Higher representation theory in algebra and geometry: Lecture VIII

Transcription

1 Hgher representaton theory n algebra and geometry: Lecture VIII Ben Webster UVA Aprl 8, 2014 Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

2 References For ths lecture, useful references nclude: B.W., Knot nvarants and hgher representaton theory The sldes for the talk are on my webpage at: You can also fnd some proofs that I ddn t feel lke gong through n class at: Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

3 The future So, there are 3 class meetngs left. In what tme s left, I want to try to cover two nterestng applcatons of the theory we ve dscussed. the constructon of knot nvarants usng ths theory. We ve already dscussed one specal case of ths, usng sl 2 -categorfcatons to obtan the Jones polynomal. Ths generalzes to other types. There s also a dual constructon of these knot nvarants for sl n, whch we ll lkely get to n Lecture 9. Ths also ncludes some nterestng connectons to algebrac geometry. the perspectve on the representaton theory of Cherednk algebras afforded by hgher representaton theory. Ths s s, of course, an enormous topc, but I thnk t s an exctng applcaton of the theory, and one worth dscussng a bt. I antcpate that wll be Lecture 10. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

4 Knot nvarants Roadmap quantum groups U q (g) rbbon category of U q (g)-reps quantum knot polynomals (Jones polynomal, etc.) Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

5 Knot nvarants Roadmap 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

6 Knot nvarants Roadmap 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

7 Knot nvarants Roadmap 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

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 (UVA) HRT : Lecture VIII Aprl 8, / 36

9 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

10 Knot nvarants Bradng Reshetkhn-Turaev nvarants 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

11 Knot nvarants Bradng Reshetkhn-Turaev nvarants 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

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)-acton). 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

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)-acton). 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

14 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)-acton). 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

15 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 natural choce of a V (I really mean functoral) 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

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 rbbon lnk results n a scalar: the Reshetkhn-Turaev nvarant for that labelng. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

17 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 rbbon lnk results n a scalar: the Reshetkhn-Turaev nvarant for that labelng. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

18 Knot nvarants Cups and caps A hstorcal nterlude 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. What I ll gve you s a unfed, pctoral constructon that should nclude all of these. For that, we need tensor products. p=proven, c=conjectured. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

19 Knot nvarants Cups and caps A hstorcal nterlude 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. What I ll gve you s a unfed, pctoral constructon that should nclude all of these. For that, we need tensor products. p=proven, c=conjectured. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

20 Knot nvarants Cups and caps A hstorcal nterlude 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. What I ll gve you s a unfed, pctoral constructon that should nclude all of these. For that, we need tensor products. p=proven, c=conjectured. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

21 Knot nvarants Cups and caps A hstorcal nterlude 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. What I ll gve you s a unfed, pctoral constructon that should nclude all of these. For that, we need tensor products. p=proven, c=conjectured. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

22 Knot nvarants Cups and caps A hstorcal nterlude 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. What I ll gve you s a unfed, pctoral constructon that should nclude all of these. For that, we need tensor products. p=proven, c=conjectured. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

23 Knot nvarants Cups and caps A hstorcal nterlude 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. What I ll gve you s a unfed, pctoral constructon that should nclude all of these. For that, we need tensor products. p=proven, c=conjectured. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

24 Knot nvarants Cups and caps A hstorcal nterlude 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. What I ll gve you s a unfed, pctoral constructon that should nclude all of these. For that, we need tensor products. p=proven, c=conjectured. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

25 Knot nvarants Cups and caps A hstorcal nterlude 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. What I ll gve you s a unfed, pctoral constructon that should nclude all of these. For that, we need tensor products. p=proven, c=conjectured. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

26 Knot nvarants Cups and caps A hstorcal nterlude 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. What I ll gve you s a unfed, pctoral constructon that should nclude all of these. For that, we need tensor products. p=proven, c=conjectured. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

27 Knot nvarants Cups and caps A hstorcal nterlude 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. p Khovanov-Rozansky ( 04): C n for sl n. p Stroppel-Mazorchuk, Sussan ( 06-07): C n for sl n. p Cauts-Kamntzer ( 06): C n for sl n. c Khovanov-Rozansky( 06): C n for so n. What I ll gve you s a unfed, pctoral constructon that should nclude all of these. For that, we need tensor products. p=proven, c=conjectured. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

