Categorification of quantum groups and quantum knot invariants

Transcription

1 Categorification of quantum groups and quantum knot invariants Ben Webster MIT/Oregon March 17, 2010 Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

2 The big picture quantum groups U q (g) ribbon category of U q (g)-reps quantum knot polynomials Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

3 The big picture quantum groups U q (g) HAVE Khovanov-Lauda/Rouquier 2-categories U ribbon category of U q (g)-reps??? quantum knot polynomials quantum knot homologies WANT Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

4 The big picture quantum groups U q (g) HAVE Khovanov-Lauda/Rouquier 2-categories U ribbon category of U q (g)-reps ribbon 2-category of U-reps??? quantum knot polynomials quantum knot homologies WANT Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

5 The big picture quantum groups U q (g) HAVE Khovanov-Lauda/Rouquier 2-categories U ribbon category of U q (g)-reps categorifications of tensor products of simples! quantum knot polynomials quantum knot homologies HAVE Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

6 Tensor products Morally, these knot invariants arise from Chern-Simons theory. They are the expectation value of the trace on a chosen representation of the holonomy around the knot for a certain probability distribution on the space of g-connections on S 3. But I d like to have a definition that didn t require are to be in quotes. What can be done is to make Chern-Simons theory an extended TQFT (attach a category to a 1-manifold, etc.) and see that the lower level is controlled by quantum groups. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

7 Reshetikhin-Turaev invariants More concretely, Reshetikhin and Turaev gave a mathematical construction of these invariants. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

8 Reshetikhin-Turaev invariants More concretely, Reshetikhin and Turaev gave a mathematical construction of these invariants. They label each component of the knot with a representation, and choose a projection of the knot. They then use the theory of quantum groups to attach maps to small diagrams like: W V V V C[q, q 1 ] V W C[q, q 1 ] V These are called the braiding, the quantum trace and the coevaluation. V Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

9 Reshetikhin-Turaev invariants More concretely, Reshetikhin and Turaev gave a mathematical construction of these invariants. They label each component of the knot with a representation, and choose a projection of the knot. They then use the theory of quantum groups to attach maps to small diagrams like: W V V V C[q, q 1 ] V W C[q, q 1 ] These are called the braiding, the quantum trace and the coevaluation. Composing these together for a given link results in a scalar: the Reshetikhin-Turaev invariant for that labeling. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29 V V

10 A historical interlude Progress has been made on categorifying these in a piecemeal fashion for a while Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

11 A historical interlude Progress has been made on categorifying these in a piecemeal fashion for a while Khovanov ( 99): Jones polynomial (C 2 for sl 2 ). Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

12 A historical interlude Progress has been made on categorifying these in a piecemeal fashion for a while Khovanov ( 99): Jones polynomial (C 2 for sl 2 ). Oszvath-Szabo, Rasmussen ( 02): Alexander polynomial (which is actually a gl(1 1) invariant, and doesn t fit into our general picture). Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

13 A historical interlude Progress has been made on categorifying these in a piecemeal fashion for a while Khovanov ( 99): Jones polynomial (C 2 for sl 2 ). Oszvath-Szabo, Rasmussen ( 02): Alexander polynomial (which is actually a gl(1 1) invariant, and doesn t fit into our general picture). Khovanov ( 03): C 3 for sl 3. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

14 A historical interlude Progress has been made on categorifying these in a piecemeal fashion for a while Khovanov ( 99): Jones polynomial (C 2 for sl 2 ). Oszvath-Szabo, Rasmussen ( 02): Alexander polynomial (which is actually a gl(1 1) invariant, and doesn t fit into our general picture). Khovanov ( 03): C 3 for sl 3. Khovanov-Rozansky ( 04): C n for sl n. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

