KLR algebras and knot homology II
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1 KLR algebras and knot homology II Ben Webster Unversty of Vrgna June 6, 013 Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
2 The case of sl So, remember, last tme I ntroduced an algebra T l attached to the sl representaton (C ) l. = + = 0 = = = + = = 0 Ben Webster (UVA) KLR algebras and knot homology II June 6, 013 / 5
3 Representatons One natural class of T l representatons s the projectves ( just fx sequence at the top, no relatons): P 0, = P 1,0, = Under the somorphsm K(T l -mod) (C ) l, we send [P a1,,a l ] F a l ([P a1,,a l 1 ] v + ) There are also standard modules S a1,,a l, the quotent of P a1,,a l by the relatons = 0 Under the somorphsm K(T l -mod) (C ) l, we send [S a1,,a l ] F a 1 v + F a l v + Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
4 Representatons Let s look agan at the structure of the case of T wth one black lne: S 1,0 P 0,1 S 0,1 P 1,0 = S 1,0 L The only module we can see n ths dagram klled by E s L. Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
5 The case of sl The cup (up to shft) s assocated to derved tensor product wth the bmodule: Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
6 The case of sl The cup (up to shft) s assocated to derved tensor product wth the bmodule: Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
7 The case of sl The cup (up to shft) s assocated to derved tensor product wth the bmodule: Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
8 The case of sl The cup (up to shft) s assocated to derved tensor product wth the bmodule: Of course, we need to have some relatons: = 0 = 0 = 0 = = Ths s nsertng two new tensor factors, and puttng n the nvarant module for those two factors. Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
9 The case of sl So, for a crcle, we get elements of the bmodule for each pcture: It s easy to check that we can smplfy so that the bubble s separate. But we have to thnk a bt harder than ths; the functor for a cup sn t exact! You need to use a projectve resoluton!
10 The case of sl So, for a crcle, we get elements of the bmodule for each pcture: It s easy to check that we can smplfy so that the bubble s separate. But we have to thnk a bt harder than ths; the functor for a cup sn t exact! You need to use a projectve resoluton!
11 The case of sl So, for a crcle, we get elements of the bmodule for each pcture: It s easy to check that we can smplfy so that the bubble s separate. But we have to thnk a bt harder than ths; the functor for a cup sn t exact! You need to use a projectve resoluton! Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
12 The case of sl We can evaluate compostons by notng that = = 0 = 0 So, we can see the relatons Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
13 The case of sl We can evaluate compostons by notng that = = 0 = 0 So, we can see the relatons Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
14 The case of sl We can evaluate compostons by notng that = = 0 = 0 So, we can see the relatons Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
15 The case of sl We can evaluate compostons by notng that = = 0 = 0 So, we can see the relatons = Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
16 The case of sl We can evaluate compostons by notng that = = 0 = 0 So, we can see the relatons Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
17 The case of sl We can evaluate compostons by notng that = = 0 = 0 So, we can see the relatons = Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
18 The acton of cobordsms Theorem (Chatav) There s an acton of the cobordsm -category on the categores D b (T n -mod). In fact, one can easly check that ths acton satsfes the Bar-Natan relatons (S,T,G,NC). Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
19 The acton of cobordsms Theorem (Chatav) There s an acton of the cobordsm -category on the categores D b (T n -mod). In fact, one can easly check that ths acton satsfes the Bar-Natan relatons (S,T,G,NC). Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
20 The acton of cobordsms Theorem (Chatav) There s an acton of the cobordsm -category on the categores D b (T n -mod). In fact, one can easly check that ths acton satsfes the Bar-Natan relatons (S,T,G,NC). Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
21 The acton of cobordsms Theorem (Chatav) There s an acton of the cobordsm -category on the categores D b (T n -mod). In fact, one can easly check that ths acton satsfes the Bar-Natan relatons (S,T,G,NC). Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
22 The acton of cobordsms Theorem (Chatav) There s an acton of the cobordsm -category on the categores D b (T n -mod). In fact, one can easly check that ths acton satsfes the Bar-Natan relatons (S,T,G,NC). Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
23 The acton of cobordsms Fun game: look at a constructon n BN world, transport t to modules over T l. Dror tells us that ( ) = Cone The cone of an surjectve map of bmodules over T l s ts kernel. The kernel s the set of dagrams where we can see a par of red lnes wth no black between them. Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
24 The acton of cobordsms Fun game: look at a constructon n BN world, transport t to modules over T l. Dror tells us that ( ) = Cone The cone of an surjectve map of bmodules over T l s ts kernel. The kernel s the set of dagrams where we can see a par of red lnes wth no black between them. To remnd that we have to go around, snap the red strands together. Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
25 Comparson to Khovanov homology These bmodules defne a functor D b (T l 1 -mod) D b (T l -mod) for every tangle connectng l 1 ponts to l ; I should say tangle projecton but we already know sotopy nvarance! Theorem The resultng knot nvarant s Khovanov homology. Note that we can make functoralty work as usual usng the acton of cobordsms. If you want to avod sgn problems, you need to use the Morrson-Walker dsorentaton scheme. ( = Cone ) ) ) Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
26 Jones-Wenzl projectors Cooper and Krushkal constructed a categorfed JW projector n Bar-Natan s category. The mage over T l s surprsngly easy to descrbe. w/ relatons a = b You can thnk of ths as realzng T (n 1,...,n l ) -mod as a quotent category of T n 1+ +n l -mod. Ths allows us to transport ther categorfcaton of colored Jones to our pcture. Actually, ths s a very specal case of the general constructon. Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
