An introduction to categorifying quantum knot invariants

Transcription

1 msp Geometry & Topology Monographs 18 (2012) An ntroducton to categorfyng quantum knot nvarants BEN WEBSTER We construct knot nvarants categorfyng the quantum knot nvarants for all representatons of quantum groups, based on categorcal representaton theory. Ths paper gves a condensed descrpton of the constructon from the author s earler papers on the subect, wthout proofs and certan constructons used only ndrectly n the descrpton of these nvarants. 57M27; 17B37, 16P10 Our am n ths paper s to gve a more accessble and compact descrpton of the knot nvarants ntroduced by the author n the papers [49; 50]. In partcular, t wll contan few new results and for the most part neglect proofs, referrng the reader to those earler papers. These nvarants are categorfcatons of quantum knot nvarants ntroduced by Reshetkhn and Turaev [48; 35]. Partcular cases of these nclude: the Jones polynomal when g D sl 2 and all strands are labeled wth the defnng representaton; the colored Jones polynomals for other representatons of g D sl 2 ; specalzatons of the HOMFLYPT polynomal for the defnng representaton of g D sl n ; the Kauffman polynomal (not to be confused wth the Kauffman bracket, a varant of the Jones polynomal) for the defnng representaton of so n. In [50], the author proves the followng: Theorem A For each smple complex Le algebra g, there s a homology theory K.L; f g/ for lnks L whose components are labeled by fnte-dmensonal representatons of g (here ndcated by ther hghest weghts ), whch assocates to such a lnk a bgraded vector space whose graded Euler characterstc s the quantum nvarant of ths labeled lnk. Ths theory concdes up to gradng shft wth Khovanov s homologes for g D sl 2 ; sl 3 when the lnk s labeled wth the defnng representaton of these algebras, and the Mazorchuk Stroppel Sussan homology for the defnng representaton of sl n. Publshed: 14 October 2012 DOI: /gtm

2 254 Ben Webster Categorcal representatons Our am s to descrbe the constructon of these nvarants and how one arrves at ther most basc propertes. They are based on a prncple whch s easy to state, but harder to mplement: Every obect or structure of any sgnfcance n the theory of quantum groups possesses a natural categorfcaton. Obvously, a prncple stated ths baldly has no hope of really beng true, but t has stll been a very successful dea. In [7], Chuang and Rouquer ntroduced the noton of a categorcal acton of sl 2 ; whle ther work had mportant consequences for the theory of symmetrc groups (provng Broué s conecture for the symmetrc group), t also showed that categorcal representatons have a remarkable structure. Whle they are more complex than representatons of sl 2 (whch, of course, have a very smple structure theory), they have the same elementary buldng blocks: for each smple representaton of sl 2, there s an essentally unque smple categorcal representaton whose Grothendeck group t s. Rouquer defned categorcal actons of other smple Le algebras and showed that the same prncple holds n that context [37]. Ths s one manfestaton of the prncple above. Khovanov and Lauda also consdered the prncple above, and arrved at 2 categores U whose Grothendeck groups concde wth the unversal envelopng algebra of any smple Le algebra. Ths was frst done for sl 2 by Lauda [26]. The general constructon for all g and the proof that the Grothendeck group concdes wth U.sl n / n that case appeared n [21]; the proof that the Grothendeck group s correct was completed n [49]. We should emphasze, whle there are mnor dfferences n formalsm and notaton, arsng from dfferng motvatons and hstorcal reasons, Rouquer s approach and Khovanov Lauda s are n essence the same; recent work by Cauts and Lauda [5] shows that under very weak techncal condtons a categorcal g acton n the sense of Rouquer gves rse to an acton of a varant of the 2 category U. Ths earler work, whch we wll largely take for granted heren, provdes a context for the statement above. In partcular, we expect any natural operaton on representatons to have an analogue on categorcal representatons, and any natural map between representatons to have an analogous functor between categorcal representatons. Now, we turn to descrbng the maps of nterest to us.

3 An ntroducton to categorfyng quantum knot nvarants 255 Quantum knot nvarants Reshetkhn and Turaev s constructon (n somewhat more modern language) s essentally the observaton that fnte-dmensonal representatons of a quantum group U q.g/ are a rbbon category (for more background on rbbon categores, see Char and Pressley [6] or Kassel [15]). Due to the extra trouble of drawng rbbons, we wll draw all pctures n the blackboard framng (that s, they should be thckened to rbbons pressed flat to the page). The category of orented rbbon tangles rb has obects gven by sequences of up and down arrows and morphsms gven by orented rbbon tangles (e tangles wth choce of framng) matchng the sequences at the top and bottom, consdered up to sotopy. It s mportant that we don t allow half-twsts n our rbbons, only full-twsts; f we thnk of each rbbon as havng a lght and dark sde, the lght sde must always be facng upward when the rbbon s attached to ts up or down arrow. In fact, one descrpton of the blackboard framng s that the lght sde of the rbbon always faces outward. The category rb s a monodal category (e, t possesses a tensor product operaton on obects), where the tensor product of obects or morphsms s ust horzontal concatenaton. We can extend ths constructon by allowng rbbons to be labeled wth obects n a category C and allowng coupons wth morphsms n the category to be placed on rbbons, whch compose n the obvous way. We denote ths category C rb. Defnton A rbbon structure s a weakly monodal functor C rb! C whch sends an upward pontng, untwsted rbbon labeled wth C 2 Ob.C/ to C, and a sngle upward pontng, untwsted rbbon wth a coupon to the labelng morphsm. Obvously, such a functor s determned by the mage of three types of rbbon tangles: a full twst n one rbbon, a crossng of two rbbons (untwsted), and cups and caps n all orentatons. Furthermore, t s possble to wrte a fnte lst of relatons that these maps must satsfy n order for them to defne a functor, all of whch are obvous manpulatons of rbbon tangles; these relatons can be found, for example, n [6]. Gven any rbbon structure on a category C, we arrve mmedately at a lnk nvarant valued n End.1/ Š ZŒq; q 1 where 1 s the tensor dentty n ths category: any rbbon