15 A historical interlude Progress has been made on categorifying these in a piecemeal fashion for a while Khovanov ( 99): Jones polynomial (C 2 for sl 2 ). Oszvath-Szabo, Rasmussen ( 02): Alexander polynomial (which is actually a gl(1 1) invariant, and doesn t fit into our general picture). Khovanov ( 03): C 3 for sl 3. Khovanov-Rozansky ( 04): C n for sl n. Stroppel-Mazorchuk, Sussan ( 06-07): i C n for sl n. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

16 A historical interlude Progress has been made on categorifying these in a piecemeal fashion for a while Khovanov ( 99): Jones polynomial (C 2 for sl 2 ). Oszvath-Szabo, Rasmussen ( 02): Alexander polynomial (which is actually a gl(1 1) invariant, and doesn t fit into our general picture). Khovanov ( 03): C 3 for sl 3. Khovanov-Rozansky ( 04): C n for sl n. Stroppel-Mazorchuk, Sussan ( 06-07): i C n for sl n. Cautis-Kamnitzer ( 06): i C n for sl n. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

17 A historical interlude Progress has been made on categorifying these in a piecemeal fashion for a while Khovanov ( 99): Jones polynomial (C 2 for sl 2 ). Oszvath-Szabo, Rasmussen ( 02): Alexander polynomial (which is actually a gl(1 1) invariant, and doesn t fit into our general picture). Khovanov ( 03): C 3 for sl 3. Khovanov-Rozansky ( 04): C n for sl n. Stroppel-Mazorchuk, Sussan ( 06-07): i C n for sl n. Cautis-Kamnitzer ( 06): i C n for sl n. Khovanov-Rozansky( 06): C n for so n. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

18 A historical interlude Progress has been made on categorifying these in a piecemeal fashion for a while Khovanov ( 99): Jones polynomial (C 2 for sl 2 ). Oszvath-Szabo, Rasmussen ( 02): Alexander polynomial (which is actually a gl(1 1) invariant, and doesn t fit into our general picture). Khovanov ( 03): C 3 for sl 3. Khovanov-Rozansky ( 04): C n for sl n. Stroppel-Mazorchuk, Sussan ( 06-07): i C n for sl n. Cautis-Kamnitzer ( 06): i C n for sl n. Khovanov-Rozansky( 06): C n for so n. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

19 A historical interlude Progress has been made on categorifying these in a piecemeal fashion for a while p Khovanov ( 99): Jones polynomial (C 2 for sl 2 ).? Oszvath-Szabo, Rasmussen ( 02): Alexander polynomial (which is actually a gl(1 1) invariant, and doesn t fit into our general picture). p Khovanov ( 03): C 3 for sl 3. c Khovanov-Rozansky ( 04): C n for sl n. p Stroppel-Mazorchuk, Sussan ( 06-07): i C n for sl n. c Cautis-Kamnitzer ( 06): i C n for sl n. c Khovanov-Rozansky( 06): C n for so n. What I want to show is a unified, pictorial construction that should include all of these. p=proven, c=conjectured. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

20 Decategorification? For our purposes, decategorification means sending Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

21 Decategorification? For our purposes, decategorification means sending a vector space V its dimension dim V. a graded vector space V its q-dimension dim q V. an abelian category C its Grothendieck group K 0 (C) a graded abelian category C its q Grothendieck group Z[q, q 1 ] Z K 0 (C). An exact functor F : C C the induced map [F] : K 0 (C) K 0 (C ). So F : C C categorifies φ : V V means there are isomorphisms K 0 (C) = V and K 0 (C ) = V, such that the map induced by F is φ. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

22 Reshetikhin-Turaev invariants The idea now is to categorify everything in sight. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

23 Reshetikhin-Turaev invariants The idea now is to categorify everything in sight. In particular, what we d like to find is graded categories V λ 1,,λ n such that K 0 q(v λ 1,,λ n ) = V λ = V λ1 V λn where V λi is the representation of U q (g) of highest weight λ. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