28 Knot nvarants Cups and caps A hstorcal nterlude 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. p Khovanov-Rozansky ( 04): C n for sl n. p Stroppel-Mazorchuk, Sussan ( 06-07): C n for sl n. p Cauts-Kamntzer ( 06): C n for sl n. c Khovanov-Rozansky( 06): C n for so n. What I ll gve you s a unfed, pctoral constructon that should nclude all of these. For that, we need tensor products. p=proven, c=conjectured. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

29 Tensor products Defnton Tensor products In the case of sl 2, we ntroduced a graphcal calculus for elements of V λ = V λ1 V λl. A downward black lne on the left means actng by F. A red lne at the left 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 Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

30 Tensor products Defnton Tensor products In the case of sl 2, we ntroduced a graphcal calculus for elements of V λ = V λ1 V λl. A downward black lne on the left means actng by F. A red lne at the left 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 Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

31 and Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36 Tensor products Tensor products Defnton Let T λ be the algebra whose elements are k-lnear combnatons of mmersed 1-manfolds wth black components orented, dotted and labeled wth Γ and red components have no ntersectons, and are labeled wth the weghts λ n order modulo the relatons λ = = λ λ λ j λ λ = = j λ λ + a+b=λ 1 a λ b any dagram wth a black lne at the far left s 0. λ λ = =

32 Dagrams Tensor products Defnton = unless = j = + j j = Q j (y 1, y 2 ) = unless = k = j ± 1 j j j k j k = 0 = Q j (y 3, y 2 ) Q j (y 1, y 2 ) y 3 y 1 j j j Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

33 Dagrams Tensor products Defnton = unless = j = + j j = Q j (y 1, y 2 ) = unless = k = j ± 1 j j j k j k = 0 = Q j (y 3, y 2 ) Q j (y 1, y 2 ) y 3 y 1 j j j Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

34 Categorcal acton Tensor products Defnton Recall, last tme, we defned the noton of a categorcal acton of g. For ths, we need functors F and E. These are nducton and restrcton functors, whch can thnk of as tensor product wth the bmodules: left acton left acton F = E = rght acton rght acton The acton of R m on the power F m s by attachng pctures at the bottom. Adjuncton s essentally automatc. The trcky part s checkng the sl 2 relatons. Ths s hard. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

35 Tensor products Defnton Grothendeck groups 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 λ (addng a red lne) to the ncluson V v hgh V V λ. But we d lke to talk about the category 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

36 Tensor products Defnton Grothendeck groups 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 λ (addng a red lne) to the ncluson V v hgh V V λ. But we d lke to talk about the category 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

37 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 the tensor product of the crystals for V λ. These gve you a dual canoncal bass. 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

38 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 the tensor product of the crystals for V λ. These gve you a dual canoncal bass. 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

39 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 the tensor product of the crystals for V λ. These gve you a dual canoncal bass. 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

40 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 the tensor product of the crystals for V λ. These gve you a dual canoncal bass. 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

41 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

42 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

43 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

44 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

45 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. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

46 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. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

47 Derved category Bradng functors Defnton What 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 be the bounded-above derved category of T λ -mod. I bet lots of you are happer wth the homotopy category, but that doesn t work so well for me. Workng n that category would requre me knowng some projectve resolutons that are very hard to wrte down. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

48 Derved category Bradng functors Defnton What 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 be the bounded-above derved category of T λ -mod. I bet lots of you are happer wth the homotopy category, but that doesn t work so well for me. Workng n that category would requre me knowng some projectve resolutons that are very hard to wrte down. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

49 Bradng functors Defnton Bradng and duals Theorem 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 these functors, and how they gve knot nvarants. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

50 Bradng functors Defnton Bradng and duals Theorem 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 these functors, and how they gve knot nvarants. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

51 Bradng functors Defnton Bradng and duals Theorem 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 these functors, and how they gve knot nvarants. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

52 Bradng functors Defnton Bradng and duals Theorem 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 these functors, and how they gve knot nvarants. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

53 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 crossngs. 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

54 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 crossngs. 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

55 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 crossngs. 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

56 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. Unfortunately, I don t understand at the moment how to wrte down ths complex explctly. In most cases, t must have nfnte length and s 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

57 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 λ j+1 Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

58 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 λ j+1 Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

59 Bradng functors Checkng propertes Bradng In partcular, 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

60 Bradng functors Checkng propertes Bradng In partcular, 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

61 Bradng functors Checkng propertes Bradng In partcular, 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

62 Bradng functors Checkng propertes Bradng In partcular, 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 (UVA) HRT : Lecture VIII Aprl 8, / 36