4 256 Ben Webster lnk gves such a map, thought of as a tangle startng and endng at the empty dagram. That s, we pck a generc heght functon on our lnk, use ths to slce t nto peces that look lke the basc tangles descrbed above, and compose the correspondng maps. In ths language, we can phrase the constructon of the Reshetkhn Turaev knot nvarants thus: Theorem B The category of fnte-dmensonal representatons of U q.g/ has a natural (but not qute canoncal) rbbon structure. Ths s the theorem that we ntend to categorfy. The most obvous guess for a categorfcaton s that there exsts a defnton of rbbon 2 category such that categorcal representatons of g are a rbbon 2 category. Ths seems to be askng too much, for reasons wll hopefully become clear later n ths paper; some extra structure on the categorcal representatons wll be necessary, such as a trangulated or dg structure. Conecture C There exsts a defnton of rbbon 2 category such that categorcal representatons of g on dg categores are a rbbon 2 category. Ths would nstantly mply that Reshetkhn Turaev nvarants possess categorfcatons, by exactly the same constructon as before. The tensor dentty n categorcal representatons wll ust be a copy of the dg category of complexes of vector spaces Vect K over a fxed base feld K. Thus, an exact autofunctor s tensor product wth a complex, whch yelds the desred knot homology. Unfortunately, ths remans unproven; nstead, we have shown that some of the consequences of Conecture C hold wthout provng the conecture tself. The frst roadblock to ths conecture s that our approach does not specfy what the tensor product of an arbtrary collecton of categorcal representatons s. Instead, we explctly gve a categorcal representaton V attached to each tensor product. We beleve that ths must arse from a unversal defnton of a tensor product; such a defnton has been announced by Rouquer [38], but at the tme of wrtng, the detals reman unpublshed. Smlarly, we construct by hand the desred functors attached to small tangles. These maps behave exactly as they would f there were a rbbon structure as suggested above. After ths constructon, the desred knot nvarants are obtaned by composng these functors. In practce, ths s stll an explct calculaton, but conceptually t s qute clean. Unfortunately, we thnk t unlkely that there s any royal road to a categorcal rbbon structure; n all lkelhood, computatons by hand lke those n the current approach wll stll be necessary n order to prove Conecture C above.

5 An ntroducton to categorfyng quantum knot nvarants 257 A bref hstory of knot homology Whle the dea of knot homologes and categorfed representaton theory appears as early as the work of Crane and Frenkel [11], the frst appearance of one n the lterature as far as the author s aware s Khovanov s categorfcaton of the Jones polynomal [16; 17]. Khovanov later extended hs pcture to the defnng representaton of sl 3 [18], and then ontly wth Rozansky, and usng very dfferent methods nvolvng matrx factorzatons, extended ths to the defnng representaton of sl n and the defnng representaton of so n [23; 22], whch was nterpreted n a sprt more lke Khovanov s orgnal constructon by Mackaay, Stošć and Vaz [29]. A degenerate case of ths constructon defnes a homology for the HOMFLYPT polynomal tself and ts colored versons, studed by Khovanov and Rozansky [19; 24], Mackaay, Stošć and Vaz [30] and the author and Wllamson [51]. The extenson of the matrx factorzaton perspectve to other fundamental representatons of sl n was consdered by Wu [53] and Yonezawa [54]. A more geometrc perspectve, usng Fukaya categores and coherent sheaves, appears n the work of Sedel Smth [39], Manolescu [31], Cauts Kamntzer [3; 4]. Fnally, the approach most lke ours s the one taken by Stroppel and Mazorchuk Stroppel [42; 32] and Sussan [46], whch studed categorfcatons of tensor products of fundamental representatons of sl n that arse as blocks of parabolc category O for type A Le algebras. In a paper stll n preparaton, Stroppel and Sussan also consder the case of the colored Jones polynomal [44] (buldng on prevous work wth Frenkel [13]). It seems lkely ther constructon s equvalent to ours va the constructons of Secton 6. Smlarly, Cooper, Hogancamp, and Krushkal have gven a categorfcaton of the 2 colored Jones polynomal n Bar-Natan s cobordsm formalsm for Khovanov homology [9]. However, there are two serous ssues wth the work mentoned above: they have only consdered mnuscule representatons (of whch there are only fntely many n each type), and n a few recent papers, the other representatons of sl 2. The work of physcsts suggests that categorfcatons for all representatons exst; one schema for defnng them s gven by Wtten [52]. The relatonshp between the nvarant presented n ths paper and those suggested by physcsts s completely unknown (at least to the author) and presents a very nterestng queston for consderaton n the future. On the other hand, the quantum nvarants whch have categorfcatons often have several competng ones: for the defnng representaton of sl n, nvarants have been defned by Khovanov Rozansky, Manolescu, Mazorchuk Stroppel Sussan, and Cauts Kamntzer, and no two of these are known to the same or dfferent for n > 3!

6 258 Ben Webster In ths paper, we solve the frst of these problems, by defnng a homology attached to every labelng of representatons. We cannot help much wth the second, but we at least do not make t worse: for defnng representatons of sl n, our nvarants agree wth those of Mazorchuk Stroppel Sussan. Our approach s essentally the same as thers; however, we need categores that have not appeared (to our knowledge) n representaton theory prevously. The constructon of these categores and ther basc propertes s done n the paper [49]. The constructon of the functors correspondng to the acton of tangles s done n [50]. Let us state our man theorem n a form desgned to match the defnton of a rbbon category. Let g rb denote the category of tangles labeled wth hghest weghts of g. Let g cat be the category consstng of categorcal g modules, wth morphsms beng somorphsm classes of functors whch weakly commute wth the categorcal g acton. Theorem D There s a functor g rb! g cat whch sends a seres of up and down arrows wth the th labeled wth f orented upward and D w 0 f orented downward to the categorfcaton V of the tensor product defned later n ths paper and a rbbon tangle between these to an explct functor defned later n ths paper usng a knot proecton whch categorfes (a slght modfcaton of) the Reshetkhn Turaev rbbon structure on the category of fnte-dmensonal U q.g/ modules. Ths would be an mmedate consequence of Conecture C and should be consdered strong evdence of ts truth. Ths theorem does mply the exstence of the desred knot homology, by the same argument. Open questons Snce we hope that ths paper wll be more accessble to grad students and other newcomers to the feld, we thought t would be approprate to menton some questons whch reman to be resolved. One expected property of knot homologes s that they wll be functoral over embedded cobordsms between knots; n partcular, ths would follow for any knot homology that arses as part of an extended TQFT. Furthermore, ths functoralty s confrmed n the cases of the defnng representaton of sl 2 and sl 3, and up to sgn n Khovanov Rozansky homology for sl n.