24 Reshetikhin-Turaev invariants The idea now is to categorify everything in sight. In particular, what we d like to find is graded categories V λ 1,,λ n such that K 0 q(v λ 1,,λ n ) = V λ = V λ1 V λn where V λi is the representation of U q (g) of highest weight λ. Graded functors categorifying: Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

25 Reshetikhin-Turaev invariants The idea now is to categorify everything in sight. In particular, what we d like to find is graded categories V λ 1,,λ n such that K 0 q(v λ 1,,λ n ) = V λ = V λ1 V λn where V λi is the representation of U q (g) of highest weight λ. Graded functors categorifying: the Chevalley generators E i, F i of U q (g). Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

26 Reshetikhin-Turaev invariants The idea now is to categorify everything in sight. In particular, what we d like to find is graded categories V λ 1,,λ n such that K 0 q(v λ 1,,λ n ) = V λ = V λ1 V λn where V λi is the representation of U q (g) of highest weight λ. Graded functors categorifying: the Chevalley generators E i, F i of U q (g). the braiding maps relating different orderings of the highest weights. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

27 Reshetikhin-Turaev invariants The idea now is to categorify everything in sight. In particular, what we d like to find is graded categories V λ 1,,λ n such that K 0 q(v λ 1,,λ n ) = V λ = V λ1 V λn where V λi is the representation of U q (g) of highest weight λ. Graded functors categorifying: the Chevalley generators E i, F i of U q (g). the braiding maps relating different orderings of the highest weights. the coevalution and quantum trace maps. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

28 Pictorial categorification One of the most successful programs of categorification has been the understanding of quantum groups, which goes back to Lusztig. Some remarkable progress on this story was made in the 80 s and 90 s. Using categorifications: Kazhdan and Lusztig defined a basis of the Hecke algebra. Lusztig defined the canonical basis of U q (g) and its representations. but this work at its heart was all geometric; there were a lot of perverse sheaves involved. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

29 Pictorial categorification One of the most successful programs of categorification has been the understanding of quantum groups, which goes back to Lusztig. Some remarkable progress on this story was made in the 80 s and 90 s. Using categorifications: Kazhdan and Lusztig defined a basis of the Hecke algebra. Lusztig defined the canonical basis of U q (g) and its representations. but this work at its heart was all geometric; there were a lot of perverse sheaves involved. While geometry has a lot of power, it s also kinda hard. Luckily, in the past few years Rouquier and Khovanov-Lauda were able to redigest this whole story combinatorially, and so I can tell you an entirely pictorial story (though there s a still a little geometry tucked away in corners). Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

30 Pictorial categorification Now we define an algebra R(g) generated by pictures consisting of strands each colored with a simple root, each labeled with any number of dots with the restrictions that strands must begin on y = 0, end on y = 1 strands can never be horizontal Product is given by stacking (and is 0 if ends don t match). Let g = sl 3 with simple roots α 1 = (1, 1, 0) and α 2 = (0, 1, 1). Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

31 The Khovanov-Lauda relations The relations for g = sl 3 are given by (keeping in mind there is an automorphism interchanging blue and green). Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

32 The Khovanov-Lauda relations The relations for g = sl 3 are given by (keeping in mind there is an automorphism interchanging blue and green). = + = Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

33 The Khovanov-Lauda relations The relations for g = sl 3 are given by (keeping in mind there is an automorphism interchanging blue and green). = + = = 0 = + Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

34 The Khovanov-Lauda relations The relations for g = sl 3 are given by (keeping in mind there is an automorphism interchanging blue and green). = + = = 0 = = + = = + Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

35 Pictorial categorification There s a natural ring map R R R given by horizontal composition : placing diagrams side by side. Extension of scalars under this map defines a monoidal structure on the category V of graded R-modules. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

36 Pictorial categorification There s a natural ring map R R R given by horizontal composition : placing diagrams side by side. Extension of scalars under this map defines a monoidal structure on the category V of graded R-modules. Theorem (Khovanov-Lauda) The category V categorifies U + q (g). Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

