Categorifying quantum knot invariants
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- Vincent Berry
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1 Categorfyng quantum knot nvarants Ben Webster U. of Oregon November 25, 2010 Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
2 Ths talk s onlne at Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
3 Roadmap Knot nvarants quantum groups U q (g) rbbon category of U q (g)-reps quantum knot polynomals (Jones polynomal, etc.) Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
4 Roadmap Knot nvarants quantum groups U q (g) HAVE Khovanov-Lauda/Rouquer 2-categores U rbbon category of U q (g)-reps??? quantum knot polynomals (Jones polynomal, etc.) quantum knot homologes WANT Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
5 Roadmap Knot nvarants quantum groups U q (g) HAVE Khovanov-Lauda/Rouquer 2-categores U rbbon category of U q (g)-reps rbbon 2-category of U-reps??? quantum knot polynomals (Jones polynomal, etc.) quantum knot homologes WANT Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
6 Roadmap Knot nvarants quantum groups U q (g) HAVE Khovanov-Lauda/Rouquer 2-categores U rbbon category of U q (g)-reps categorfcatons of tensor products of smples! quantum knot polynomals (Jones polynomal, etc.) quantum knot homologes HAVE Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
7 Knot nvarants Reshetkhn-Turaev nvarants Bradng Let me brefly ndcate how the left sde of the dagram works. Quantum groups are deformatons of unversal envelopng algebras. Perhaps the most mportant thng about them s that they deform the tensor product of U(g) representatons. Gven two reps V, W, we stll have a U q (g)-acton on V W. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
8 Knot nvarants Reshetkhn-Turaev nvarants Bradng Let me brefly ndcate how the left sde of the dagram works. Quantum groups are deformatons of unversal envelopng algebras. Perhaps the most mportant thng about them s that they deform the tensor product of U(g) representatons. Gven two reps V, W, we stll have a U q (g)-acton on V W. However, n ths new defnton, the obvous map V W W V s not a map of representatons. Luckly, ths can be fxed by changng the map a lttle bt, and multplyng by a formal sum R U q (g) U q (g) called the unversal R-matrx. b V,W : V W R V W flp W V Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
9 Knot nvarants Reshetkhn-Turaev nvarants Bradng Proposton The maps b V,W make U q (g) nto a braded monodal category. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
10 Knot nvarants Reshetkhn-Turaev nvarants Bradng Proposton The maps b V,W make U q (g) nto a braded monodal category. One way to thnk about ths fact s that f you represent b V,W Then the maps nduced by swtchng factors of bg tensor products satsfy the brad relatons. (1 W b U,V )(b U,W 1 V )(1 U b V,W ) (b V,W 1 U )(1 V b U,W )(b U,V 1 W ) = On the other hand b V,W b W,V 1, as the pcture above suggests. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
11 Knot nvarants Reshetkhn-Turaev nvarants Cups and caps The other mportant structure on representatons of a quantum group s takng dual of representatons. As wth swtchng tensor factors, we have to be careful about left and rght. There s a contravarant functor V V called rght dual (there s also left dual whch s the same vector space wth a dfferent U q (g)-actons). Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
12 Knot nvarants Reshetkhn-Turaev nvarants Cups and caps The other mportant structure on representatons of a quantum group s takng dual of representatons. As wth swtchng tensor factors, we have to be careful about left and rght. There s a contravarant functor V V called rght dual (there s also left dual whch s the same vector space wth a dfferent U q (g)-actons). The category of U q (g)-representatons has canoncal maps evaluaton V V C(q), represented by coevaluaton C(q) V V, represented by If you want the maps the other way, you need to take left dual. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