7 An ntroducton to categorfyng quantum knot nvarants 259 At the moment, we have not proven that the theores defned n ths paper are functoral, but we do have a proposal for the map assocated to a cobordsm when the weghts are all mnuscule. As usual n knot homology, ths proposed functoralty map s constructed by pckng a Morse functon on the cobordsm, and assocatng smple maps to the addton of handles. At the moment, we have no proof that ths defnton s ndependent of Morse functon and we antcpate that provng ths wll be qute dffcult. For all hghest weghts of sl 2, Hogancamp has proven functoralty of the homology from [9] for surfaces wthout local maxma, whch allows us to avod an aduncton whch only exsts n the mnuscule case; obvously, we hope that one can extend ths to all types. One very nterestng consequence of the functoralty of Khovanov homology s Rasmussen s nvarant [34], a concordance nvarant of lnks whch s a lower bound for slce genus; ths was generalzed to Khovanov Rozansky homology by Lobb [27], and also has an analogue n knot Floer homology, called the nvarant. Obvously, t would be very nterestng to generalze ths work to the nvarants dscussed here. More generally, Rasmussen (buldng on hs ont work wth Dunfeld and Gukov [12]) constructed a large number of spectral sequences relatng Khovanov Rozansky homologes for dfferent ranks [33]. It s hard to say what an approprate generalzaton of these results would be, but t would seem very surprsng f there were none. As mentoned earler, Wtten has suggested an approach to homologcal knot nvarants usng Morse cohomology for certan solutons of PDEs [52]. Whle the connecton of hs pcture to ours s not at all clear, t presents a very nterestng possblty for future research. Notaton We let g be a fnte-dmensonal smple complex Le algebra, whch we wll assume s fxed for the remander of the paper. In future work, we wll nvestgate tensor products of hghest and lowest weght modules for arbtrary symmetrzable Kac Moody algebras, hopefully allowng us to extend the contents of Sectons 3, 4 and 5 to ths case. We fx an order on the smple roots of g, whch we wll smply denote wth < for two nodes ;. Ths choce s purely auxlary, but wll be useful for breakng symmetres.

8 260 Ben Webster Consder the weght lattce Y.g/ and root lattce X.g/, and the smple roots and coroots _. Let c D _. / be the entres of the Cartan matrx. Let D be the determnant of the Cartan matrx. For techncal reasons, t wll often be convenent for us to adont a Dth root of q, whch we denote q 1 =D. We let h ; denote the symmetrzed nner product on Y.g/, fxed by the fact that the shortest root has length p 2 and 2 h ; D _./: h ; As usual, we let 2d D h ;, and for 2 Y.g/, we let D _./ D h ; =d : We let be the unque weght such that _./ D 1 for all and _ the unque coweght such that _. / D 1 for all. Snce 2 1=2X and _ 2 1=2Y, for any weght, the numbers h; and _./ are not necessarly ntegers, but 2h; and 2 _./ are (not necessarly even) ntegers. Throughout the paper, we wll use D. 1 ; : : : ; `/ to denote an ordered ` tuple of domnant weghts, and always use the notaton D P. We let U q.g/ denote the deformed unversal envelopng algebra of g; that s, the assocatve C.q 1 =D / algebra gven by generators E, F, K for and 2 Y.g/, subect to the relatons: () K 0 D 1, K K 0 D K C 0 for all ; 0 2 Y.g/, () () acbd K E D q _./ E K for all 2 Y.g/, K F D q _./ F K for all 2 Y.g/, zk (v) E F F E D ı zk, where q d q d zk D K d, (v) For all X. 1/ a E.a/ E E.b/ X D 0 and. 1/ a F.a/ F F.b/ D 0: c C1 acbd c C1 Ths s a Hopf algebra wth coproduct on Chevalley generators gven by.e / D E 1 C zk E.F / D F zk C 1 F and antpode on these generators defned by S.E / D zk E ; S.F / D F zk : We should note that ths choce of coproduct concdes wth that of Lusztg [28], but s opposte to the choce n some of our other references, such as [6; 40]. In partcular, we

9 An ntroducton to categorfyng quantum knot nvarants 261 should not use the formula for the R matrx gven n these references, but that arsng from Lusztg s quas-r matrx. There s a unque element 2 8 Uq.g/ U C q.g/ n a sutable completon of the tensor product such that.u/ D x.u/, where x.e / D E 1 C zk E x.f / D F zk C 1 F : If we let A be the operator whch acts on weght vectors by A.v w/dq hwt.v/;wt.w/ v w, then as noted by Tngley [47, 2.10], R D A 1 s a unversal R matrx for the coproduct (whch Tngley denotes op ). Ths s the opposte of the R matrx of [6] (for example). We let Uq Z.g/ denote the Lusztg (dvded powers) ntegral form generated over ZŒq 1 =D ; q 1 =D by En Œn q! ; F n Œn q for all ntegers n of ths quantum group. The ntegral form of the representaton of hghest weght over ths quantum group wll! be denoted by V Z, and V Z D V Z 1 ZŒq 1=D ;q 1 =D ZŒq 1 =D ;q 1 =D V Z. We let ` V D V Z ZŒq 1 =D ;q 1 =D Z..q 1 =D // be the tensor product wth the rng of nteger valued Laurent seres n q 1 =D ; ths s the completon of V Z n the q adc topology. Acknowledgments Of course, f I were to thank everyone who helped wth the orgnal papers, my lst would be qute long and redundant. Let me ust settle for thankng Mke for havng a helpfully tmed brthday, and all the organzers of the conference, for makng t so lovely. 1 Categorfcaton of quantum groups Categores In ths paper, our notaton bulds on that of Khovanov and Lauda, followed by Cauts and Lauda [5] who gve a graphcal verson of the 2 quantum group, whch we denote U (leavng g understood). These constructons could also be rephrased n terms of Rouquer s descrpton and we have strven to make the paper readable followng ether [21] or [37]. However, we wll use the relatons gven n [5] whch are a varaton on the formalsms gven n those papers. The dfference between ths category and the categores defned by Rouquer n [37] s qute subtle; t concerns precsely whether the nverse to a partcular map s formally added, or mposed to be a partcular composton of other generators n the category. Most mportantly for our purposes,

