Tannaka Reconstruction of Weak Hopf Algebras in Arbitrary Monoidal Categories

T annak a Reconstruction of W eak Hopf Algebras in Arbitrary Monoidal Categories Micah Blak e McCurdy Marc h 2, 2009 Abstract W e in tro duce a v ariant on the graphical calculus of Cock ett and Seely[2] for monoidal functors and illustrate it with a discussion of T annak a reconstruction, some of which is kno wn and some of whic h is new. The new portion is: giv en a separable F rob enius functor F : A − → B from a monoidal category A to a suitably complete or co complete braided autonomous category B , the usual formula for T annak a reconstruction gives a weak bialgebra in B ; if, moreov er, A is autonomous, this weak bialgebra is in fact a w eak Hopf algebra. 1 In tro duction Broadly sp eaking, T annak a dualit y describ es the relationship b etw een algebraic ob jects and their represen- tations, for an excellent introduction, see [7]. On the one hand, given an algebraic ob ject H in a monoidal category B (for instance, a hopf ob ject in the category V ec k of v ector spaces ov er a field k ), one can consider the functor which takes algebraic ob jects of the given type to their category of representations, rep B H , for whic h there is a canonical forgetful functor back to B . This pro cess is r epr esentation and it can b e defined in a great v ariety of situations, with very mild assumptions on B . F or instance: • If J is a bialgebra in a braided linearly distributiv e category B , then rep B J is linearly distributive [2]. • If H is a Hopf algebra in a braided star-autonomous category B , then rep B H is star-autonomous [2]. Note that the first example includes the familiar notion of a bialgebra in a monoidal category giving rise to a monoidal category of representations, since every monoidal category is degenerately linearly distributive b y taking b oth of the monoidal pro ducts to b e the same. On the other hand, given a suitable functor F : A − → B , we can try to use the properties of F (which of course include those of A and B ) to build an algebraic ob ject in B ; this is called (T annak a) r e c onstruction , since historically the algebraic ob jects hav e b een considered primitive. W e denote the reconstructed ob ject as E F , following [12]. This requires making more stringent assumption on B ; certainly it must b e braided; it must b e autonomous, and it is assumed that B admits certain ends or co ends which cohere with the monoidal pro duct. F or instance, under these assumptions: • If F is an y functor, then E F is a monoid in B . • If A is monoidal and F is a monoidal functor, then E F is a bialgebra in B . • If A is autonomous and F is a strong monoidal functor, then E F is a Hopf algebra in B ([11], [8]). T o this we add the following: • If A is monoidal and F is a separable F rob enius functor, then E F is a weak bialgebra in B . • If A is autonomous and F is a separable F rob enius functor, then E F is a weak Hopf algebra in B . 1 Note that the notion of weak Hopf algebra considered by Haring-Oldb erg [6] is a different notion, the sense of weak we use here is that of [1]. It should b e noted that a sp ecial case of the weak Hopf algebra result has b een obtained b y Pfeiffer[10], where A is taken to b e a mo dular category and B is taken to b e V ec k ; see also [13]. In fav ourable circumstances, reconstruction is left adjoint to representation, for instance: Monoids( B ) Cat /B rep B −   Cat /B Monoids( B ) E − J J a Bialgebras( B ) MonCat Strong /B rep B −   MonCat Strong /B Bialgebras( B ) E − J J a Hopf ( B ) AutCat Strong /B rep B −   AutCat Strong /B Hopf ( B ) E − J J a In the weak cases which I discuss in the sequel, ho wev er, we do not hav e these adjunctions. 2 Graphical Notation for Monoidal and Comonoidal F unctors Before we discuss the reconstruction itself, we discuss notations for monoidal and comonoidal functors. The original notion for graphically depicting monoidal functors as transparent b oxes in string diagrams is due to Co c kett and Seely[2], and has recently b een revived and p opularized by Mellies[9] with prettier graphics and an excellent pair of example calculations which nicely show the w orth of the notation. How ever, a small mo dification impro v es the notation considerably . F or a monoidal functor F : A − → B , we ha ve a pair of maps, F x ⊗ F y − → F ( x ⊗ y ) and e − → F e , which we notate as follows: Similarly , for a comonoidal F , we hav e maps F ( x ⊗ y ) − → F x ⊗ F y and F e − → e which we notate in the ob vious dual w ay , as follows: Graphically , the axioms for a monoidal functor are depicted as follows: 2 where, once again, the similar constraints for a comonoidal functor are exactly the ab ov e with comp osition read right-to-left instead of left-to-right. It is curious and pleasing that the t wo unit axioms b ear a sup erficial resemblence to triangle-identies b eing applied