On endomorphism algebras of functors with non-compact domain

On endomorphism algebras of functors with non-compact domain Brian Da y No v e m ber 20, 2018 Abstract As a development o f [2] and [3], we construct a “VN- core” in V ect k for eac h k -linear split-semigroupal functor from a suitable monoidal category C to V ect k . The main ai m here is to a v oid the customary compactness assumption on th e set o f generators of the domain category C (cf. [3]). 1 In tro duction W e prop ose the cons truction o f a VN-cor e as so ciated to each ( k -linear) split semigroupa l functor U from a suitable monoidal categor y C to V ect k , wher e all our categor ie s, functors, and natur al t ransforma tions are assumed to b e k - linear, fo r a fixed field k . E s sentially , the catego ry C must b e e quipped with a small “ U -generato r” A carry ing some extra duality infor mation and with U A still b eing finite dimensiona l for all A in A . W e shall use the term “VN-core” (in V ect k ) to mean a (usual) k -semibialgebr a E together with a k -linear endomor phism S suc h that µ ( µ ⊗ 1)(1 ⊗ S ⊗ 1)(1 ⊗ δ ) δ = 1 : E → E . The VN-core is called “an tipo dal” if S ( xy ) = S y S x (and S (1 ) = 1) for all x, y ∈ E . This minimal type of structure is intro duced her e in o r der to av oid compactness assumptions on the ge nerator A ⊂ C and, at the s ame time, retain the “fusion” op erator , namely ( µ ⊗ 1 )(1 ⊗ δ ) : E ⊗ E → E ⊗ E , satisfying the usual fusion equa tion [7]. Note that here the fusion op era tor alwa ys has a partial inverse (see [1]). In § 2 we establish sufficient conditions o n a functor U in o rder that End ∨ U = Z A ( U A ) ∗ ⊗ U A be a VN-core in V ect k (following [2 ]). This co re can be completed to a VN-core End ∨ U ⊕ k with a unit elemen t. In § 3 w e give several ex amples o f suitable functors U for the theory . 1 2 The construction of End ∨ U Let C = ( C , ⊗ , I ) b e a monoidal categ ory and let U : C → V ect be a functor with b oth a semigr oupal structure, denoted r = r C,D : U C ⊗ U D → U ( C ⊗ D ) , and a cosemigro upal structure, deno ted i = i C,D : U ( C ⊗ D ) → U C ⊗ U D , such that r i = 1. W e shall supp ose also that ther e exists a small full sub categor y A of C with the prop erties : 1. U A is finite dimensiona l for all A ∈ A , 2. U -densit y; the canonical ma p α C : Z A C ( A, C ) ⊗ U A → U C is an isomo rphism for all C ∈ C , 3. there is an “a nt ipo de” functor ( − ) ∗ : A op → A with a (“ca nonical”) map e A : A ⊗ A ∗ ⊗ A → A in C for each A ∈ A , 4. there is a natur al iso morphism u = u A : U ( A ∗ ) ∼ = − → U ( A ) ∗ , 5. the fo llowing diagrams defining ˜ τ , ˜ ρ b oth c omm ute U A ⊗ U ( A ) ∗ ⊗ U A U A ⊗ U ( A ∗ ) ⊗ U A U ( A ⊗ A ∗ ⊗ A ) U A ˜ τ / / 1 ⊗ u − 1 ⊗ 1 9 9 r r r r r r r r r r r r 3 % % L L L L L L L L L L L e U A % % L L L L L L L L L L L L U e A y y r r r r r r r r r r r r and U A ⊗ U ( A ) ∗ ⊗ U A U A ⊗ U ( A ∗ ) ⊗ U A U ( A ⊗ A ∗ ⊗ A ) U A ˜ ρ o o 1 ⊗ u ⊗ 1 y y r r r r r r r r r r r i 3 e e L L L L L L L L L L L e U A % % L L L L L L L L L L L L U e A y y r r r r r r r r r r r r where e U A = 1 ⊗ ev in V ect , a nd r 3 i 3 = 1 . 