10 262 Ben Webster the 2 category U receves a canoncal map from each of Rouquer s categores A and A 0, so a representaton of t s a representaton n Rouquer s sense as well. Snce the constructon of these categores s rather complex, we gve a somewhat abbrevated descrpton. The most mportant ponts are these: an obect of ths category s a weght 2 Y ; a 1 morphsm! s a formal sum of words n the symbols E and F where ranges over of weght, E and F havng weghts. In [37], the correspondng 1 morphsms are denoted E ; F, but we use these for elements of U q.g/. Composton s smply concatenaton of words. In fact, we wll take dempotent completon, and thus add a new 1 morphsm for every proecton from a 1 morphsm to tself (once we have added 2 morphsms). By conventon, F D F n F 1 f D. 1 ; : : : ; n / (ths somewhat dyslexc conventon s desgned to match prevous work on cyclotomc quotents by Khovanov Lauda and others). In Cauts and Lauda s graphcal calculus, ths 1 morphsm s represented by a sequence of dots on a horzontal lne labeled wth the sequence. We should warn the reader, ths conventon requres us to read our dagrams dfferently from the conventons of [26; 21; 5]; n our dagrammatc calculus, 1 morphsms pont from the left to the rght, not from the rght to the left as ndcated n [26, Secton 4]. For reasons of conventon, the 2 category U we defne s the 1 morphsm dual of Cauts and Lauda s 2 category: the obects are the same, but all the 1 morphsms are reversed. The practcal mplcaton wll be that our relatons are the reflecton through a vertcal lne of Cauts and Lauda s (wthout changng the labelng of regons). 2 morphsms are a certan quotent of the K span of certan mmersed orented 1 manfolds carryng an arbtrary number of dots whose boundary s gven by the doman sequence on the lne y D 1 and the target sequence on y D 0. We requre that any component begn and end at lke-colored elements of the 2 sequences, and that they be orented upward at an E and downward at an F. We wll descrbe ther relatons momentarly. We requre that these 1 manfolds satsfy the same genercty assumptons as proectons of tangles (no trple ponts or tangences), but ntersectons are not over- or under-crossngs; our dagrams are genunely planar. We consder these up to sotopy whch preserves ths genercty. We draw these 2 morphsms n the style of Khovanov Lauda, by labelng the regons of the plane by the weghts (obects) that the 1 morphsms are actng on. By Morse theory, we can see that these are generated by

11 An ntroducton to categorfyng quantum knot nvarants 263 a cup W E F! or 0 W F E! D C a cap 0 W! E F or W! F E 0 D 0 D D C a crossng W F F! F F D a dot yw F! F y D As n Cauts and Lauda, we wll actually consder a famly of these categores. Through out the paper, we fx a matrx of polynomals Q.u; v/ for 2 (by conventon Q D 0) valued n K. We assume that each polynomal s homogeneous of degree h ; D 2d c D 2d c where we gve the varable u degree 2d and v degree 2d. We always assume that the leadng order of Q n u s c, and that Q.u; v/ D Q.v; u/. In order to match wth [5], we take X Q.u; v/ D t u c C t v c C up v q : qc Cpc Dc c s pq Khovanov and Lauda s category s the choce Q D u c C v c. We have not yet wrtten the relatons, but we frst descrbe a gradng on the 2 morphsm spaces; the degrees are gven by deg D h ; deg Dh ; deg D h ; deg Dh ;

12 264 Ben Webster deg deg D h; d deg D h; d deg D h; d D h; d : The relatons satsfed by the 2 morphsms nclude: The cups and caps are the unts and counts of a baduncton. The morphsm y s cyclc, whereas the morphsm s double rght dual to t =t (see [5] for more detals). Any bubble of negatve degree s zero; any bubble of degree 0 s equal to 1. We must add formal symbols called fake bubbles whch are bubbles labeled wth a negatve number of dots (these are explaned n [21, Secton 3.1.1]); gven these, we have the nverson formula for bubbles, as shown n Fgure 1. C X C1 kd 1 k k D ( 1 D 2 0 > 2 Fgure 1: Bubble nverson relatons; all strands are colored wth. The 2 relatons connectng the crossng wth cups and caps, shown n Fgure 2. Oppostely orented crossngs of dfferently colored strands smply cancel, shown n Fgure 3. The endomorphsms of words only usng F (or by dualty only E s) satsfy the relatons of the quver Hecke algebra R, shown n Fgure 4. Ths categorfcaton has analogues of the postve and negatve Borels gven by the representatons of quver Hecke algebras, the algebra gven by dagrams where all strands are orented downwards, wth the relatons n Fgure 4 appled; ths algebra s dscussed n [37, Secton 4] and an earler paper of Khovanov and Lauda [20]. We denote these 2 categores U C and U. The Grothendeck group of a 2 category s a 1 category n an obvous way; the obects are exactly the same, and a morphsm s a class n the Grothendeck group of the Hom category. We wll abuse notaton by dentfyng a 1 category wth the sum of ts Hom spaces, whch s naturally an algebra, wth dstngushed dempotents gven by the denttes of the varous obects. Theorem 1.1 [49, 1.9] The obvous map nduces an somorphsm Kq 0.U/ Š PU q.g/. Ths map ntertwnes the graded Euler form on Kq 0.U/ wth Lusztg s nner product on PU q.g/.

13 An ntroducton to categorfyng quantum knot nvarants 265 D X acbd 1 b a D X acbd 1 b a D C X acbccd 1 c a b D C X acbccd 1 c a b Fgure 2: Cross and cap relatons; all strands are colored wth. By conventon, a negatve number of dots on a strand whch s not closed nto a bubble s 0. 2 The tensor product algebras 2.1 Defnton and basc propertes We now proceed to the categorfcatons of tensor products mentoned n the ntroducton. Just as wth the unversal envelopng algebra tself, we defne a pctoral category, whch s a representaton of the 2 category U. Ths means we must defne a category for each weght, a functor between these categores for each 1 morphsm and a natural transformaton for each 2 morphsm. Frst, fx a lst D. 1 ; : : : ; `/. We let L ı be the category The obects of ths category are words n E, F and new symbols I where the symbols appear n order from rght to left; thus accordng to our dyslexc conventons, when graphcally represented, they wll read left to rght.