37 Pictorial categorification There s a natural ring map R R R given by horizontal composition : placing diagrams side by side. Extension of scalars under this map defines a monoidal structure on the category V of graded R-modules. Theorem (Khovanov-Lauda) The category V categorifies U + q (g). In fact, this is a combinatorial version of Lusztig s construction: Theorem (Vasserot-Varagnolo) For g simply laced, the category of graded modules over R(g) is derived equivalent to Lusztig s categorification of U + (g) and the canonical basis is categorified by the indecomposible projective modules. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

38 Irreducible representations Fix a highest weight λ of g. We want a module category over R(g) -mod generated by a highest weight object which I draw as a red line. Thus, if I act horizontally with R(g), I ll get pictures like λ But I need to impose relations to get a finite dimensional Grothendieck group: = α 1 (λ) = 0 = α 2 (λ) = 0 λ λ λ λ Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

39 Irreducible representations Fix a highest weight λ of g. We want a module category over R(g) -mod generated by a highest weight object which I draw as a red line. Thus, if I act horizontally with R(g), I ll get pictures like λ But I need to impose relations to get a finite dimensional Grothendieck group: = α 1 (λ) = 0 = α 2 (λ) = 0 λ λ λ λ Theorem (Lauda-Vazirani) The category V λ with its V action categorifies the irreducible representation V λ. The canonical basis is categorified by the indecomposible projective modules. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

40 Tensor products Now, some of you might think: Wait, why is looking at the category of 2-representations hard? Can t you just take tensor product of the categories? Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

41 Tensor products Now, some of you might think: Wait, why is looking at the category of 2-representations hard? Can t you just take tensor product of the categories? There are a host of reasons why this is a bad idea. For one, the whole point of quantum groups is that they treat the two sides of the tensor product inequitably. We shouldn t expect a democratic construction, but one slanted toward one tensor factor or another. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

42 Tensor products Now, some of you might think: Wait, why is looking at the category of 2-representations hard? Can t you just take tensor product of the categories? There are a host of reasons why this is a bad idea. For one, the whole point of quantum groups is that they treat the two sides of the tensor product inequitably. We shouldn t expect a democratic construction, but one slanted toward one tensor factor or another. Also, the canonical bases give us hints of the structure of the categorifications of things, and the canonical basis of the tensor product is not the tensor product of canonical bases. In fact, we ll see that there are objects categorifying the tensor product of the canonical bases, but they are not projective. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

43 Tensor products Now, some of you might think: Wait, why is looking at the category of 2-representations hard? Can t you just take tensor product of the categories? There are a host of reasons why this is a bad idea. For one, the whole point of quantum groups is that they treat the two sides of the tensor product inequitably. We shouldn t expect a democratic construction, but one slanted toward one tensor factor or another. Also, the canonical bases give us hints of the structure of the categorifications of things, and the canonical basis of the tensor product is not the tensor product of canonical bases. In fact, we ll see that there are objects categorifying the tensor product of the canonical bases, but they are not projective. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

44 Tensor product algebras Now we define an algebra generated by pictures consisting of red strands, colored with an g-rep non-red strands, colored with simple roots, each labeled with any number of dots Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

45 Tensor product algebras Now we define an algebra generated by pictures consisting of red strands, colored with an g-rep non-red strands, colored with simple roots, each labeled with any number of dots with the restrictions that strands must begin on y = 0, end on y = 1 and can never be horizontal red strands can never cross Product is given by stacking (and is 0 if ends don t match). Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

46 Tensor product algebras Now we define an algebra generated by pictures consisting of red strands, colored with an g-rep non-red strands, colored with simple roots, each labeled with any number of dots with the restrictions that strands must begin on y = 0, end on y = 1 and can never be horizontal red strands can never cross Product is given by stacking (and is 0 if ends don t match). λ 1 λ 2 λ 3 Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

