B r B,B = B B B B m B ∆ B r B,B e B,B r B,B ⊗ A B e B ⊗ A B ,B e B,B r B,B α − 1 B,B ,B = B B B B m B r B,B ⊗ A B e B ⊗ A B,B r B ⊗ A B,B r B,B e B,B ⊗ A B e B,B ∆ B = B B B B m B r B,B ∆ B e B,B (5.7) The first equalit y is the assertion that B is a F ro b enius algebra in C A | A , the second one implemen ts the definition of α − 1 B ,B ,B . In the last step the resulting idemp oten ts are mo v ed up o r down, up on whic h they can b e dropp ed. A parallel argumen t establishes the second iden tit y . Similarly one che c ks that sp ecialness and symmetry o f ˙ B are transp orted to C as well. T o apply t he results of the previous section to the algebra A we also need to deal with Morita equiv alence in C A | A . W e start with the follow ing observ ation. Lemma 5.3. L et B b e a symmetric sp e cial F r ob enius algebr a in C A | A , and C and D b e algebr as in C A | A and let ( C M B , ρ C ,  B ) b e a C - B - b imo dule and ( B N D , ρ B ,  D ) a B - D -bim o dule. Then the tensor pr o duct M ⊗ B N in C A | A is is o m orphic, as a ˙ C - ˙ D -bimo d ule in C , to the tensor pr o d uct ˙ M ⊗ ˙ B ˙ N over the algebr a ˙ B in C . Pr o of. Since B is symmetric sp ecial F rob enius in C A | A , the idempo ten t P B M ,N corresp onding to the tensor pro duct of M and N o v er B is w ell defined in C A | A . Explicitly it reads (  B ⊗ A ρ B ) ◦ α − 1 M ,B , B ⊗ A N ◦ (id M ⊗ A α B ,B ,N ) ◦ (id M ⊗ A (∆ B ◦ η B ) ⊗ A id N )) ◦ (id M ⊗ A λ A ( N ) − 1 ) . (5.8) One calculates that this equals the morphism g iv en b y the following morphism in C : M ⊗ A N M ⊗ A N  B ρ B η B ∆ B e M ,N e B,B r B,N r M ,B r M ,N (5.9) Comp osing with the mor phisms e M ,N and r M ,N for the t ensor pro duct ov er A , we see that P ˙ B ˙ M , ˙ N = e M ,N ◦ P B M ,N ◦ r M ,N = e M ,N ◦ e B M ,N ◦ r B M ,N ◦ r M ,N . This furnishes a differen t decomp osition 28 of P ˙ B ˙ M , ˙ N in to a monic and an epi, hence there is an isomorphism f : M ⊗ B N ∼ = → ˙ M ⊗ ˙ B ˙ N of the images of P B M ,N and P ˙ B ˙ M , ˙ N , suc h that f ◦ r B M ,N ◦ r M ,N = r ˙ B ˙ M , ˙ N and e M ,N ◦ e B M ,N = e ˙ B ˙ M , ˙ N ◦ f . Now the left action of C on M ⊗ B N is given by r B M ,N ◦ ( ρ C ⊗ A id N ) ◦ α − 1 C,M ,N ◦ (id C ⊗ A e B M ,N ) = C ⊗ A ( M ⊗ B N ) M ⊗ B N ρ C r B M ,N e C ⊗ A M ,N r C ⊗ A M ,N e C,M ⊗ A N r C,M ⊗ A N e M ,N r C,M r M ,N e B M ,N e C,M ⊗ B N = C ⊗ A ( M ⊗ B N ) M ⊗ B N ρ C r B M ,N e M ,N r C,M r M ,N e B M ,N e C,M ⊗ B N (5.10) Comp ose this morphism from the righ t with r C,M ⊗ B N and drop the resulting idemp o t ent to get the transp orted left action of ˙ C . No w comp o sing with f from the left and replacing f ◦ r B M ,N ◦ r M ,N b y r ˙ B ˙ M , ˙ N and e M ,N ◦ e B M ,N b y e ˙ B ˙ M , ˙ N ◦ f sho ws that f also intert wines the left actions of ˙ C . Similar ly one sho ws that f is also an isomorphism o f ˙ D -right mo dules. Corollary 5.4. Assume that B and C ar e symmetric sp e cial F r ob