13 Knot nvarants Reshetkhn-Turaev nvarants Cups and caps The other mportant structure on representatons of a quantum group s takng dual of representatons. As wth swtchng tensor factors, we have to be careful about left and rght. There s a contravarant functor V V called rght dual (there s also left dual whch s the same vector space wth a dfferent U q (g)-actons). The category of U q (g)-representatons has canoncal maps evaluaton V V C(q), represented by coevaluaton C(q) V V, represented by If you want the maps the other way, you need to take left dual. Not all s lost! After all, we have a map whch swtches tensor factors. But should we take or? Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
14 Knot nvarants Reshetkhn-Turaev nvarants Cups and caps Of course, we can t play favortes. Instead we should take the geometrc mean. If V s rreducble, there s a unque constant a V C(q) (actually a power of q) such that 1 av = a V. A functoral choce of a V s called a rbbon structure. The reason for the name s that f we nterpret the dagrams as drawn wth rbbon, then they are wth a left and rght twst added, respectvely. Defnton Ths map s called quantum trace and ts vertcal flp s called quantum cotrace. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
15 Knot nvarants Reshetkhn-Turaev nvarants Cups and caps Ths allows us to assocate a map for any orented tangle labeled wth representatons, by assocatng the bradng to a crossng and approprate trace or evaluaton to cups: W V V V C[q, q 1 ] V W C[q, q 1 ] V V Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
16 Knot nvarants Reshetkhn-Turaev nvarants Cups and caps Ths allows us to assocate a map for any orented tangle labeled wth representatons, by assocatng the bradng to a crossng and approprate trace or evaluaton to cups: W V V V C[q, q 1 ] V W C[q, q 1 ] V V Composng these together for a gven lnk results n a scalar: the Reshetkhn-Turaev nvarant for that labelng. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
17 A hstorcal nterlude Knot nvarants Cups and caps Progress has been made on categorfyng these n a pecemeal fashon for a whle Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
18 A hstorcal nterlude Knot nvarants Cups and caps Progress has been made on categorfyng these n a pecemeal fashon for a whle Khovanov ( 99): Jones polynomal (C 2 for sl 2 ). Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
19 A hstorcal nterlude Knot nvarants Cups and caps Progress has been made on categorfyng these n a pecemeal fashon for a whle Khovanov ( 99): Jones polynomal (C 2 for sl 2 ). Oszvath-Szabo, Rasmussen ( 02): Alexander polynomal (whch s actually a gl(1 1) nvarant, and doesn t ft nto our general pcture). Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
20 A hstorcal nterlude Knot nvarants Cups and caps Progress has been made on categorfyng these n a pecemeal fashon for a whle Khovanov ( 99): Jones polynomal (C 2 for sl 2 ). Oszvath-Szabo, Rasmussen ( 02): Alexander polynomal (whch s actually a gl(1 1) nvarant, and doesn t ft nto our general pcture). Khovanov ( 03): C 3 for sl 3. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
21 A hstorcal nterlude Knot nvarants Cups and caps Progress has been made on categorfyng these n a pecemeal fashon for a whle Khovanov ( 99): Jones polynomal (C 2 for sl 2 ). Oszvath-Szabo, Rasmussen ( 02): Alexander polynomal (whch s actually a gl(1 1) nvarant, and doesn t ft nto our general pcture). Khovanov ( 03): C 3 for sl 3. Khovanov-Rozansky ( 04): C n for sl n. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
22 A hstorcal nterlude Knot nvarants Cups and caps Progress has been made on categorfyng these n a pecemeal fashon for a whle Khovanov ( 99): Jones polynomal (C 2 for sl 2 ). Oszvath-Szabo, Rasmussen ( 02): Alexander polynomal (whch s actually a gl(1 1) nvarant, and doesn t ft nto our general pcture). Khovanov ( 03): C 3 for sl 3. Khovanov-Rozansky ( 04): C n for sl n. Stroppel-Mazorchuk, Sussan ( 06-07): C n for sl n. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