47 Tensor product algebras First, we impose the Khovanov-Lauda relations from before. Also, we also need some relations involving red lines. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

48 Tensor product algebras First, we impose the Khovanov-Lauda relations from before. Also, we also need some relations involving red lines. = + a+b=α 1 (λ) 1 a b = = = Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

49 Tensor product algebras First, we impose the Khovanov-Lauda relations from before. Also, we also need some relations involving red lines. = + a+b=α 1 (λ) 1 a b = = = = α 1 (λ) = α 2 (λ) λ λ λ λ = α 1 (λ) = α 2 (λ) λ λ λ λ Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

50 Tensor product algebras First, we impose the Khovanov-Lauda relations from before. Also, we also need some relations involving red lines. = + a+b=α 1 (λ) 1 a b = = = λ = = α 1 (λ) λ α 1 (λ) λ = = α 2 (λ) λ α 2 (λ) Any diagram where a blue or green strand is to the left of all red strands is 0. λ λ λ λ Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

51 Comparison to Lauda-Vazirani If there s only red line, then we only get one new interesting relation: α2 (λ) = = 0 λ λ If there s only one red line labeled with λ, then we just get back the category for a simple representation. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

52 Tensor product algebras For a sequence of representations λ = (λ 1,..., λ l ), let E λ be the subalgebra where the red lines are labeled with λ in order. Theorem K(E λ -mod) = V λ1 V λl = V λ. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

53 Tensor product algebras For a sequence of representations λ = (λ 1,..., λ l ), let E λ be the subalgebra where the red lines are labeled with λ in order. Theorem K(E λ -mod) = V λ1 V λl = V λ. We can also describe the action of U q (g) on V λ using this categorification: Theorem The horizontal action of R(g) -mod induces the usual action of U q (g) on V λ. This is the first sanity check: the grading shifts necessary to get to quantum coproduct are built in. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

54 Standard modules The proof is by constructing a set of modules which categorify pure tensors. We call these standard modules. Consider the right ideal generated by all pictures where all red/black crossings are negative. negative crossing positive crossing Let S λ denote the right E λ -module given by quotient by this ideal. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

55 Standard modules The proof is by constructing a set of modules which categorify pure tensors. We call these standard modules. Consider the right ideal generated by all pictures where all red/black crossings are negative. negative crossing positive crossing Let S λ denote the right E λ -module given by quotient by this ideal. Proposition End(S λ ) = E λ 1 E λ l. So, the failure of S λ to be projective is exactly what encodes the difference between our tensor product category, and the naive tensor product. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

56 Standard modules We also obtain a functor from the naive product category to our category given by L E λ 1 E λ l S λ : V λ 1;...;λ l V λ. Proposition This functor induces V λ = K(V λ 1 ) K(V λ l) = K(V λ ). Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

57 Standard modules We also obtain a functor from the naive product category to our category given by L E λ 1 E λ l S λ : V λ 1;...;λ l V λ. Proposition This functor induces V λ = K(V λ 1 ) K(V λ l) = K(V λ ). To see why this is so, consider F i (S λ ). This has a filtration given by elements λ 1 i λ m λ 1 λ m i whose successive quotients match the terms of the coproduct (l) (F) = 1 F i + + F i K 1 i K 1 i. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

58 Braiding So, now we need to look for braiding functors. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

59 Braiding So, now we need to look for braiding functors. Consider the bimodule B i over E λ and E (i,i+1) λ given by exactly the same sort of diagrams, but with a single crossing inserted between the ith and i + 1st crossings. λ 1 λ 3 λ 2 λ 1 λ 2 λ 3 Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