14 266 Ben Webster D t D t Fgure 3: The cancellaton of oppostely orented crossngs wth dfferent labels We label the regons between the dots representng the letters n these words as follows: the regon at far left s labeled 0, and each tme we pass a dot t contrbutes ts weght. We assgn E weght, F weght and I weght. The morphsms are mmersed 1 manfolds matchng up wth the graphcal representaton of the obects at the top and bottom. Some strands of ths mmersed manfold are colored red and some are colored black; the black strands carry an orentaton, and as before are allowed to carry dots. The strands attached to E s must be black and orented up, those connected to F are drawn black and orented down, whle red strands connect I to I wthout any crossngs wth other red strands (crossngs wth black are fne) or self-ntersectons. The composton ab s the pcture b stacked on top of a (of course, assumng the morphsms are composable). We wll only ever be nterested n these pctures up to sotopes preservng the condtons above. Thus we are allowed to have local pctures lke:

15 An ntroducton to categorfyng quantum knot nvarants 267 = unless D = + = + = 0 and = Q.y 1 ; y 2 / k = k unless D k = C Q.y 3 ; y 2 / Q.y 1 ; y 2 / y 3 y 1 Fgure 4: The relatons of the quver Hecke algebra; these relatons are nsenstve to labelng of the plane. In order to match [5] take r D 1.

16 268 Ben Webster The black strands satsfy all the relatons of the category U, usng the labels on regons ndcated above. We must also nclude new relatons nvolvng red lnes whch are: All black crossngs and dots can pass through red lnes, wth a correcton term smlar to Khovanov and Lauda s (for the latter two relatons, we also nclude ther mrror mages): = X C b acbc1d ı ; a (1) = = The cost of a separatng a red and a black lne s addng D _ the black strand:./ dots to = (2) = If at any pont n the dagram any black lne s to the left of all reds (e, there s a value a such that the left-most ntersecton of y D a wth a strand s wth a black strand), then the dagram s 0. We wll refer to such a strand as volatng.

17 An ntroducton to categorfyng quantum knot nvarants 269 a non-volatng strand a volatng strand Fgure 5: An example of a volatng and non-volatng strand Defnton 2.1 of Lı. We let L be the dempotent completon (that s, Karouban envelope) Followng Brundan and Kleshchev, we wll sometmes use yw F! F to represent multplcaton by a dot on the a black strand, and W F F! F F to represent a crossng. Theorem 2.2 [49, 2.11] There s a representaton of U n K lnear categores sendng 7! L, wth E and F sent to addng ths symbol at the left to our lst (thus, graphcally, we add a strand at the rght), and dagrams to the obvous correspondng dagrams. We wll fnd t convenent to represent an obect n L n terms of the sequence of smple roots whch occur on black strands D. 1 ; : : : ; n /; for hstorcal reasons, we use to stand n for F and to stand n for E the weakly ncreasng map W Œ1; `! Œ0; n that sends k to m f the k th red strand s between the mth and m C 1st black strands (where by conventon, the 0-th strand s at 1 and the ` C 1-th strand s at 1). We denote the correspondng obect n L by smply.; /. Under decategorfcaton, the proectve.; / s sent to the vector F n F.`/..F.3/ F.2/C1.F.2/ F 1 v 1 / v 2 / v`/; where v 2 V s a fxed hghest weght vector; we state a more precse theorem along these lnes n Secton 2.2. Of course, as wth any dempotent complete category wth fntely many obects and fnte-dmensonal Hom spaces, L s equvalent to the category of proectve modules over the algebra M T D End.; / ;

18 270 Ben Webster where we let range over all postve strngs of roots, and over weakly ncreasng maps as above. Obvously, many readers wll be more comfortable workng wth modules over ths algebra. Gradng Ths category s graded wth degrees gven by a black/black crossng: h ;, a black dot: h ; D 2d a red/black crossng: h ; D d. Ths category s also self-dual; there s an somorphsm Hom L.A; B/! Hom L.B; A/ gven by reflectng dagrams n the horzontal axs. We wll denote ths map by. /. Representatons Recall that a fnte-dmensonal representaton of a K lnear category C s a K lnear addtve functor C! Vect K to the category of fnte-dmensonal K vector spaces. Smlarly, a graded representaton s a homogeneous functor C! gvect K. For each obect.; /, the Yoneda embeddng gves a correspondng graded representaton Hom L..; /; / whch we denote by P ;. Defnton 2.3 We let V be the category of fnte dmensonal graded representatons of L, and V be the category of fnte-dmensonal graded representatons of L. These are equvalent to the correspondng representaton categores over the algebra T. We can defne a U module structure on V usng that on L. If M s an obect n V, then we can defne E M.N / WD M.F N / F M.N / WD M.E N /. h ; C d /: Whle ths defnton may ntally look strange, t s chosen so that the Yoneda embeddng s U equvarant, snce E and F are badont. If D./ s a sngle element, then ths category has been prevously studed; V s the module category of the cyclotomc quver Hecke algebra, as descrbed by Khovanov and Lauda [20].

19 An ntroducton to categorfyng quantum knot nvarants Examples If the algebra g D sl n, then these algebras have a well-understood combnatoral structure. For one thng, we need only consder the case where conssts of fundamental weghts; all other cases arse from breakng the appearng weghts up nto fundamentals (n any order) and only consderng sequences where each of those blocks comes all together wth no black strands separatng them. We can consder (now assumed to consst only of fundamental weghts) as a lst p 1 ; : : : ; p` of ntegers between 1 and n 1, such that s the hghest weght of V p C n. Recall that an ` mult-partton s an ordered ` tuple of parttons, and that a tableau on such a partton s a numberng of the boxes by ntegers from ` dfferent alphabets, we denote 1 1 ; 2 1 ; 3 1 ; : : : ; 1 2 ; 2 2 ; : : : ; 1`; 2`; : : : ; we order these as lsted, frst accordng to whch alphabet t les n, and then by the usual order on each alphabet. A tableau s standard f all rows and columns are strctly ncreasng; k cannot appear after the k th partton; each symbol appears at most once, and f k appears, then m k appears for all m <. Each dagram carres a superstandard tableau dstngushed by only usng the k th alphabet n the k th pece of the partton, and t s the unque tableau wth the row readng word beng totally ordered. There s a standard colorng of the dagram of our multpartton wth the smple roots of sl n ; the box n the th row and k th column of the mth dagram s colored wth pm Ck. Attached to each standard tableau, we have a sequence consstng of 1, followed by the root for the box labeled 1 1, then for the box labeled 2 1, etc. After all boxes labeled wth that alphabet have been read, we add 2 and read all boxes correspondng to 1 2 ; 2 2 ; : : :, add 3 and so on. So, for n D 4, and D.! 3 ;! 1 / then the labelng by roots s gven by for example, we assocate: (! 2 ; 3;! 1 ; 1; 2; 2) 1