enius alg ebr as in C A | A , and that B P ,P ′ ← → C is a Morita c on text i n C A | A . Th e n ˙ B ˙ P , ˙ P ′ ← → ˙ C is a Morita c o ntext in C . Pr o of. F ollo ws fro m lemma 5.3 ab o v e. The commutativit y of the diagra ms (2.15) in C follow s from t he commutativit y of their coun terparts in C A | A . W e are now in a p osition to generalise the results of the previous section to t he case of algebras in arbitrary Morita classes. Prop osition 5.5. L e t H ≤ Pic ( C A | A ) b e a finite s ub gr oup of the Pic ar d gr oup of C A | A and assume that dim A ( L h ∈ H L h ) 6 = 0 for r epr esentatives L h of H in C A | A . Then ther e exists an algebr a A ′ in C and a Morita c ontext A P ,P ′ ← → A ′ in C such that Π P ′ ,P maps H in to the ima ge of Ψ A ′ in Pic( C A ′ | A ′ ) . In other wor ds, for any h ∈ H ther e is an alg e b r a automorphism β h of A ′ such that the twiste d bimo dule id A ′ β h is iso m orphic to ( P ′ ⊗ A L h ) ⊗ A P . Pr o of. Applying prop osition 4.3 to the tensor unit of D yields, b y the equiv alence D ≃ C A | A , a symmetric sp ecial F rob enius algebra B in C A | A and a Morita contex t A Q,Q ′ ← → B in C A | A , suc h that there are automorphisms β h of B and B -bimo dule isomorphisms F h : ( Q ′ ⊗ A L h ) ⊗ A Q ∼ = − → id B β h for all h ∈ H . By prop o sition 5 .2 and coro llary 5 .4 this giv es rise to a Morita con text ˙ A P ,P ′ ← → ˙ B in C , where P = ˙ Q and P ′ = ˙ Q ′ . The morphisms F h remain isomorphisms of bimo dules when transp orted to C , see prop osition 5.2. It follow s that ( P ′ ⊗ A L h ) ⊗ A P ∼ = id ˙ B β h as ˙ B -bimo dules in C . 29 Since the algebra ˙ B migh t hav e a larger auto morphism g roup in C , its image under Ψ ˙ B migh t b e larger in Pic( C ˙ B | ˙ B ). So we cannot conclude that H ∼ = im(Ψ ˙ B ) in this case. But as w e ha ve Aut A | A ( B ) ≤ Aut C ( ˙ B ) a s subgroups, w e can still generalise t heorem 4.1 2. This furnishes the main result of this pap er: Theorem 5.6. L et C b e a skeletal ly smal l sover eign ab elian monoidal c ate gory with simple and absolutely sim p le tensor unit that is enriche d ove r V ect k , with k a field of ch ar acteristic zer o. L et A b e a simple and absolutely simple symmetric sp e cial F r ob enius algebr a in C , and let ψ b e a normali se d thr e e-c ocycle describing the asso ciator of the Pic ar d c ate g ory o f C A | A . L et H b e an admissible s ub gr oup of Pic( C A | A ) (cf. definition 4.10). Then ther e exist a symm e tric sp e cial F r ob enius al g ebr a A ′ in C and a Morita c ontext A P ,P ′ ← → A ′ such that for e ach trivialisation ω of ψ on H (cf. definition 4.7) the fol lo w ing holds. (i) Ther e is an inje ctive homomorp h ism α ω : H → Aut( A ′ ) such that Π P ,P ′ ◦ Ψ A ′ ◦ α ω = id H . Th e assignment ω 7→ α ω is inje ctive. (ii) The fixe d alge b r a o f im( α ω ) ≤ Aut( A ′ ) is isomorp h ic to ˙ Q ( H , ω ) , wher e ˙ Q ( H , ω ) is the a lgebr