60 Braiding So, now we need to look for braiding functors. Consider the bimodule B i over E λ and E (i,i+1) λ given by exactly the same sort of diagrams, but with a single crossing inserted between the ith and i + 1st crossings. λ 1 λ 3 λ 2 Theorem λ 1 λ 2 The derived tensor product L B i : V λ V (i,i+1) λ categorifies the braiding map R i : V λ V (i,i+1) λ. The inverse functor is given by RHom(B i, ). λ 3 Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

61 Coevalution and quantum trace We also need functors corresponding to the cups and caps in our theory. First, consider the case where we have two highest weights λ and w 0 λ = λ. In this case, pick a reduced expression w 0 = s 1 s n with corresponding roots α 1,, α n. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

62 Coevalution and quantum trace We also need functors corresponding to the cups and caps in our theory. First, consider the case where we have two highest weights λ and w 0 λ = λ. In this case, pick a reduced expression w 0 = s 1 s n with corresponding roots α 1,, α n. There s a unique simple module L λ not killed by the idempotent for the sequence of weights 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. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

63 Coevalution and quantum trace We also need functors corresponding to the cups and caps in our theory. First, consider the case where we have two highest weights λ and w 0 λ = λ. In this case, pick a reduced expression w 0 = s 1 s n with corresponding roots α 1,, α n. There s a unique simple module L λ not killed by the idempotent for the sequence of weights 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. The coevaluation functor is categorified by the functor V = Vect V λ,λ sending C L λ. The quantum trace functor is categorified by RHom(L λ, )[2ρ (λ)](2 λ, ρ ): V λ,λ V = Dfd (Vect). Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

64 Coevalution and quantum trace In particular, the algebra (which is the invariant of the circle) A λ = Ext (L λ, L λ )[2ρ (λ)](2 λ, ρ ) has graded dimension given by the quantum dimension of V λ. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

65 Coevalution and quantum trace In particular, the algebra (which is the invariant of the circle) A λ = Ext (L λ, L λ )[2ρ (λ)](2 λ, ρ ) has graded dimension given by the quantum dimension of V λ. Has anyone seen this algebra before? Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

66 Coevalution and quantum trace In particular, the algebra (which is the invariant of the circle) A λ = Ext (L λ, L λ )[2ρ (λ)](2 λ, ρ ) has graded dimension given by the quantum dimension of V λ. Has anyone seen this algebra before? One interesting candidate is the algebra structure that Feigin, Frenkel and Rybnikov put on V λ using the quantum shift of argument algebra at a principal nilpotent. Another tantalizing possibility is that it is related to the geometry of Gr λ. Perhaps a ring structure on intersection cohomology? Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

67 Coevalution and quantum trace To do this in general, you can construct natural bimodules K λ. Rather than give a definition, let me just draw the picture. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

68 Coevalution and quantum trace To do this in general, you can construct natural bimodules K λ. Rather than give a definition, let me just draw the picture. λ 1 α µ 2 (µ) α 1 + α 2 (µ) α 1 (µ) µ λ l λ 1 L µ λ l Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

69 Coevalution and quantum trace To do this in general, you can construct natural bimodules K λ. Rather than give a definition, let me just draw the picture. λ 1 α µ 2 (µ) α 1 + α 2 (µ) α 1 (µ) µ λ l λ 1 L µ λ l Theorem Tensor product with this bimodule categorifies coevaluation, and Hom with it categorifies quantum trace. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

70 Ribbon structure Now, I should have drawn all these pictures as ribbon knots, since framing matters in our picture. Moreover, I need to associate an actual functor to the ribbon twist. = Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

71 Ribbon structure Now, I should have drawn all these pictures as ribbon knots, since framing matters in our picture. Moreover, I need to associate an actual functor to the ribbon twist. = The ribbon functor is just M M( 2λ, ρ )[2ρ (λ)]. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

72 Ribbon structure Now, I should have drawn all these pictures as ribbon knots, since framing matters in our picture. Moreover, I need to associate an actual functor to the ribbon twist. = The ribbon functor is just M M( 2λ, ρ )[2ρ (λ)]. Note: this is a strange ribbon element! (It appeared in work of Snyder and Tingley on half-twist elements.) But that won t change things very much. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