2 W e now define the semibialgebra str uctur e (End ∨ U, µ, δ ) on End ∨ U = Z A U ( A ) ∗ ⊗ U A as in [2] § 2, with the isomo rphism of k -linear spaces S = σ : End ∨ U → End ∨ U given (as in [2] § 3 ) by the usual comp onents U ( A ) ∗ ⊗ U A U ( A ∗ ) ∗ ⊗ U ( A ∗ ) U ( A ) ∗ ⊗ U ( A ) ∗∗ U ( A ∗ ) ⊗ U ( A ∗ ) ∗ σ A / / u − 1 ⊗ u ∗ / / 1 ⊗ d   c O O where d is the canonical map from a v ector space to it s double dual. F urther- more, each map e U A = 1 ⊗ ev : U A ⊗ U A ∗ ⊗ U A → U A satisfies b oth the conditions U A U A ⊗ U A ∗ ⊗ U A U A n ⊗ 1 ? ?        e U A   ? ? ? ? ? ? ? 1 / / (E1) commutes, and U A ∗ U A ∗ ⊗ U A ⊗ U A ∗ U A ∗ ⊗ U A ∗∗ ⊗ U A ∗ 1 ⊗ n : : t t t t t t t t t 1 ⊗ d ⊗ 1 $ $ J J J J J J J J e ∗ U A / / (E2) commutes, where n = coe v : 1 → U A ⊗ U A ∗ in V ect . Then we o btain: Theorem 2.1. The structu r e (End ∨ U, µ, δ, S ) is a VN-c or e in V ect k which c an b e c omplete d to the VN- c or e (End ∨ U ) ⊕ k . Pr o of. The von Neuma nn axiom µ 3 (1 ⊗ S ⊗ 1) δ 3 = 1 3 bec omes the dia g ram (in which we hav e omitted “ ⊗ ” ): U ( A ) ∗ U A U ( A ) ∗ U A U ( A ) ∗ U A U ( A ) ∗ U A U ( A ∗ ) ∗ U ( A ∗ ) U ( A ) ∗ U A U ( A ) ∗ U A U ( A ) ∗ U ( A ) ∗∗ U ( A ) ∗ U A U ( A ) ∗ U A U ( A ∗ ) U ( A ∗ ) ∗ U ( A ) ∗ U A U ( A ) ∗ U ( A ∗ ) ∗ U ( A ) ∗ U A U ( A ∗ ) U A U ( A ) ∗ U ( A ) ∗∗ U ( A ) ∗ U A U ( A ) ∗ U A U ( A ) ∗ U ( A ) U ( A ) ∗ U A U ( A A ∗ A ) ∗ U ( A ) U ( A ) ∗ U A U ( A A ∗ A ) ∗ U ( A A ∗ A ) U ( A ) ∗ U ( A A ∗ A ) U ( A ) ∗ U ( A ) U ( A ) ∗ U A Z C U ( C ) ∗ U ( C ) ( ∗ ) (E1) 1 1 1 d 1 1   * * * * * * * * * * * * * * * * * * * * 1 1 S 1 1 / / 1 1 u − 1 u ∗ 1 1 r r r r r r r 9 9 r r r r r r r 1 c 1 / / 1 1 c 1 1 9 9 r r r r r r r r r r r r r r r ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q U ( e ) ∗ 1 / / 1 u ∗ 1 1 u − 1 1 O O 1 c 1 / / 1 1 1 n 1 J J 1 ˜ τ / / 1 ˜ τ * * V V V V V V V V V V V V V V V V V V V V V V V V V V V V U ( e ) ∗ 1 5 5 l l l l l l l l l l l l l l l l l l l l l 1 U e w w n n n n n n n n n n n n n n n n n n 1 n 1 D D                        1 / / 1 e   4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 ˜ ρ ∗ ˜ τ            cop C = A A ∗ A   cop C = A / / δ 3 I I e ∗ 1 1 1 2 2 d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d where ( ∗ ) is the exter ior of the diagra m U ( A ) ∗ ⊗ U A ⊗ U ( A ) ∗ ⊗ U A ⊗ U ( A ) ∗ ⊗ U A U ( A ) ∗ ⊗ U A ⊗ U ( A ) ∗ ⊗ U ( A ) ∗∗ ⊗ U ( A ) ∗ ⊗ U A U ( A ) ∗ ⊗ U A ⊗ U ( A ) ∗ ⊗ U ( A ) ⊗ U ( A ) ∗ U ( A ) ∗ ⊗ U A ⊗ U ( A ) ∗ ⊗ U ( A ) U ( A ) ∗ ⊗ U ( A ) ∗∗ ⊗ U ( A ) ∗ ⊗ U ( A ) ⊗ U ( A ) ∗ ⊗ U A U ( A ) ∗ ⊗ U A (E2) 1 ⊗ 1 ⊗ 1 ⊗ d ⊗ 1 ⊗ 1 / / 1 ⊗ c ⊗ 1 ) ) S S S S S S S S S S S S S S S S S S S S S S 1 ⊗ c ⊗ 1   1 ⊗ d ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1 ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 1 ⊗ n ⊗ 1 ⊗ 1 ⊗ 1 5 5 k k k k k k k k k k k k k k k k k k k k k k e ∗ ⊗ 1 ⊗ 1 ⊗ 1 / / 1 ⊗ 1 ⊗ 1 ⊗ n ⊗ 1 O O 1 ⊗ n ⊗ 1 O O which commutes using (E2) a nd commutativit y o f · · · · · n 9 9 s s s s s 1 1 n / / c   n % % K K K K K n 1 1 / / 4 3 Examples 3.1 Example The first type o f example is der ived from the idea of a (contra v ariant) inv olution on a (small) comonoidal c a tegory D . This includes the doubles D = B op + B and D = B op ⊗ B with their resp ective “switch” maps (wher e B is a given small comonoidal V ect k -categor y), or an y small comonoidal and compact-monoida l V ect k -categor y D (such as the category M at k of finite matr ices o v er k ) with the tensor duals of ob jects now providing an a ntipo de on the comonoidal as p ect of the structure rather than on the monoidal part, or an y ∗ -algebra structure on a given k -bialgebr a (e.g., a C ∗ -bialgebra ) with the ∗ -op eratio n providing the antipo de. In ea ch cas e , a n even functor f rom D to V ect is defined to b e a ( k -linear) functor F equipp ed w ith a (chosen) dinatural isomorphism F ( D ∗ ) ∼ = F ( D ) . If we take the morphisms of even functors to b e all the natural tra nsformations betw een them then we o bta in a ca teg ory E = E ( D , V ect ) . Let A = E ( D , V ect fd ) fs be the full sub categ o ry of E consisting of the finitely v alued functors of finite supp o rt. While this categ o ry is g enerally no t compac t, it has o n it a natural antipo de de r ived from those on D and V ect fd , namely A ∗ ( D ) := A ( D ∗ ) ∗ . Of cour se, there are also examples where A is actually co mpact, such as tho s e where D is a Ho pf alge br oid, in the sense of [4], with antipo de ( − ) ∗ = S , in which cas e each A from D to V ect has a symmetry str uc tur e on it. Now let C be the full sub categor y of E consis ting of the sma ll copr o ducts in E of ob jects from A . This categor y C is e a sily seen to b e monoida l under the po in t wise conv olution structure from D , and the inclusion A ⊂ C is U -dense for the functor U : C → V ect k given by U ( C ) = X D C ( D ) which is split semigroupa l with U A finite dimensional for all A ∈ A . Mor eov er, U ( A ∗ ) = M D A ∗ ( D ) = M D A ( D ) ∗ = U ( A ) ∗ , for all A ∈ A . The conditions o f (5) are easily verified if we de fine maps e : A ⊗ A ∗ ⊗ A → A 5 by commutativit y o f the diag rams A ( D ) ⊗ A ∗ ( D ) ⊗ A ( D ) A ( D ) A ( D ) ⊗ A ( D ) ∗ ⊗ A ( D ) , e D / / ∼ =   1 ⊗ ev 4 4 j j j j j j j j j j j j j j j where the e xterior of A ∗ ( D ) ⊗ A ( D ) A ( D ) ∗ ⊗ A ( D ) A ∗ ( E ) ⊗ A ( D ) A ( E ) ∗ ⊗ A ( D ) k A ( E ) ∗ ⊗ A ( E ) A ∗ ( E ) ⊗ A ( E ) ∼ =   A ∗ ( f ) ⊗ 1 z z t t t t t t t t t t t t t t t t t t t t t ˆ e $ $ H H H H H H H H H H H H H H H H H H H H H ∼ = / / A ( f ) ∗ ⊗ 1 O O ev ) ) T T T T T T T T T T T T T T T T 1 ⊗ A ( f )   ev 5 5 j j j j j j j j j j j j j j j j ∼ =   1 ⊗ A ( f ) $ $ J J J J J J J J J J J J J J J J J J J J J ˆ e : : v v v v v v v v v v v v v v v v v v v v v commutes for a ll maps f : D → E in D so that e = 1 ⊗ ˆ e : A ⊗ A ∗ ⊗ A → A ⊗ k ∼ = A is a genuine ma p in C when C is g iven the p oint wise monoida l s tr ucture from D . This completes the details of the g eneral example. 3.2 Example In the case wher e k = C and D ha s just o ne ob ject D whose endomorphism algebra is a C ∗ -bialgebra , we hav e a o ne-ob ject comonoidal categor y D with a C -co njugate-linear antipo de given by the ∗ - op eration. Then the c onv olution [ D , Hilb fd ], where [ D , Hilb fd ] ⊂ [ D , V ect C ] , is a monoidal categor y , with a C -linear a ntipo de given by F ∗ ( D ) = F ( D ∗ ) ◦ where H ◦ denotes the conjugate-tr a nsp o se of H ∈ Hilb fd . W e now interpret an even functor F to b e a functor equipp ed with a dinatural is o morphism F ( D ∗ ) ∼ = F ( D ) in D ∈ D which is C -line ar , so that F ∗ ( D ) ∼ = F ( D ) ◦ for such a functor. T ake A = E ( D , Hilb fd ) and let C be the class of s mall co pro ducts in [ D , V ect C ] of the underlying [ D , V ect C ]-representations of A ’s in A (with the appropria te maps). E ach map e : A ⊗ A ∗ ⊗ A → A in C is defined by the C -linear comp onents e : A ( D ) ⊗ A ∗ ( D ) ⊗ A ( D ) 1 ⊗ ˆ e − − → A ( D ) , 6 where ˆ e : A ∗ ( D ) ⊗ A ( D ) → C in V ect C comes from the C -bilinear compo s ite of tw o maps which are both C -linear in the first v ar iable a nd C -linear in the seco nd, namely A ∗ ( D ) × A ( D ) C A ( D ) ◦ × A ( D ) . / / ∼ =   h− , −i 7 7 o o o o o o o o o o o o The rema inder o f this ex a mple is as see n b efore in Example 3.1. 3.3 Example Let V = ( V , ⊗ , I ) b e a (small) braided monoida l catego ry a nd let B b e the k - linearization o f Semicoalg ( V ) with the mo noidal struc tur e induced fro m that on V . By analo gy with [5], let X ⊂ B b e a finite full s ubca tegory of B with I ∈ X and X op promonoida l when p ( x, y , z ) = B ( z , x ⊗ y ) j ( z ) = B ( z , I ) for x, y , z ∈ X . F or exa mple (cf. [5]), one could take X to be a (finite) set of non-isomo r phic “basic” o b jects in some braided mono idal categor y V , where each x ∈ X has a coasso cia tiv e diagona l map δ : x → x ⊗ x . Howev er, we won’t nee d the category X to be discrete o r lo ca lly finite in the following. Now let C b e the conv olution [ X op , V ect ] and let A = [ X op , V ect fd ]. The functor U : C → V ect is defined by U ( C ) = M x C ( x ) , and the obvious inc lus ion A ⊂ C is U -dense . If there is a cano nic a l (natural) retraction p ( x, y , z ) = B ( z , x ⊗ y ) B ( z , x ) ⊗ B ( z , y ) , i x,y / / r x,y o o derived fro m the semico algebra structures on