63 Bradng functors Coevaluton and quantum trace Checkng propertes We also need functors correspondng to the cups and caps n our theory. Frst, consder the case where we have two hghest weghts λ and w 0 λ = λ. We must frst defne an somorphsm between V λ and V λ. That s to say, a parng V λ V λ C(q). We start wth a chosen hghest weght vector of both representatons v λ, v λ (ths comes from the rrep n T λ λ -mod = k -mod). So, a parng s fxed by a choce of lowest weght vector. Pck a reduced expresson w 0 = s 1 s n wth correspondng roots α 1,, α n. Then we have a lowest weght vector of the form v low = F (α n (s n 1 s 1 λ)) n We wll always choose ths one. F (α 2 (s 1λ)) 2 F (α 1 (λ)) 1 v λ Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

64 Bradng functors Coevaluton and quantum trace Checkng propertes We also need functors correspondng to the cups and caps n our theory. Frst, consder the case where we have two hghest weghts λ and w 0 λ = λ. We must frst defne an somorphsm between V λ and V λ. That s to say, a parng V λ V λ C(q). We start wth a chosen hghest weght vector of both representatons v λ, v λ (ths comes from the rrep n T λ λ -mod = k -mod). So, a parng s fxed by a choce of lowest weght vector. Pck a reduced expresson w 0 = s 1 s n wth correspondng roots α 1,, α n. Then we have a lowest weght vector of the form v low = F (α n (s n 1 s 1 λ)) n We wll always choose ths one. F (α 2 (s 1λ)) 2 F (α 1 (λ)) 1 v λ Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

65 Invarants Bradng functors Checkng propertes We should look for a categorfcaton of the unque nvarant vector c V λ V λ. We can actually guess qute easly what ths should be. The space of nvarants s orthogonal under the Euler form to all projectves of the form F M for any. We know by countng arguments that all but one ndecomposable projectve s a summand of a F M. We actually know exactly what ths remanng projectve P λ s; t corresponds to the sequence of weghts and roots (λ, α (α 1 (λ)) 1, α (α 2 (s 1λ)) 2,..., α (α n (s n 1 s 1 λ)) n, λ ). So, an element of nvarants s gven by the smple quotent of P λ. Denote ths L λ. It s pretty easy to check by hand that L λ s klled by all E. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

66 Invarants Bradng functors Checkng propertes We should look for a categorfcaton of the unque nvarant vector c V λ V λ. We can actually guess qute easly what ths should be. The space of nvarants s orthogonal under the Euler form to all projectves of the form F M for any. We know by countng arguments that all but one ndecomposable projectve s a summand of a F M. We actually know exactly what ths remanng projectve P λ s; t corresponds to the sequence of weghts and roots (λ, α (α 1 (λ)) 1, α (α 2 (s 1λ)) 2,..., α (α n (s n 1 s 1 λ)) n, λ ). So, an element of nvarants s gven by the smple quotent of P λ. Denote ths L λ. It s pretty easy to check by hand that L λ s klled by all E. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

67 Bradng functors Checkng propertes Coevaluton and evaluaton The coevaluaton functor s categorfed by the functor V = Vect V λ,λ sendng C L λ. The evaluaton functor s categorfed by RHom(L λ, )[2ρ (λ)](2 λ, ρ ): V λ,λ V = Dfd (Vect). Now, we know that f we want quantum trace, we should compromse between L λ [2ρ (λ)](2 λ, ρ ) and L λ [ 2ρ (λ)]( 2 λ, ρ ) Defnton The postve rbbon twst acts on the category by [2ρ (λ)](2 λ, ρ ). Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

68 Bradng functors Checkng propertes Coevaluton and evaluaton The coevaluaton functor s categorfed by the functor V = Vect V λ,λ sendng C L λ. The evaluaton functor s categorfed by RHom(L λ, )[2ρ (λ)](2 λ, ρ ): V λ,λ V = Dfd (Vect). Now, we know that f we want quantum trace, we should compromse between L λ [2ρ (λ)](2 λ, ρ ) and L λ [ 2ρ (λ)]( 2 λ, ρ ) Defnton The postve rbbon twst acts on the category by [2ρ (λ)](2 λ, ρ ). Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