20 272 Ben Webster The superstandard tableau of the same shape gves: (! 2 ; 3; 2;! 1 ; 1; 2) 1 Ths rule leads to a bass for T ndexed by pars of sem-standard tableaux of the same shape on ` mult-parttons such that the dagram of the th partton fts n a p n p box (e has at most p rows and n p columns). Assocated to each standard tableau S, we have a dagram B S where the sequence at one end s the sequence descrbed above, and the other s the sequence assocated to the superstandard tableau of the same shape. Ths dagram follows the smple rule of connect elements correspondng to the same boxes. So, n the example above, we swtch the second and thrd black lnes to obtan! 2 3! In general, ths s not unque: t essentally depends on pckng a reduced expresson for the correspondng permutaton. We fx one of these n a completely arbtrary manner. Let C S;T D B S B T be product of the dagram assocated to one of the tableaux, tmes the reflecton of the dagram assocated to the other (these match along the dempotent of the superstandard tableau). Theorem 2.4 [45, 5.11] The elements C S;T for S and T a par of sem-standard tableaux of the same shape, whch fts nto boxes as above, are a graded cellular bass of T. For example, f n D 2, then one s only allowed shapes whch ft n a 1 1 box, so they are ether empty or a sngle box, and the dempotents correspondng to these have no two consecutve black lnes (and ths s the only restrcton on them). Smlarly, for a sem-standard tableau of ths shape, the frst restrcton s vacuous, and thus the only restrcton s that we must not use the k th alphabet after the k th dagram, or use the same symbol twce. For example, for D.! 1 ;! 1 ;! 1 / and D 1, the elements B S assocated to tableaux are shown n Fgure 6. Of course, the bass s easly constructed from these by matchng pars of the dagrams shown, and ther reflectons., but for reasons of space we do not explctly wrte t out.

21 An ntroducton to categorfyng quantum knot nvarants Decategorfcaton Fgure 6: The lower halves of bass vectors n one example Let v 2 V be defned nductvely by f.`/ D n, then v D v v` where v` s the hghest weght vector of V `, and s the restrcton to Œ1; ` 1 ; f.`/ n, then v D F n v, where D. 1 ; : : : ; n 1 /. Theorem 2.5 [49, 3.6] There s a canoncal somorphsm W K 0.T /! V Z gven by ŒP 7! v. Descrpton n terms of functors Ths category actually arses n a very natural way nsde the categorfcaton of the correspondng smple representaton of hghest weght D 1 C C `.

22 274 Ben Webster If and 0 are domnant weghts so that 0 s also domnant, then there s a natural functor L! L 0 commutng wth the acton of F s; one smply sends 7! (note that ths does not commute wth E s; ths s unsurprsng snce there s also a map of n modules between these representatons of the same form, and no such map of g modules). On representatons, of course, ths nduces a functor n the other drecton I 0 W V0! V commutng wth E. Theorem 2.6 [49, 3.22] There s a fully fathful functor L! V sendng.; / 7! F n F.`/I `F.`/ 1 F.` 1/I ` 1 F.` 1/ 1 (that s, by replacng all I s n the correspondng word by the functors I ). Thus, these representatons have an nterestngly recursve structure; n a sense that s mpossble on the decategorfed level, the tensor products already appear as soon as you consder hghest weght representatons and very natural functors between them. 3 Bradng functors 3.1 Bradng Recall that the category of ntegrable U q.g/ modules (of type I) s a braded category; that s, for every par of representatons V; W, there s a natural somorphsm V;W W V W! W V satsfyng varous commutatve dagrams (see, for example, [6, 5.2B], where the name quas-tensor category s used nstead). Ths bradng s descrbed n terms of an R matrx R 2 5 U.g/ U.g/, where we complete the tensor square wth respect to the kernels of fnte-dmensonal representatons, as usual. As we mentoned earler, we were left at tmes wth dffcult decsons n terms of reconclng the dfferent conventons whch have appeared n prevous work. One whch we seem to be forced nto s to use the opposte R matrx from the one used by many authors (for example, n [6]). Thus, we must be qute careful about matchng formulas wth references such as [6]. Our frst task s to descrbe a functor categorfyng the bradng. To begn wth, let B k W L! V be the functor whch sends a sequence.; / to the functor where B k..; /I. 0 ; 0 // s the K span of pctures lke the morphsms of L, begnnng at.; /, endng wth. 0 ; 0 / and havng exactly one crossng between the k th and k C 1st red strands, wth certan relatons we descrbe below. We can thnk of ths as a functor L L! gvect K. From ths perspectve, the morphsms n L act by attachng at the bottom of ths dagram, and morphsms n L act at the top.

23 An ntroducton to categorfyng quantum knot nvarants kc1 k ` 1 k kc1 ` Fgure 7: An example of an element of B k As before, we need to mod out by relatons: We mpose all local relatons from Secton 2, ncludng planar sotopy. Furthermore, we have to add the relatons (along wth ther mrror mages) whch along a black strand to slde through a red crossng: k k 1 k k 1 = k 1 k k 1 k k k 1 k k 1 = k 1 k k 1 k Defnton 3.1 Let B k be the derved functor LB k W D.V /! D.V k /. For any word n the brad generators 1 1 n n, we let B. 1 1 n n / D B 1 1 B n n Here, D.V / refers to the bounded above derved category of V. Theorem 3.2 [50, 1.5-8] The functors B are equvalences and defne a weak brad groupod acton on our categores; that s, for any brad D 1 1 n n, the functor B. 1 1 n n / only depends on the resultng brad up to somorphsm.