a Q ( H , ω ) i n C A | A as describ e d in pr op osition 4.8, tr a n sp orte d to C . Pr o of. (i) Denote aga in by D the strictification of the bimo dule category C A | A . By pro p ositions 4.3 and 4.8 and theorem 4.1 2 w e find a symmetric sp ecial F ro b enius algebra B in C A | A and a Morita context A Q,Q ′ ← → B in C A | A suc h that there is a homomo r phism φ ω : H → Aut A | A ( B ) with Π Q,Q ′ ◦ Ψ B : Aut A | A ( B ) → Pic( C A | A ) as one-sided in v erse. According to prop osition 5.2, ˙ B is a symmetric sp ecial F rob enius algebra in C and A P ,P ′ ← → ˙ B is a Morita con text in C with P = ˙ Q and P ′ = ˙ Q ′ . F urther, w e can extend Ψ B to Aut C ( ˙ B ) by putting Ψ ˙ B ( γ ) = [ id ˙ B γ ] for γ ∈ Aut C ( ˙ B ). Since w e hav e Aut A | A ( B ) ⊆ Aut C ( ˙ B ) as a subgroup, φ ω then give s a homomorphism α ω : H → Aut C ( ˙ B ) that ha s Π P ,P ′ ◦ Ψ ˙ B as one-sided in v erse. As w as seen in theorem 4.12, the assignmen t ω 7→ φ ω is a bijection, and hence the assignmen t ω 7→ α ω is still injectiv e. (ii) Put β h := α ω ( h ). F ro m theorem 4.13 w e know that the algebra Q ( H , ω ) is isomorphic to the fixed algebra B H in C A | A . It comes together with a n algebra morphism i : Q ( H, ω ) → B and a coalgebra morphism s : B → Q ( H , ω ) suc h that s ◦ i = id Q ( H,ω ) and i ◦ s = | H | − 1 P h ∈ H β h . No w let f ∈ Hom C ( X , ˙ B ) b e a morphism ob eying β h ◦ f = f for all h ∈ H . Then for ¯ f := s ◦ f : X → ˙ Q ( H , ω ) w e find i ◦ ¯ f = i ◦ s ◦ f = 1 | H | P h ∈ H β h ◦ f = f , i.e. ˙ Q ( H , ω ) satisfies the univ ersal prop ert y of the fixed algebra in C . So ˙ B H ∼ = ˙ Q ( H , ω ) as ob jects in C . By prop osition 5.2, i is still a morphism of F rob enius algebras when t r ansp orted to C . It follows that ˙ B H ∼ = ˙ Q ( H , ω ) a s F rob enius alg ebras in C . 30 A App endix Here w e collect some facts ab out the category of bimo dules ov er an algebra in a monoida l category . Let ( C , ⊗ , 1 ) b e a an ab elian so v ereign strict monoidal category , enric hed o v er V ect k with k a field of c haracteristic zero, and with simple and a bsolutely simple tensor unit. Let A b e an alg ebra in C . The tensor pro duct X ⊗ A Y of t w o A -bimo dules X ≡ ( X , ρ X ,  X ) and Y ≡ ( Y , ρ Y ,  Y ) is defined to b e the cokernel of the morphism (  X ⊗ id Y − id X ⊗ ρ Y ) ∈ Hom( X ⊗ A ⊗ Y , X ⊗ Y ). In the follo wing w e will only deal with tensor pro ducts ov er symmetric sp ecial F rob enius algebras. In this case the notion of tensor pro duct can equiv alen tly b e describ ed as follows. Let ( A, m, η , ∆ , ε ) b e a symmetric normalised sp ecial F rob enius algebra. Consider the morphism P X,Y := Y Y X X ∈ Hom A | A ( X ⊗ Y , X ⊗ Y ) . (A.1) Since A is symmetric sp ecial F rob enius, P X,Y is an idemp o t en t. W riting P X,Y = e X,Y ◦ r X,Y as a comp osition of a monic e X,Y and an epi r X,Y , one can c heck that the morphism r X,Y satisfies the