69 Rbbon structure Bradng functors Checkng propertes So ths decategorfes to ( 1) 2ρ (λ) q 2 λ,ρ. Note: ths s a strange rbbon element! (It appeared n work of Snyder and Tngley on half-twst elements.) For each rbbon element, there s a noton of quantum dmenson, and n ths pcture, qdmv q=1 = ( 1) 2ρ (λ) dm V. For example, n sl 2, qdmv n = ( 1) n qn+1 q n 1 q q 1. From now on, all my knots are rbbon knots (n the blackboard framng), and I ll really get nvarants of rbbon knots (but twsts just gve gradng shfts). = Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

70 Bradng functors Checkng propertes Coevaluton and quantum trace In partcular, the algebra (whch s the nvarant of the crcle) A λ = Ext (L λ, L λ )[2ρ (λ)](2 λ, ρ ) has graded Euler characterstc gven by the quantum dmenson of V λ. If V λ s mnscule, then everythng works beautfully. The dmenson of A λ s really the dmenson of V λ. In partcular, f λ = ω for g = sl n, then A λ = H (Grass(, n)). Conjecture If λ s mnscule, A λ = H (Gr λ ). On the other hand, f λ s not mnscule, thngs blow up. For example, f g = sl 2 and λ = 2, then Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

71 Bradng functors Checkng propertes Coevaluton and quantum trace In partcular, the algebra (whch s the nvarant of the crcle) A λ = Ext (L λ, L λ )[2ρ (λ)](2 λ, ρ ) has graded Euler characterstc gven by the quantum dmenson of V λ. If V λ s mnscule, then everythng works beautfully. The dmenson of A λ s really the dmenson of V λ. In partcular, f λ = ω for g = sl n, then A λ = H (Grass(, n)). Conjecture If λ s mnscule, A λ = H (Gr λ ). On the other hand, f λ s not mnscule, thngs blow up. For example, f g = sl 2 and λ = 2, then Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

72 Bradng functors Checkng propertes Coevaluton and quantum trace In partcular, the algebra (whch s the nvarant of the crcle) A λ = Ext (L λ, L λ )[2ρ (λ)](2 λ, ρ ) has graded Euler characterstc gven by the quantum dmenson of V λ. If V λ s mnscule, then everythng works beautfully. The dmenson of A λ s really the dmenson of V λ. In partcular, f λ = ω for g = sl n, then A λ = H (Grass(, n)). Conjecture If λ s mnscule, A λ = H (Gr λ ). On the other hand, f λ s not mnscule, thngs blow up. For example, f g = sl 2 and λ = 2, then,j ( t)j dm q A j λ q 2 t q 2 t 2 Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

73 Bradng functors Checkng propertes Coevaluton and quantum trace In partcular, the algebra (whch s the nvarant of the crcle) A λ = Ext (L λ, L λ )[2ρ (λ)](2 λ, ρ ) has graded Euler characterstc gven by the quantum dmenson of V λ. If V λ s mnscule, then everythng works beautfully. The dmenson of A λ s really the dmenson of V λ. In partcular, f λ = ω for g = sl n, then A λ = H (Grass(, n)). Conjecture If λ s mnscule, A λ = H (Gr λ ). On the other hand, f λ s not mnscule, thngs blow up. For example, f g = sl 2 and λ = 2, then,j ( t)j dm q A j λ = q 2 t q 2 t 2 + q2 q 2 t 1 t 2 q 4 Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

74 Bradng functors Checkng propertes Coevaluton and quantum trace In partcular, the algebra (whch s the nvarant of the crcle) A λ = Ext (L λ, L λ )[2ρ (λ)](2 λ, ρ ) has graded Euler characterstc gven by the quantum dmenson of V λ. If V λ s mnscule, then everythng works beautfully. The dmenson of A λ s really the dmenson of V λ. In partcular, f λ = ω for g = sl n, then A λ = H (Grass(, n)). Conjecture If λ s mnscule, A λ = H (Gr λ ). On the other hand, f λ s not mnscule, thngs blow up. For example, f g = sl 2 and λ = 2, then,j ( 1)j dm q A j λ = q q 2 Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

75 Bradng functors Coevaluton and quantum trace Checkng propertes To do ths n general, you can construct natural bmodules K µ. Ths s gven by the pcture. α 1 (µ) α n (s n 1 s 1 µ) λ 1 λ l µ n 1 λ 1 L µ λ l Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