along the b oundary of the “ F -region”. The ab ov e axioms seem to indicate some sort of “in v ariance under con tin uous deformation of F -regions”. F or a functor whic h is b oth monoidal and comonoidal, pursuing this line of thinking leads one to consider the following pair of axioms: A functor with these prop erties has b een called a F rob enius monoidal functor by Day and Pastro [4]; more- o ver, it is the notion which results from considering linear functors in the sense of [2] b et w een degenerate linearly distributive categories. They are more common than strong monoidal functors yet still share the k ey prop ert y of preserving duals. Let F : A − → B b e a (mere) functor b etw een monoidal categories, where B is assumed to b e left closed. Then define E F = Z a ∈ A [ F a, F a ] where I assume that A and B are such that the indicated end exists. As Richard Garner imimitably asked at the 2006 PSSL in Nice, “Ha ve y ou considered enriching ev erything?”. I do not discuss the matter here, but it has b een considered by Brian Day[3]. There is a canonical action of E F on F x for eac h ob ject x in A , whic h w e denote as α = α x : E F ⊗ F x − → F x . This is defined as: E F ⊗ F x =  Z a ∈ A [ F a, F a ]  ⊗ F x π x ⊗ F x − − − − → [ F x, F x ] ⊗ F x ev − − − → F x using the x ’th pro jection from the end follow ed b y the ev aluation of the monoidal c losed structure of B . The dinaturalit y of the end in a gives rise to the naturality of the action on F a in a , which we notate as: 3 Let us no w assume that the closed structure of B is given by left duals, that is, [ a, b ] = b ⊗ La . If we also assume that B is braided and that the tensor pro duct coheres with the ends in B , then we obtain canonical actions of E n F on F x 1 ⊗ . . . ⊗ F x n , written α n x . T aking α 1 = α , w e define α n recursiv ely as follo ws: E n F ⊗   n O j =1 F x j   = E n − 1 F ⊗ E F ⊗   n − 1 O j =1 F x j   ⊗ F x n braid − − − → E n − 1 F ⊗   n − 1 O j =1 F x j   ⊗ E F ⊗ F x n α n − 1 ⊗ α 1 − − − − − − → n O i =1 F x i F or an y map f : X − → E n F , w e ma y paste together f with α n along N n i =1 F x i to obtain a “discharged form” of f : X ⊗ F x 1 ⊗ . . . ⊗ F x n − → F x 1 ⊗ F x 1 ⊗ · · · ⊗ F x n Tw o maps are equal if and only if they hav e the same discharged forms. Man y treatmen ts instead consider E F = Z a ∈ A [ F a, F a ] Under dual assumptions to the ones ab ov e – namely , that the co end exists and coheres with the tensor pro duct in B – we obtain canonical coactions of E F on F x for each x ∈ A , and iterated coactions, &c. This approac h has certain tec hnical b enefits; among others, that the tensor pro duct in B = V ect coheres with co ends but not with ends. How ever, the notation we use cov ers b oth cases, E F and E F . F or the former, one m ust read comp osition left-to-right, and for the latter, from right-to-left. W e write E F as a lab el for con venience, preferring for conv enience to read in the conv entional English w ay , but it is a crucial feature of the notation that in fact no choice is made. Without assuming that F bears a monoidal structure, one can define a monoid structure on E F , as follo ws: Note that this monoidal structure is asso ciative and unital, without any assumption on F . F urthermore, if F is known to b e (lax) monoidal and comonoidal (without at the momen t assuming any coherence b etw een these structures) we can define a comonoid structure on E F . 4 Finally , if A is kno wn to hav e (left, say) duals, we can define a canonical map S : E F − → E F whic h we think of as a candidate for an antipo de. Notice in particular how the monoidal and comonoidal structures on F permit one to consider the application of F as not merely “b oxes” but more lik e a flexible sheath. No w, the ab ov e is the ra w data for t wo different structures, namely , Hopf algebras and w eak Hopf algebras, whic h differ only in axioms. It has b een remark ed b efore that requiring F to b e strong (that is, in our treatment, demanding that the monoidal and comnoidal structures b e mutually inv erse) mak es the ab o v e data into a Hopf algebra. Before we discuss the Hopf data (that is, the antipo de), let us first consider the bialgebra data. A bialgebra in a braided monoidal category satisfies the following four axioms. First, the unit follow ed by the counit must b e the iden tity: Second and third, the unit and counit must resp ect the comultiplication and multiplication, resp ectively: F ourthly , the multiplication must cohere with the comultiplication, with the help of the braiding: 5 Some easy calculations sho w ho w the strength of F features crucially in showing all four of these axioms. F or the first of these, we calculate: and we see that this comp osite is the iden tity on e precisely when e − → F e − → e is the identit y . F or the second bialgebra axiom, we hav e the following tw o calculations: and so we see