23 A hstorcal nterlude Knot nvarants Cups and caps Progress has been made on categorfyng these n a pecemeal fashon for a whle Khovanov ( 99): Jones polynomal (C 2 for sl 2 ). Oszvath-Szabo, Rasmussen ( 02): Alexander polynomal (whch s actually a gl(1 1) nvarant, and doesn t ft nto our general pcture). Khovanov ( 03): C 3 for sl 3. Khovanov-Rozansky ( 04): C n for sl n. Stroppel-Mazorchuk, Sussan ( 06-07): C n for sl n. Cauts-Kamntzer ( 06): C n for sl n. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
24 A hstorcal nterlude Knot nvarants Cups and caps Progress has been made on categorfyng these n a pecemeal fashon for a whle Khovanov ( 99): Jones polynomal (C 2 for sl 2 ). Oszvath-Szabo, Rasmussen ( 02): Alexander polynomal (whch s actually a gl(1 1) nvarant, and doesn t ft nto our general pcture). Khovanov ( 03): C 3 for sl 3. Khovanov-Rozansky ( 04): C n for sl n. Stroppel-Mazorchuk, Sussan ( 06-07): C n for sl n. Cauts-Kamntzer ( 06): C n for sl n. Khovanov-Rozansky( 06): C n for so n. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
25 A hstorcal nterlude Knot nvarants Cups and caps Progress has been made on categorfyng these n a pecemeal fashon for a whle Khovanov ( 99): Jones polynomal (C 2 for sl 2 ). Oszvath-Szabo, Rasmussen ( 02): Alexander polynomal (whch s actually a gl(1 1) nvarant, and doesn t ft nto our general pcture). Khovanov ( 03): C 3 for sl 3. Khovanov-Rozansky ( 04): C n for sl n. Stroppel-Mazorchuk, Sussan ( 06-07): C n for sl n. Cauts-Kamntzer ( 06): C n for sl n. Khovanov-Rozansky( 06): C n for so n. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
26 A hstorcal nterlude Knot nvarants Cups and caps Progress has been made on categorfyng these n a pecemeal fashon for a whle p Khovanov ( 99): Jones polynomal (C 2 for sl 2 ).? Oszvath-Szabo, Rasmussen ( 02): Alexander polynomal (whch s actually a gl(1 1) nvarant, and doesn t ft nto our general pcture). p Khovanov ( 03): C 3 for sl 3. c Khovanov-Rozansky ( 04): C n for sl n. p Stroppel-Mazorchuk, Sussan ( 06-07): C n for sl n. c Cauts-Kamntzer ( 06): C n for sl n. c Khovanov-Rozansky( 06): C n for so n. What the categorfcaton pcture of ths talk gves s a unfed, pctoral constructon that should nclude all of these. For that, we need tensor products.. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
27 A hstorcal nterlude Knot nvarants Cups and caps Progress has been made on categorfyng these n a pecemeal fashon for a whle p Khovanov ( 99): Jones polynomal (C 2 for sl 2 ).? Oszvath-Szabo, Rasmussen ( 02): Alexander polynomal (whch s actually a gl(1 1) nvarant, and doesn t ft nto our general pcture). p Khovanov ( 03): C 3 for sl 3. c Khovanov-Rozansky ( 04): C n for sl n. p Stroppel-Mazorchuk, Sussan ( 06-07): C n for sl n. c Cauts-Kamntzer ( 06): C n for sl n. c Khovanov-Rozansky( 06): C n for so n. What the categorfcaton pcture of ths talk gves s a unfed, pctoral constructon that should nclude all of these. For that, we need tensor products. p=proven, c=conjectured. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
28 Tensor products Tensor products Defnton We ntroduce a graphcal calculus for elements of V λ = V λ1 V λl. A downward black lne on the rght means actng by F. A red lne at the rght labeled by λ corresponds to v λ, where v λ s the hghest weght vector of V λ. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
29 Tensor products Tensor products Defnton We ntroduce a graphcal calculus for elements of V λ = V λ1 V λl. A downward black lne on the rght means actng by F. A red lne at the rght labeled by λ corresponds to v λ, where v λ s the hghest weght vector of V λ. So, we obtan a spannng set of V λ consstng of vectors lke F (v λ1 F j v λ2 ) λ 2 λ 2 j λ 2 α j λ 1 λ 1 + λ 2 α j λ 1 + λ 2 α j + α Morphsms between these look lke λ 2 λ 2 α λ 1 α α j λ 1 + λ 2 α j + α λ 2 λ 2 j λ 2 α j λ 1 λ 1 + λ 2 α j λ 1 + λ 2 α j + α Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
30 Tensor product algebras Tensor products Defnton Now we defne an algebra T λ generated by pctures consstng of red strands, colored wth the hghest weghts λ from left to rght non-red strands, colored wth smple roots, each labeled wth any number of dots Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