24 276 Ben Webster Furthermore, the map B. 1 1 n n / nduces on the level of Grothendeck groups s exactly the map nduced on tensor products of the representatons by the braded structure on representatons of U q.g/. 4 Rgdty structures 4.1 Coevaluaton and evaluaton for a par of representatons Now, we must consder the cups and caps n our theory. The most basc case of ths s D.; /, where we use D w 0 to denote the hghest weght of the dual representaton to V. It s mportant to note that V Š V, but ths somorphsm s not canoncal. In fact, the representaton K 0.T / comes wth more structure, snce t s an ntegral form V Z. In partcular, t comes wth a dstngushed hghest weght vector v h, the class of the unque smple n V whch s 1 dmensonal and concentrated n degree 0. Thus, n order to fx the somorphsm above, we need only fx a lowest weght vector v l of V, and take the unque nvarant parng such that hv h ; v l D 1. Proposton 4.1 [50, 2.3] There exsts a unque self-dual smple module L 2 V ; 0 whch s U nvarant; t s klled by all E and all F. The class of ths smple n V V can be regarded as an somorphsm V! V, whch we wll fx from now on. Recall that the coevaluaton Z..q//! V ; s the map sendng 1 to the canoncal element of the parng we have fxed, and evaluaton s the map nduced by the parng V ;! Z..q//. Defnton 4.2 Let K ; W D.gVect/! V ; be the functor RHom K. PL ; /.2h; /Œ 2 _./ and E ; W V ;! D.gVect/ be the functor L T P L Proposton 4.3 [50, 2.3] the evaluaton. The functor K ; categorfes the coevaluaton, and E ; We represent these functors as leftward orented cups as s done for the coevaluaton and evaluaton n the usual dagrammatc approach to quantum groups; ths s shown n Fgure 8.

25 An ntroducton to categorfyng quantum knot nvarants 277 K ; E ; 4.2 Rbbon structure Fgure 8: Pctures for the coevaluaton and evaluaton maps Now, n order to defne quantum knot nvarants, we must also have quantum trace and cotrace maps, whch can only be defned after one has chosen a rbbon structure. The Hopf algebra U q.g/ does not have a unque rbbon structure; n fact, topologcal rbbon elements form a torsor over the characters Y=X! f 1g. Essentally, ths acton s by multplyng quantum dmenson by the value of the character. The standard conventon s to choose the rbbon element so that all quantum dmensons are Laurent polynomals n q wth postve coeffcents; however, a smple calculaton above shows that ths choce s not compatble wth our categorfcaton! Proposton 4.4 B 1 L Š L Œ 2 _./. 2h; h; /. By Proposton 4.4, we have B 2 L D L Œ 4 _./. 4h; 2h; /: Thus, f we wsh to defne a rbbon functor R to satsfy the equatons B 2 L Š R 2 1 L D R 2 2 L D R 1 1 R 1 2 L ; whch are necessary for topologcal nvarance (as we depct n Fgure 9). Defnton 4.5 The rbbon functor R s defned by R M D M Œ2 _. /.2h ; C h ; /: Takng Grothendeck group, we see that we obtan the rbbon element n U q.g/ unquely determned by the fact that t acts on the smple representaton of hghest weght by. 1/ 2_./ q h;c2h;. Ths s the nverse of the rbbon element constructed by Snyder and Tngley n [40]; we must take nverse because Snyder and Tngley use the opposte choce of coproduct from ours. See Theorem 4.6 of that paper for a proof that ths s a rbbon element. From now on, we wll term ths the ST rbbon element.

26 278 Ben Webster = Fgure 9: The compatblty of double twst and the rbbon element It may seem strange that ths element seems more natural from the perspectve of categorfcaton than the standard rbbon element, but t s perhaps not so surprsng; the ST rbbon element s closely connected to the brad group acton on the quantum group, whch also played an mportant role n Chuang and Rouquer s early nvestgatons on categorfyng sl 2 n [7]. It s not surprsng at all that we are forced nto a choce, snce rbbon structures depend on the ambguty of takng a square root; whle numbers always have 2 or 0 square roots n any gven feld (of characterstc 2), a functor wll often only have one. Ths dfferent choce of rbbon element wll not serously affect our topologcal nvarants; we smply multply the nvarants from the standard rbbon structure by a sgn dependng on the framng of our lnk and the Frobenus Schur ndcator of the label, as we descrbe precsely n Proposton 5.6. = Fgure 10: Changng the orentaton of a cap Proposton 4.6 [50, 2.11] The quantum trace and cotrace for the ST rbbon structure are categorfed by the functors C ; W D.gVect/! V ; gven by RHom. PL ; /.2h; /Π2 _./ and T ; W V ;! D.gVect/ gven by T P L :

27 An ntroducton to categorfyng quantum knot nvarants 279 C ; T ; Fgure 11: Pctures for the quantum (co)trace 4.3 Coevaluaton and quantum trace n general More generally, whenever we are presented wth a sequence and a domnant weght, we wsh to have a functor relatng the categores and C D. 1 ; : : : ; 1 ; ; ; ; : : : ; `/. Ths wll be gven by left tensor product wth a partcular bmodule. The coevaluaton bmodule K C 1 s generated by the dagrams of the form v ` k k k ` 1.s k 1 / k where v s an element of L and dagrams only nvolvng the strands between and act n the obvous way, modulo the relaton (and ts mrror mage). v v = One can thnk of the relaton above as categorfyng the equalty.f v/ K D F.v K/ for any nvarant element K.

28 280 Ben Webster Defnton 4.7 The coevaluaton functor s K C D RHom T.KC ; /.2h; /Œ 2_./ W V! V C : Smlarly, the quantum trace functor s the left adont to ths gven by T C D L T C KC W VC! V : The evaluaton and quantum cotrace are defned smlarly. Snce K C s proectve as a rght module, Hom wth t gves an exact functor. The quantum trace functor, however, s very far from beng exact n the usual t -structure. Proposton 4.8 [50, 2.13] trace. K C C categorfes the coevaluaton and T the quantum 5 Knot nvarants 5.1 Constructng knot and tangle nvarants Now, we wll use the functors from the prevous secton to construct tangle nvarants. Usng these as buldng blocks, we can assocate a functor ˆ.T /W V! V to any dagram of an orented labeled rbbon tangle T wth the bottom ends gven by D f 1 ; : : : ; `g and the top ends labeled wth D f 1 ; : : : ; m g. As usual, we choose a proecton of our tangle such that at any heght (fxed value of the x coordnate) there s at most a sngle crossng, sngle cup or sngle cap. Ths allows us to wrte our tangle as a composton of these elementary tangles. For a crossng, we gnore the orentaton of the knot, and separate crossngs nto postve (rght-handed) and negatve (left-handed) accordng to the upward orentaton we have chosen on R 2. To a postve crossng of the and C 1st strands, we assocate the bradng functor B. To a negatve crossng, we assocate ts adont B 1 are somorphc, snce B s an equvalence). (the left and rght adonts For the cups and caps, t s necessary to consder the orentaton, usng the conventons shown n Fgures 8 and 11. To a clockwse orented cup, we assocate the coevaluaton.