76 Bradng functors Coevaluton and quantum trace Checkng propertes To do ths n general, you can construct natural bmodules K µ. Ths s gven by the pcture. α 1 (µ) α n (s n 1 s 1 µ) λ 1 λ l µ n 1 λ 1 L µ λ l Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

77 Bradng functors Checkng propertes Coevaluton and quantum trace There s exactly one nterestng relaton here, whch says that µ µ 1 n µ µ 1 n = L µ L µ F v c λ = F (v c λ ). Theorem Tensor product wth ths bmodule categorfes coevaluaton/quantum cotrace, and Hom wth t categorfes evaluaton/quantum trace. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

78 Bradng functors Checkng propertes Knot nvarants Now, we start wth a pcture of our knot (n red), cut t up nto these elementary peces, and compose these functors n the order the elementary peces ft together. For a lnk L, we get a functor F L : V = D(Vect) V = D(Vect). So F L (C) s a complex of vector spaces (actually graded vector spaces). Theorem The cohomology of F L (C) s a knot nvarant, and fnte-dmensonal n each homologcal and each graded degree. The graded Euler characterstc of ths complex s J V,L (q). As usual, we can take a generatng seres of F L (C). Ths wll not be a polynomal, but t should be a ratonal functon. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

79 Bradng functors Checkng propertes Knot nvarants Now, we start wth a pcture of our knot (n red), cut t up nto these elementary peces, and compose these functors n the order the elementary peces ft together. For a lnk L, we get a functor F L : V = D(Vect) V = D(Vect). So F L (C) s a complex of vector spaces (actually graded vector spaces). Theorem The cohomology of F L (C) s a knot nvarant, and fnte-dmensonal n each homologcal and each graded degree. The graded Euler characterstc of ths complex s J V,L (q). As usual, we can take a generatng seres of F L (C). Ths wll not be a polynomal, but t should be a ratonal functon. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

80 Bradng functors Checkng propertes Knot nvarants V V V V V V V V V V V V V V V V V V V V A 7 = RHom(K 1,2 V, B 6) A 6 = RHom(K 2,3 V, B 5) A 5 = B 4 B 3 A 4 = B 3 B 1 A 3 = RHom(B, B 2 ) A 2 = B 1 K 1,2 V A 1 = C K 1,2 V Knot homology! Replace wth njectve resoluton B 6 Replace wth njectve resoluton B 5 Replace wth projectve resoluton B 4 Replace wth projectve resoluton B 3 Replace wth njectve resoluton B 2 Replace wth projectve resoluton B 1 Start wth C. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

81 Bradng functors Checkng propertes Knot nvarants V V V V V V V V V V V V V V V V V V V V A 7 = RHom(K 1,2 V, B 6) A 6 = RHom(K 2,3 V, B 5) A 5 = B 4 B 3 A 4 = B 3 B 1 A 3 = RHom(B, B 2 ) A 2 = B 1 K 1,2 V A 1 = C K 1,2 V Knot homology! Replace wth njectve resoluton B 6 Replace wth njectve resoluton B 5 Replace wth projectve resoluton B 4 Replace wth projectve resoluton B 3 Replace wth njectve resoluton B 2 Replace wth projectve resoluton B 1 Start wth C. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

82 Bradng functors Checkng propertes Knot nvarants V V V V V V V V V V V V V V V V V V V V A 7 = RHom(K 1,2 V, B 6) A 6 = RHom(K 2,3 V, B 5) A 5 = B 4 B 3 A 4 = B 3 B 1 A 3 = RHom(B, B 2 ) A 2 = B 1 K 1,2 V A 1 = C K 1,2 V Knot homology! Replace wth njectve resoluton B 6 Replace wth njectve resoluton B 5 Replace wth projectve resoluton B 4 Replace wth projectve resoluton B 3 Replace wth njectve resoluton B 2 Replace wth projectve resoluton B 1 Start wth C. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

83 Bradng functors Checkng propertes Knot nvarants V V V V V V V V V V V V V V V V V V V V A 7 = RHom(K 1,2 V, B 6) A 6 = RHom(K 2,3 V, B 5) A 5 = B 4 B 3 A 4 = B 3 B 1 A 3 = RHom(B, B 2 ) A 2 = B 1 K 1,2 V A 1 = C K 1,2 V Knot homology! Replace wth njectve resoluton B 6 Replace wth njectve resoluton B 5 Replace wth projectve resoluton B 4 Replace wth projectve resoluton B 3 Replace wth njectve resoluton B 2 Replace wth projectve resoluton B 1 Start wth C. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