31 Tensor product algebras Tensor products Defnton Now we defne an algebra T λ generated by pctures consstng of red strands, colored wth the hghest weghts λ from left to rght non-red strands, colored wth smple roots, each labeled wth any number of dots wth the restrctons that strands must begn on y = 0, end on y = 1 and can never be horzontal (so, we stop drawng down arrows). red strands can never cross Product s gven by stackng (and s 0 f ends don t match). Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
32 Tensor product algebras Tensor products Defnton Now we defne an algebra T λ generated by pctures consstng of red strands, colored wth the hghest weghts λ from left to rght non-red strands, colored wth smple roots, each labeled wth any number of dots wth the restrctons that strands must begn on y = 0, end on y = 1 and can never be horzontal (so, we stop drawng down arrows). red strands can never cross Product s gven by stackng (and s 0 f ends don t match). λ 1 λ 2 λ 3 Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
33 Tensor products Tensor products Defnton Now, we need to mpose some relatons: λ = = λ λ λ j = = j + a+b=λ 1 a b any dagram wth a black lne at the far left s 0. λ λ = = Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
34 Tensor products Defnton Tensor products = unless = j = + j j = 0 = unless = k j j k j k = Q j (y 1, y 2 ) = + Qj(y3, y2) Qj(y1, y2) y3 y1 j j j j j Q j (u, v) = Q j (v, u) = { 1 j ± 1 au + bv = j + 1 Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
35 An example Tensor products Defnton In the case where g = sl 2 and λ = (1, 1), the algebras T λ α are easly descrbed as follows: Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
36 An example Tensor products Defnton In the case where g = sl 2 and λ = (1, 1), the algebras Tα λ are easly descrbed as follows: T (1,1) 2 = k: t s multples of the dagram whch s just a par of red lnes. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
37 An example Tensor products Defnton In the case where g = sl 2 and λ = (1, 1), the algebras Tα λ are easly descrbed as follows: T (1,1) 2 = k: t s multples of the dagram whch s just a par of red lnes. T (1,1) 0 s spanned by,,,, Ths s a regular block of category O for sl 2. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
38 An example Tensor products Defnton In the case where g = sl 2 and λ = (1, 1), the algebras Tα λ are easly descrbed as follows: T (1,1) 2 = k: t s multples of the dagram whch s just a par of red lnes. T (1,1) 0 s spanned by,,,, Ths s a regular block of category O for sl 2. = End(k 3 ). T (1,1) 2 Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
39 Tensor products Comparson to Lauda-Vazran Defnton If there s only red lne, then we only get one new nterestng relaton: α (λ) = = 0 λ λ If there s only one red lne labeled wth λ, then we just get back the category for a smple representaton. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
40 Grothendeck groups Tensor products Defnton Theorem The GG of T λ -pmod s the Lusztg ntegral form of V λ, sendng the functor F to the acton of F, and the functor I λ (addng a red lne) to the ncluson V v hgh V V λ. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
41 Grothendeck groups Tensor products Defnton Theorem The GG of T λ -pmod s the Lusztg ntegral form of V λ, sendng the functor F to the acton of F, and the functor I λ (addng a red lne) to the ncluson V v hgh V V λ. But we d lke to talk about the category of all module T λ -mod, whch doesn t have the same Grothendeck group: the map K 0 (T λ -pmod) K 0 (T λ -mod) s njectve, but not surjectve, snce not all smple modules have fnte projectve resolutons. (Thnk about k[x]/(x 2 )). However, ths map s an somorphsm after tensorng wth C((q)), so every fnte dmensonal T λ -module defnes a class n V λ. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
42 Back to geometry Tensor products Defnton Ths category has a conjectural descrpton n terms of geometry of quver varetes. In my talk on Monday, I descrbed how to assocate a category O to a C -acton on a symplectc varety; attached to a tensor product, we have an assocated quver varety wth C -acton, and thus a category O λ. Conjecture D + (O λ ) = T λ -dgmod. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