27 Dagrams So, let s move on to the general case. Let T λ be the algebra generated by dagrams of red and black lnes; red lnes are labeled by weghts, black by smple roots. λ 1 j n deg = α j, α j 1 j j+1 λ l deg = α j, α j+1 1 j λ k n deg = α j, λ k Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
28 Relatons = + = a j + a j j j j = 0 j = j + a a+b=a j 1 j b =α (λ) λ λ λ = λ + a a+b=α (λ) 1 λ b = α (λ) = 0 λ λ Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
29 Dagrams Theorem (W.) The Grothendeck group of the category C(λ) := D(T λ -mod) s the tensor product V λ = V λ1 V λl ; the categorcal g-acton nduces the usual acton of g on V λ. So, I want to defne a 4-d extended TQFT that assgns C(λ) to a dsk, and Inv(C(λ)) to a sphere labeled wth λ 1,..., λ l. Now, we want to assgn functors to cobordsms wth Wlson lnes (.e. a tangle). Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
30 Brad functors I already told you what these functors are, but let me tell you a lttle more carefully. A postve crossng s sent to the bmodule B j where the jth and j + 1st red lnes cross exactly once. All relatons nvolvng the red-red crossng are nave sotopes. λ j λ j+1 λ j λ j+1 Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
31 Brad operators Theorem The functors L B : C(λ) C(s λ) are equvalences that satsfy the brad relatons. The nduced maps on Grothendeck groups V λ V λ s the bradng/r-matrx tmes flp/map on Hlbert spaces nduced by cobordsm. Ths helps us address a naggng pont, whch s that the category depends on an orderng of the weghts λ, whereas on the surface they are allowed to move around. The brad group s exactly the mappng class group of a punctured dsk (fxng the boundary), so we expect t to act on the category C(λ). Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
32 Tangle nvarants Assocated to a cup, we also have a functor. Lke the sl case, we just draw the cup n the red lnes, but we to have somethng pop out when we ht the mnmum. Just as before, you have to nsert a copy of the unque nvarant smple. It s a bt more unpleasant to say exactly what that means; bascally you take black lnes correspondng to gong from the hghest to lowest weght vector n V µ, and get 0 f pull a strand out of the cup. Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
33 Tangle nvarants Assocated to a cup, we also have a functor. Lke the sl case, we just draw the cup n the red lnes, but we to have somethng pop out when we ht the mnmum. µ µ µ µ v Just as before, you have to nsert a copy of the unque nvarant smple. It s a bt more unpleasant to say exactly what that means; bascally you take black lnes correspondng to gong from the hghest to lowest weght vector n V µ, and get 0 f pull a strand out of the cup. Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
34 Cups and caps Ths pcture and ts mrror mage defne two bmodules. Theorem The functors attached to cups and caps categorfy the (co)evaluaton and quantum (co)trace maps on tensor products (that s, those assocated to the Wlson operators). These do satsfy the expected adjunctons, so we get an nvarant of flat tangles; however, we don t seem to get an acton of cobordsms n general. We only get ths when λ s are all mnuscule. Outsde the mnuscule case, we won t even get a fnte dmensonal nvarant for a crcle! The problem s that the nvarant smple mght not have a fnte resoluton. Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
35 Tangle nvarants Theorem For every rbbon tangle τ n D I, we have a functor F τ : C(λ) C(λ ) between the categores assocated to the top and bottom, such that F τ τ F τ F τ, whch categorfes the Reshetkhn-Turaev constructon. In partcular, we have a doubly graded nvarant attached to lnks wth components labeled by g-representatons. The Euler characterstc of ths homology s the nvarant obtaned from Chern-Smons theory. Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
36 The case of sl 3 For sl, I could rewrte everythng n terms of Dror s constructon. For sl 3, t s gong to be slghtly harder. In addton to the cups and caps, we need some trvalent vertces The relatons are qute smlar to those of the cup: = = 0 = ± Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
37 The case of sl 3 For sl, I could rewrte everythng n terms of Dror s constructon. For sl 3, t s gong to be slghtly harder. In addton to the cups and caps, we need some trvalent vertces The relatons are qute smlar to those of the cup: = = 0 = ± Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
38 The case of sl 3 The same argument as n sl shows that we have an exact sequence Usng adjunctons, one can also show that These look pretty remnscent of Khovanov s approach to sl 3 homology. Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
39 Foams Khovanov descrbed an approach to categorfyng the RT nvarant for C 3 as a module over sl 3 whch s remnscent of Dror s pcture, but harder. You have to allow cobordsms wth sheets labeled by fundamental representatons, and sngulartes lke our bmodules, called foams. Theorem There s an acton of sl 3 foams on these bmodules, whch matches the bradng bmodules wth the expected complexes of foams. In partcular, the knot homologes we ve constructed match the Khovanov sl 3 homology. Ben Webster (UVA) KLR algebras and knot homology II June 6, 013 / 5
40 The case of sl n There are smlar descrptons for sl n, though consderably more complcated. You end up wth resolutons lke For experts: ths s a Koszul dual categorcal acton of sl e categorfyng skew Howe dualty, and the bradng functors are exactly the Chuang-Rouquer brad functors for ths dual acton. Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
41 The case of sl n There s an already known nvarant for the defnng representaton C n of sl n : Khovanov-Rozansky homology (wth extensons to other fundamentals by Wu and Yonezawa). One queston that has bugged me for a long tme s whether ths matches the nvarants already dscussed. You can easly match my approach wth earler work of Mazorchuk and Stroppel n ths case Based on recent work of Mackaay and Yonezawa, I can now assert: Theorem? They match! I consder the queston mark to be an excess of cauton, but at the moment, the argument s unecessarly crcutous, and should be made more drect. Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
42 Thanks for your attenton. Ben Webster (UVA) KLR algebras and knot homology II June 6, / 5
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