that these tw o are equal precisely when F x ⊗ F y − → F ( x ⊗ y ) − → F x ⊗ F y is the identit y . F or the third bialgebra axiom, we hav e the following tw o calculations: and w e see that for these tw o to b e equal, it suffices to hav e F e − → e − → F e b e the identit y , the use of whic h b et ween the tw o actions in the first calculation gives the result. Finally , for the final bialgebra axiom, the calculations sho wn in figure (1) compute the discharged forms as 6 whic h sho ws that it suffices to request that F ( x ⊗ y ) − → F x ⊗ F y − → F ( x ⊗ y ) should b e the identit y . Demanding that F b e strong imp oses the follo wing four conditions on the monoidal/comonoidal structure: Notice that precisely one of them preserves the n umber of connected comp onen ts of F , namely , the one whic h is used in the pro of of the coherence of the multiplication with the comultiplication. A monoidal functor satisfying this axiom has b een called “separable” b y some. T o mov e from a (strong) bialgebra to a w eak one, this axiom is the only one whic h is retained. The coherence of the unit with the counit is discarded en tirely , and the second and third axioms are replaced with the following four axioms: 7 W e first examine the unit axioms. In discharged form, the first unit expression is calculated as: The calculations in figure (2) show that the second and third unit expressions hav e the following dis- c harged forms: F or these unit axioms, we see that it suffices to assume that F is F rob enius. As for the counit axioms, the disc harged form of the first of these is easily calculated: The discharged forms of the second and third counit expression are computed in figure (3) and are, of course the same. Examing this figure shows that the counit axioms follo w merely from F b eing b oth monoidal and comonoidal, without requiring F robenius or separable. This assymmetry (b etw een unit and counit axioms) results from defining E F using an end, had w e instead used a co end, the situation would b e rev ersed. As for the antipo de axioms, we can also consider the pair of strong antipo de axioms or the trio of weak an tip o de axioms. The strong antipo de axioms request the follo wing tw o equations: 8 On the other hand, the weak antipo de axioms request the following three equations: Both sets of axioms inv olve the tw o conv olutions of the antipo de with the (comp ositional) iden tity , and so we calculate these t w o quantities explicitly . The pair of calculations in figure (4) show that the disc harged forms of S ? E F and E F ? S are the following: In the (strong) case, b oth of these conv olutions are supp osed to equal the comp osite E F  − − − → e η − − − → E F , the discharged form of which we compute: 9 Therefore, w e see that, in the usual Hopf algebra case, it suffices to take F e − → e − → F e equal to the iden tity . F or the weak case, these conv olutions are instead set equal to the expressions which are computed in figure (5). So we see that for these tw o axioms it suffices to take F to b e monoidal and comonoidal. There is one additional antipo de axiom which is imp osed for a weak Hopf algebra, namely , that the con volution S ? E F ? S should equal S . F or this, we compute the left hand side in figure (6), the last diagram of which is the definition of S , as desired. An up dated version of this pap er will treat reconstruction of braided w eak bialgebras and braided w eak Hopf algebras when A is known to b e a braided category . References [1] Bohm, Gabriella, Florian Nill, Kornel Szlac hanyi, We ak Hopf Algebr as I. Inte gr al The ory and C ∗ - structur e , Journal of Algebra v221 (1999), p385-438, av ailable as ar χ iv:math/9805116v3. [2] Cock ett, J.R.B., and R.A.G. Seely , Line arly Distributive F unctors , JP AA v143 (1999), p155-203, av ail- able from the website of the second author. [3] Da y , Brian J., Enriche d T annaka r e c onstruction , JP AA v108-1 (1996), p17-22 [4] Da y , Brian J., and Craig Pastro, Note on F r ob enius Monoidal F unctors , New Y ork Journal of Math- ematics v14 (2008), p733-742, a v ailable from the w ebsite of the second author, also a v ailable as ar χ iv:0801.4107v2. [5] Etingof, P av el, and Olivier Sc hiffman, L e ctur es on Quantum Gr oups , International Press, 1998. 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[13] Szlac hanyi, Kornel, “Adjoin table Monoidal F unctors and Quantum Group oids”, in Hopf algebr as in nonc ommutative ge ometry and physics , Lecture Notes in Pure and Applied Mathematics v239, Dekker, New Y ork, (2005), p291-307, av ailable as ar χ iv:math/0301253v1. 10 Figure 1: Coherence of the multiplication with the comultiplication 11 Figure 2: W eak unit calculations 12 Figure 3: W eak counit calculations 13 Figure 4: Calculations of S ? E F and E F ? S 14 Figure 5: “Source” and “T arget” maps. 15 Figure 6: The calculation showing S ? E F ? S = S 16