43 Back to geometry Tensor products Defnton Ths category has a conjectural descrpton n terms of geometry of quver varetes. In my talk on Monday, I descrbed how to assocate a category O to a C -acton on a symplectc varety; attached to a tensor product, we have an assocated quver varety wth C -acton, and thus a category O λ. Conjecture D + (O λ ) = T λ -dgmod. Whle ths s not proven, a rather smlar result holds usng Zheng s categores D λ, whch are constructed as localzatons of the category of sheaves on Lusztg s quver varetes. Proposton D λ = T λ -dgmod. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
44 Bases Tensor products Defnton What does the representaton theory of ths algebra look lke? Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
45 Bases Tensor products Defnton What does the representaton theory of ths algebra look lke? Projectves are just summands of the modules P κ = T λ e(, κ) where e(, κ) s the sequence correspondng to a partcular orderng of red and black dots. The ndecomposables gve you a canoncal bass (Lusztg s f g s symmetrc type). Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
46 Bases Tensor products Defnton What does the representaton theory of ths algebra look lke? Projectves are just summands of the modules P κ = T λ e(, κ) where e(, κ) s the sequence correspondng to a partcular orderng of red and black dots. The ndecomposables gve you a canoncal bass (Lusztg s f g s symmetrc type). Smple modules are endowed wth a crystal structure (exactly as n Lauda and Vazran), whch s somorphc to the tensor product of the crystals for V λ. These gve you a dual canoncal bass. At the moment, I don t know how to descrbe ths somorphsm; I just know that t exsts on abstract grounds. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
47 Bases Tensor products Defnton What does the representaton theory of ths algebra look lke? Projectves are just summands of the modules P κ = T λ e(, κ) where e(, κ) s the sequence correspondng to a partcular orderng of red and black dots. The ndecomposables gve you a canoncal bass (Lusztg s f g s symmetrc type). Smple modules are endowed wth a crystal structure (exactly as n Lauda and Vazran), whch s somorphc to the tensor product of the crystals for V λ. These gve you a dual canoncal bass. At the moment, I don t know how to descrbe ths somorphsm; I just know that t exsts on abstract grounds. These objects both gve bases of the Grothendeck group whch are not very compatble wth the tensor product structure. If we re ever gong to do any calculatons, we re gong to need objects that correspond to pure tensors. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
48 Standard modules Tensor products Standard modules Well, how would we construct the pure tensor v 1 F v 2? We have modules correspondng to F (v 1 v 2 ) = v 1 F v 2 + q λ F v 1 v 2 λ 1 λ 2 and F v 1 v 2 λ 1 λ 2 Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
49 Standard modules Tensor products Standard modules Well, how would we construct the pure tensor v 1 F v 2? We have modules correspondng to F (v 1 v 2 ) = v 1 F v 2 + q λ F v 1 v 2 λ 1 λ 2 and F v 1 v 2 λ 1 λ 2 So we d lke to subtract the former from the latter. Of course, n categores you can t subtract, but you can look for submodules. As t happens, the map gven by s njectve, so moddng out by ts mage gves a module wth the rght class n the Grothendeck group. Can ths phenomenon be generalzed? Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
50 Standard modules Tensor products Standard modules a left crossng a rght crossng Defnton The standard module Sλ κ s the quotent of Pκ λ by the submodule generated by all dagrams wth at least one left crossng as above, and no rght crossngs. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