84 Bradng functors Checkng propertes Knot nvarants V V V V V V V V V V V V V V V V V V V V A 7 = RHom(K 1,2 V, B 6) A 6 = RHom(K 2,3 V, B 5) A 5 = B 4 B 3 A 4 = B 3 B 1 A 3 = RHom(B, B 2 ) A 2 = B 1 K 1,2 V A 1 = C K 1,2 V Knot homology! Replace wth njectve resoluton B 6 Replace wth njectve resoluton B 5 Replace wth projectve resoluton B 4 Replace wth projectve resoluton B 3 Replace wth njectve resoluton B 2 Replace wth projectve resoluton B 1 Start wth C. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

85 Bradng functors Checkng propertes Knot nvarants V V V V V V V V V V V V V V V V V V V V A 7 = RHom(K 1,2 V, B 6) A 6 = RHom(K 2,3 V, B 5) A 5 = B 4 B 3 A 4 = B 3 B 1 A 3 = RHom(B, B 2 ) A 2 = B 1 K 1,2 V A 1 = C K 1,2 V Knot homology! Replace wth njectve resoluton B 6 Replace wth njectve resoluton B 5 Replace wth projectve resoluton B 4 Replace wth projectve resoluton B 3 Replace wth njectve resoluton B 2 Replace wth projectve resoluton B 1 Start wth C. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

86 Bradng functors Checkng propertes Knot nvarants V V V V V V V V V V V V V V V V V V V V A 7 = RHom(K 1,2 V, B 6) A 6 = RHom(K 2,3 V, B 5) A 5 = B 4 B 3 A 4 = B 3 B 1 A 3 = RHom(B, B 2 ) A 2 = B 1 K 1,2 V A 1 = C K 1,2 V Knot homology! Replace wth njectve resoluton B 6 Replace wth njectve resoluton B 5 Replace wth projectve resoluton B 4 Replace wth projectve resoluton B 3 Replace wth njectve resoluton B 2 Replace wth projectve resoluton B 1 Start wth C. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

87 Bradng functors Checkng propertes Knot nvarants V V V V V V V V V V V V V V V V V V V V A 7 = RHom(K 1,2 V, B 6) A 6 = RHom(K 2,3 V, B 5) A 5 = B 4 B 3 A 4 = B 3 B 1 A 3 = RHom(B, B 2 ) A 2 = B 1 K 1,2 V A 1 = C K 1,2 V Knot homology! Replace wth njectve resoluton B 6 Replace wth njectve resoluton B 5 Replace wth projectve resoluton B 4 Replace wth projectve resoluton B 3 Replace wth njectve resoluton B 2 Replace wth projectve resoluton B 1 Start wth C. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

88 4d TQFT Bradng functors Checkng propertes One of the nspratons for studyng categorfcatons s the connectons between hgher categores and quantum feld theory. The quantum knot nvarants arse from a 3-d TQFT: Chern-Smons theory. You can thnk of ths as bult up from attachng the category of U q (g) representaton to a crcle and buldng the 2-and 3-dmensonal layers from that. Can one make a 4-dmensonal TQFT of some knd out the category of 2-representatons of ths categorfed quantum group? Gukov and other physcsts have done work on ths, but as far as I know, nothng mathematcally rgorous has appeared. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

89 4d TQFT Bradng functors Checkng propertes One of the nspratons for studyng categorfcatons s the connectons between hgher categores and quantum feld theory. The quantum knot nvarants arse from a 3-d TQFT: Chern-Smons theory. You can thnk of ths as bult up from attachng the category of U q (g) representaton to a crcle and buldng the 2-and 3-dmensonal layers from that. Can one make a 4-dmensonal TQFT of some knd out the category of 2-representatons of ths categorfed quantum group? Gukov and other physcsts have done work on ths, but as far as I know, nothng mathematcally rgorous has appeared. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36

90 Next tme Bradng functors Checkng propertes Next tme I ll talk about how to relate ths constructon to the other ones I ve mentoned, especally those of Khovanov-Rozansky and Cauts-Kamntzer. Dong that wll also requre some dscusson of connectons to the geometry of quver varetes. Ben Webster (UVA) HRT : Lecture VIII Aprl 8, / 36