x, y , z , then U b ecomes a split semigroupa l functor v ia the str uc tur e maps U ( C ) ⊗ U ( D ) U ( C ⊗ D ) L x C ( x ) ⊗ L y D ( y ) L z Z xy p ( x, y , z ) ⊗ C ( x ) ⊗ D ( y ) L z C ( z ) ⊗ D ( z ) L z Z xy B ( z , x ) ⊗ B ( z , y ) ⊗ C ( x ) ⊗ D ( y ) , r / / i o o ∆ O O ∆ ∗   “ r ” O O “ i ”   ∼ = o o 7 where the is omorphism follows fro m the Y oneda lemma , and r i = 1. If X als o ha s on it a duality ( − ) ∗ : X → X op such that x ∼ = x ∗∗ , then, on defining A ∗ ( x ) = A ( x ∗ ) ∗ , we obtain U ( A ∗ ) = M x A ∗ ( x ) = M x A ( x ∗ ) ∗ ∼ = M x A ( x ) ∗ since x ∼ = x ∗∗ ∼ = U ( A ) ∗ , for A ∈ A , in acco r dance with the fourth requirement o n U . Finally , to obtain a suitable map e = 1 ⊗ ˆ e : A ⊗ A ∗ ⊗ A → A ⊗ I ∼ = A, where ˆ e : A ∗ ⊗ A → I , we supp ose each A in A has o n it a “dual coupling” χ = χ xy : A ( x ) ∗ ⊗ A ( y ) → B ( x ∗ ⊗ y , I ) . By consider ing the Y oneda expans ion A ( x ) ∼ = Z z A ( z ) ⊗ X ( x, z ) of the v arious functors A in A = [ X op , V ect fd ], s uc h a coupling exists on each A if we supp ose mer ely that X itself is “c oupled” by a na tural transfor mation χ : X ( y , z ) → X ( x, z ) ⊗ B ( x ∗ ⊗ y , I ); or simply χ : X ( x, z ) ∗ ⊗ X ( y , z ) → B ( x ∗ ⊗ y , I ) , if X is lo cally finite. Then, the co mpo s ite natural tra ns formation A ( x ∗ ) ∗ ⊗ A ( y ) ⊗ B ( z , x ⊗ y ) B ( x ∗∗ ⊗ y , I ) ⊗ B ( z , x ⊗ y ) B ( x ⊗ y , I ) ⊗ B ( z , x ⊗ y ) B ( z , I ) χ ⊗ 1   ∼ =   comp’n   8 yields the ma p A ∗ ⊗ A I Z xy A ∗ ( x ) ⊗ A ( y ) ⊗ p ( x, y , − ) B ( − , I ) ˆ e / / / / bec ause p ( x, y , − ) = B ( − , x ⊗ y ) (by definition). Th us suita ble conditions on the coupling χ give (5). R emark. Actually , this las t example in which the basic promonoida l s tructure o ccurs as a cano nical retra ct of a comonoida l struc tur e is typical of many other examples which ca n b e tre a ted along similar lines . References [1] Brian Day . Note on the fusion map and Hopf a lgebras, arXiv:09 02.225 9 v3 [math.CT], 20 09. [2] Brian Day and Craig P astro. On endomorphism algebras of separable monoidal functors , Theory and Applications o f Catego ries 22 (2009) 77– 96. [3] Brian Day a nd Craig Pastro. Note on endomor phis m a lgebras o f separa ble monoidal functors , arXiv:090 7.3259v1 [math.CT], 2009 . [4] Brian Day and Ross Street. Monoidal bicatego r ies and Hopf a lgebroids, Ad- v ances in Mathematics 129 (19 97) 9 9–157 . [5] Reinhard H¨ aring-O lden burg. Reconstruction of weak quasi- Hopf algebra s, Journal of Algebra 194 (1997) 14– 3 5. [6] Saunders Mac Lane. Categ o ries for the W or king Mathematician, Graduate T exts in Mathematics 5 (Second edition, Springer , 1998 ). [7] Ross Stre et. F usion op er a tors and c o cycloids in monoida l categor ies, Applied Categoric a l Structure s 6 (199 8) 17 7–191 . Department of Mathematics Macquarie U nivers ity NSW, 2109, Australia 9