51 Standard modules Tensor products Standard modules a left crossng a rght crossng Defnton The standard module Sλ κ s the quotent of Pκ λ by the submodule generated by all dagrams wth at least one left crossng as above, and no rght crossngs. Put another way, we can assocate a composton to the module P κ by countng the number of black strands between each par of reds, and we mod out by the mages of all maps from projectves strctly hgher n domnance order. In the example of the last slde, we just use that (1, 0) > (0, 1). Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
52 Standard modules Tensor products Standard modules As you may have guessed Proposton [S κ ] = F κ(1) 1 F 1 v 1 F n F κ(l) v n Ths makes standard modules nvaluable as test objects for functors to see that they behave correctly on the Grothendeck group. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
53 Standard modules Tensor products Standard modules As you may have guessed Proposton [S κ ] = F κ(1) 1 F 1 v 1 F n F κ(l) v n Ths makes standard modules nvaluable as test objects for functors to see that they behave correctly on the Grothendeck group. For example, F S κ has a fltraton whch categorfes the usual formula (l) (F ) = F K K F and smlarly for E S κ. If λ are all mnscule, then T λ s quas-heredtary. Otherwse, t s just standardly stratfed. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
54 Derved category Bradng functors Defnton What about functors? Well, we had a whole lot of maps earler, correspondng to any tangle (though t was enough to defne them for small pctures). Unfortunately, f we want to categorfy these usng the yoga we ve used thus far, we run nto a problem: the coeffcents aren t postve. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
55 Derved category Bradng functors Defnton What about functors? Well, we had a whole lot of maps earler, correspondng to any tangle (though t was enough to defne them for small pctures). Unfortunately, f we want to categorfy these usng the yoga we ve used thus far, we run nto a problem: the coeffcents aren t postve. If you want to have a drect mnus n a category, you have to use some knd of category of complexes. We let V λ = D + (T λ -mod) be the bounded-below derved category of T λ -mod. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
56 Bradng functors Defnton Bradng and duals Theorem (W.) Gven any sequence λ: Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
57 Bradng functors Defnton Bradng and duals Theorem (W.) Gven any sequence λ: For any l-strand brad σ, we have a functor V λ V σλ whch nduces the usual braded structure on the GG. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
58 Bradng functors Defnton Bradng and duals Theorem (W.) Gven any sequence λ: For any l-strand brad σ, we have a functor V λ V σλ whch nduces the usual braded structure on the GG. For any λ, and λ + gven by addng an adjacent par of dual hghest weghts, we have functors V λ+ V λ nducng evaluaton and quantum trace on GG, and dually for coevaluaton and quantum cotrace (but for a funny rbbon structure!). Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
59 Bradng functors Defnton Bradng and duals Theorem (W.) Gven any sequence λ: For any l-strand brad σ, we have a functor V λ V σλ whch nduces the usual braded structure on the GG. For any λ, and λ + gven by addng an adjacent par of dual hghest weghts, we have functors V λ+ V λ nducng evaluaton and quantum trace on GG, and dually for coevaluaton and quantum cotrace (but for a funny rbbon structure!). My goal for the rest of ths talk s to descrbe the frst pont. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
60 Bradng Bradng functors Defnton So, now we need to look for bradng functors. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
61 Bradng Bradng functors Defnton So, now we need to look for bradng functors. Consder the bmodule B over T λ and T (,+1) λ gven by exactly the same sort of dagrams, but wth a sngle crossng nserted between the th and + 1st red strands. λ 1 λ 3 λ 2 λ 1 λ 2 λ 3 Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