univ ersal prop ert y of the cok ernel of (  X ⊗ id Y − id X ⊗ ρ Y ). The ob ject X ⊗ A Y is in the bimo dule category C A | A again. Indeed, rig ht and left a ctio ns of A on X ⊗ A Y can b e defined b y X ⊗ A Y X ⊗ A Y A r X,Y e X,Y and X ⊗ A Y X ⊗ A Y A r X,Y e X,Y (A.2) The t ensor pro duct o v er A of t w o morphisms f ∈ Hom A | A ( X , X ′ ) a nd g ∈ Hom A | A ( Y , Y ′ ) is defined as f ⊗ A g := r X ′ ,Y ′ ◦ ( f ⊗ g ) ◦ e X,Y . (A.3) So in pa rticular the morphisms e X,Y , r X,Y and f ⊗ A g are morphisms of A -bimo dules. F or an y three A - bimo dules X , Y , Z one defines α X,Y ,Z := X ⊗ A ( Y ⊗ A Z ) ( X ⊗ A Y ) ⊗ A Z r X,Y ⊗ A Z e X,Y r Y ,Z e X ⊗ A Z,Y ∈ Hom A | A  ( X ⊗ A Y ) ⊗ A Z , X ⊗ A ( Y ⊗ A Z )  , (A.4) 31 whic h is a morphism of A -bimo dules. These morphisms ar e in fact isomorphisms and ha ve the follo wing prop erties: X ⊗ A ( Y ⊗ A Z ) X Y Z r X ⊗ A Y ,Z α X,Y ,Z r X,Y = X ⊗ A ( Y ⊗ A Z ) X Y Z r X,Y ⊗ A Z r Y ,Z and ( X ⊗ A Y ) ⊗ A Z X Y Z e X,Y ⊗ A Z α X,Y ,Z e Y ,Z = ( X ⊗ A Y ) ⊗ A Z X Y Z e X ⊗ A Y ,Z e X,Y (A.5) This can b e seen b y writing out α X,Y ,Z and letting the o ccurring idemp oten ts disapp ear using the prop erties of A as a symmetric sp ecial F rob enius alg ebra. An easy , alb eit length y , calculation then sho ws that the isomorphisms α X,Y ,Z ob ey the p en ta gon condition for the asso ciativity constraints in a monoidal catego ry , i.e. one has (id U ⊗ A α V , W,X ) ◦ α U,V ⊗ A W,X ◦ ( α U,V ,W ⊗ A id X ) = α U,V ,W ⊗ A X ◦ α U ⊗ A V , W,X (A.6) for an y quadruple U , V , W , X of A -bimo dules. F or an A -bimo dule M , unit constrain ts are giv en b y ρ A ( M ) = M M ⊗ A A e M ,A and λ A ( M ) = M A ⊗ A M e A,M (A.7) with in v erses ρ A ( M ) − 1 = M M ⊗ A A r M ,A and λ A ( M ) − 1 = M A ⊗ A M r A,M (A.8) This turns the category C A | A in to a (non-strict) monoida l category . W e will no w see that C A | A is sov ereign. Let M b e an A -bimo dule and M ∨ its dual as an ob ject of C . M ∨ b ecomes a n A -bimo dule by defining left and r ig h t a ctio ns o f A a s M ∨ M ∨ A := M ∨ M ∨ A and M ∨ M ∨ A := M ∨ M ∨ A (A.9) 32 The structural morphisms of left and righ t dua lit ies for C A | A are giv en b y b A M = M ⊗ A M ∨ A r M ,M ∨ , d A M = M ∨ ⊗ A M e M ∨ ,M A ˜ d A M = M ⊗ A M ∨ e M ,M ∨ A , ˜ b A M = M ∨ ⊗ A M r M ∨ ,M A (A.10) One c heck s that this indeed furnishes dualities o n C A | A , and that they coincide with those of C not only on ob jects, but a lso on morphisms. Thus if C is sov ereign, t hen so C A | A . One also easily v erifies that C A | A is ab elian and t ha t its morphism gr oups a re k -v ector spaces with Hom A | A ( M , N ) ⊂ Hom C ( M , N ) as subs paces, and hence dim k Hom A | A ( M , N ) ≤ dim k Hom C ( M , N ). F or M an A -bimo dule, w e denote its left and righ t dimension as an o b ject of C A | A b y dim l A ( M ) and dim r A ( M ), resp ectiv ely . 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