29 An ntroducton to categorfyng quantum knot nvarants 281 To a clockwse orented cap, we assocate the quantum trace. To a counter-clockwse cup, we assocate the quantum cotrace. To a counter-clockwse cap, we assocate the evaluaton. Theorem 5.1 [50, 3.1] The functor ˆ.T / does not depend (up to somorphsm) on the proecton of T. The map nduced by ˆ.T /W V! V on the Grothendeck groups V! V s that assgned to a rbbon tangle by the structure maps of the category of U q.g/ modules wth the ST rbbon structure. In partcular, the complex ˆ.T /.K/ for a closed lnk n an nvarant of K and ts graded Euler characterstc of s the quantum knot nvarant for the ST rbbon element. Theorem 5.2 [50, 3.2] The cohomology of ˆ.T /.K/ s fnte-dmensonal n each homologcal degree, and each graded degree s a complex wth fnte-dmensonal total cohomology. In partcular the bgraded Poncaré seres '.T /.q; t/ D X. t/ dm q H.ˆ.T /.K// s a well-defned element of ZŒq 1 =D ; q 1 =D..t//. The only case where the nvarant s known to be fnte-dmensonal s when the weghts are all mnuscule; recall that a domnant weght s called mnuscule f every weght wth a non-zero weght space n V s n the Weyl group orbt of. Proposton 5.3 [50, 3.3] s fnte-dmensonal. If all are mnuscule, then the cohomology of ˆ.T /.K/ Unfortunately, the cohomology of the complex ˆ.T /.K/ s not always fnte-dmensonal. Ths can be seen n examples as smple as the unknot U for g D sl 2 and label 2. In fact: Proposton 5.4 [50, 3.4] ' 2.U / D q 2 t 2 C 1 C q 2 t 2 C q 2 q 2 t 1 t 2 q 4. It s easy to see that the Euler characterstc s q 2 C 1 C q 2 D Œ3 q, the quantum dmenson of V 2. As ths example shows, nfnte-dmensonalty of nvarants s extremely typcal behavor. Ths same phenomenon of nfnte dmensonal vector spaces categorfyng ntegers has also appeared n the work of Frenkel, Sussan and Stroppel [13], and n fact, ther work could be translated nto the language of ths paper usng the equvalences of [49, Secton 5]; t would be qute nterestng to work out ths

30 282 Ben Webster correspondence n detal. Smlar nfnte-dmensonal behavor appeared n the work of Cooper, Hogancamp and Krushkal on colored Jones polynomals and Jones Wenzl proectors usng Khovanov s dagrammatc calculus [10; 9]. Conecture 5.5 The nvarant ˆ.L/ for a lnk L s only fnte-dmensonal f all components of L are labeled wth mnuscule representatons. Some care must be exercsed wth the normalzaton of these nvarants, snce as we noted n Secton 4.2, they are the Reshetkhn Turaev nvarants for a slghtly dfferent rbbon element from the usual choce. However, the dfference s easly understood. Let L be a lnk drawn n the blackboard framng, and let L be ts components, wth L labeled wth. Recall that the wrthe wr.k/ of a orented rbbon knot s the lnkng number of the two edges of the rbbon; ths can be calculated by drawng the lnk the blackboard framng and takng the dfference between the number of postve and negatve crossngs. Here we gve a slght extenson of the proposton of Snyder and Tngley relatng the nvarants for dfferent framngs [40, Theorem 5.21]: Proposton 5.6 [50, 3.8] The nvarants attached to L by the standard and Snyder Tngley rbbon elements dffer by the scalar Q. 1/2_. /.wr.l / 1/. Snce one of the man reasons for nterest n these quantum nvarants of knots s ther connecton to Chern Smons theory and nvarants of 3 manfolds, t s natural to ask: Queston 5.7 Can these nvarants glue nto a categorfcaton of the Wtten Reshetkhn Turaev nvarants of 3 manfolds? Remark 5.8 The most nave ansatz for categorfyng Chern Smons theory, followng the development of Reshetkhn and Turaev [36] would assocate such that a category C. / to each surface, and an obect n C. / to each somorphsm of wth the boundary of a 3 manfold the nvarants K we have gven are the Ext spaces of ths obect for a knot complement wth fxed generatng set of C.T 2 / labeled by the representatons of g, and the categorfcaton of the WRT nvarant of a Dehn fllng s the Ext space of ths obect wth another assocated to the torus fllng. Whle some hnts of ths structure appear n the constructons of ths paper, t s far from clear how they wll combne.

31 An ntroducton to categorfyng quantum knot nvarants Functoralty One of the most remarkable propertes of Khovanov homology s ts functoralty wth respect to cobordsms between knots; ths functoralty was conectured by Khovanov [16], shown to exst up to sgn by Jacobsson [14] and gven a more explct topologcal descrpton by Bar-Natan [2]; fnally, a way around the sgn problems was gven by Clark, Morrson and Walker [8]. Ths property s not only theoretcally satsfyng but also played an mportant role n Rasmussen s proof of the unknottng number of torus knots [34]. Thus, we certanly hope to fnd a smlar property for our knot homologes. Whle we cannot present a complete pcture at the moment, there are promsng sgns, whch we explan n ths secton. We must restrct ourselves to the case where the weghts are mnuscule, snce even the basc results we prove here do not hold n general. We wll assume ths hypothess throughout ths subsecton. The weakest form of functoralty s puttng a Frobenus structure on the vector space assocated to a crcle. Ths vector space, as we recall, s A D Ext.L ; L /Œ2 _./.2h; /: Ths algebra s naturally bgraded by the homologcal and nternal gradngs. The algebra structure on t s that nduced by the Yoneda product. Theorem 5.9 [50, 3.11] For mnuscule weghts, we have a canoncal somorphsm SL Š L. 4h; /Œ 4 _./ : Thus, the functors K and T are badont up to shft. In partcular, Ext 4h;.L ; L / Š Hom.L ; L /, and the dual of the unt W Ext 4h;.L ; L /! K s a symmetrc Frobenus trace on A of degree 4h; One should consder ths as an analogue of Poncaré dualty, and thus a pece of evdence for A s relatonshp to cohomology rngs. It would be enough to show that ths algebra s commutatve to establsh the functoralty for flat tangles; we smply use the usual translaton between 1+1 dmensonal TQFTs and commutatve Frobenus algebras (for more detals, see the book by Kock [25]). At the moment, not even ths very weak form of functoralty s known. Queston 5.10 Is there another nterpretaton of the algebra A? Is t the cohomology of a space?