62 Bradng Bradng functors Defnton So, now we need to look for bradng functors. Consder the bmodule B over T λ and T (,+1) λ gven by exactly the same sort of dagrams, but wth a sngle crossng nserted between the th and + 1st red strands. λ 1 λ 3 λ 2 Theorem λ 1 λ 2 λ 3 The derved tensor product L T λ B : V λ V (,+1) λ categorfes the bradng map R : V λ V (,+1) λ. The nverse functor s gven by RHom(B, ). Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
63 Bradng Bradng functors Defnton So, frstly, what does derved tensor product mean? It means, amongst other thngs, that we could take a projectve resoluton of B as a bmodule. Ths wll be a complex n the category T λ T λ -pmod whch s unque up to homotopy. Tensor product just means that one takes an object or complex M T λ -pmod and the left half of the bmodule eats t to get a vector space (T λ e M = em) the rght half tensored wth the vector space gves a new complex n T λ -pmod. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
64 Bradng Bradng functors Defnton So, frstly, what does derved tensor product mean? It means, amongst other thngs, that we could take a projectve resoluton of B as a bmodule. Ths wll be a complex n the category T λ T λ -pmod whch s unque up to homotopy. Tensor product just means that one takes an object or complex M T λ -pmod and the left half of the bmodule eats t to get a vector space (T λ e M = em) the rght half tensored wth the vector space gves a new complex n T λ -pmod. Unfortunately, I don t understand at the moment how to wrte down ths complex explctly. It s probably nfnte length and qute complex, but t would facltate computaton qute a bt. On the other hand, part of the magc of homologcal algebra s that you can fgure some thngs out wthout knowng ths. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
65 Bradng Bradng functors Checkng propertes In partcular, how does one check that t actually acts as the bradng? By lookng at test objects. Note that V λ1 V λ2 s generated over U q (g) by vectors of the form v v hgh and under the bradng, these are sent to q? v hgh v. As we know, these vectors are categorfed by standard modules of the form S (0,n). Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
66 Bradng Bradng functors Checkng propertes In partcular, how does one check that t actually acts as the bradng? By lookng at test objects. Note that V λ1 V λ2 s generated over U q (g) by vectors of the form v v hgh and under the bradng, these are sent to q? v hgh v. As we know, these vectors are categorfed by standard modules of the form S (0,n). Proposton B 1 L S (0,n) = S (0,0) (?) λ j+1 Proof: λ j λ j+1 Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28 λ j
67 Bradng Bradng functors Checkng propertes How does one check that t actually gves a brad groupod acton? Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
68 Bradng Bradng functors Checkng propertes How does one check that t actually gves a brad groupod acton? The postve and negatve twsts are nverse because they are adjont and derved equvalences. (Not easy! Must show that half twst sends projectves to tltngs.) Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
69 Bradng Bradng functors Checkng propertes How does one check that t actually gves a brad groupod acton? The postve and negatve twsts are nverse because they are adjont and derved equvalences. (Not easy! Must show that half twst sends projectves to tltngs.) Homologcal algebra song and dance: for reduced expresson n the symmetrc group, ts postve lft to a brad sends projectves to modules. Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
70 Bradng Bradng functors Checkng propertes How does one check that t actually gves a brad groupod acton? The postve and negatve twsts are nverse because they are adjont and derved equvalences. (Not easy! Must show that half twst sends projectves to tltngs.) Homologcal algebra song and dance: for reduced expresson n the symmetrc group, ts postve lft to a brad sends projectves to modules. So we just have to check that as modules B T B +1 T B = B+1 T B T B +1 Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
71 Bradng functors Checkng propertes Next tme: from brads to knots! Ben Webster (U. of Oregon) Categorfyng quantum knot nvarants November 25, / 28
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