73 Knot invariants Now, we start with a picture of our knot (in red), cut it up into these elementary pieces, and compose these functors in the order the elementary pieces fit together. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

74 Knot invariants Now, we start with a picture of our knot (in red), cut it up into these elementary pieces, and compose these functors in the order the elementary pieces fit together. For a link L, we get a functor F L : V = D fd (Vect) V = D fd (Vect). So F L (C) is a complex of vector spaces (actually graded vector spaces). Theorem The cohomology of F L (C) is a knot invariant. The graded Euler characteristic of this complex is J V,L (q). Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

75 Knot invariants Start with C. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

76 Knot invariants V V A 1 = C K 1,2 V Replace with projective resolution B 1 Start with C. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

77 Knot invariants V V V V V V A 2 = B 1 K 1,2 V A 1 = C K 1,2 V Replace with injective resolution B 2 Replace with projective resolution B 1 Start with C. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

78 Knot invariants V V V V V V V V V V A 3 = RHom(B i, B 2 ) A 2 = B 1 K 1,2 V A 1 = C K 1,2 V Replace with projective resolution B 3 Replace with injective resolution B 2 Replace with projective resolution B 1 Start with C. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

79 Knot invariants V V V V V V V V V V V V V V A 4 = B 3 B 1 A 3 = RHom(B i, B 2 ) A 2 = B 1 K 1,2 V A 1 = C K 1,2 V Replace with projective resolution B 4 Replace with projective resolution B 3 Replace with injective resolution B 2 Replace with projective resolution B 1 Start with C. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

80 Knot invariants V V V V V V V V V V V V V V V V V V A 5 = B 4 B 3 A 4 = B 3 B 1 A 3 = RHom(B i, B 2 ) A 2 = B 1 K 1,2 V A 1 = C K 1,2 V Replace with injective resolution B 5 Replace with projective resolution B 4 Replace with projective resolution B 3 Replace with injective resolution B 2 Replace with projective resolution B 1 Start with C. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

81 Knot invariants V V V V V V V V V V V V V V V V V V V V 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 i, B 2 ) A 2 = B 1 K 1,2 V A 1 = C K 1,2 V Replace with injective resolution B 6 Replace with injective resolution B 5 Replace with projective resolution B 4 Replace with projective resolution B 3 Replace with injective resolution B 2 Replace with projective resolution B 1 Start with C. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

82 Knot invariants 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 i, B 2 ) A 2 = B 1 K 1,2 V A 1 = C K 1,2 V Knot homology! Replace with injective resolution B 6 Replace with injective resolution B 5 Replace with projective resolution B 4 Replace with projective resolution B 3 Replace with injective resolution B 2 Replace with projective resolution B 1 Start with C. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

83 Functoriality? It s not known at the moment if this is functorial in cobordisms between knots. How would one construct a functoriality map? Cobordisms of knots can be cut (using a Morse function) into the moves of circle destruction saddle move circle creation Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

84 Functoriality? It s not known at the moment if this is functorial in cobordisms between knots. How would one construct a functoriality map? Cobordisms of knots can be cut (using a Morse function) into the moves of circle destruction saddle move circle creation Being able to define these maps requires that the cap and cup functors are biadjoint (they re clearly adjoint one way). However, one has to prove that this map does not depend on the handle decomposition. Not easy! Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

85 Open questions Can it be used practically to distinguish knots? Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

86 Open questions Can it be used practically to distinguish knots? Is there any way of doing computations efficiently? Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

87 Open questions Can it be used practically to distinguish knots? Is there any way of doing computations efficiently? What about Alexander polynomial? Could this prescription be modified to give Knot Floer homology? Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

88 Open questions Can it be used practically to distinguish knots? Is there any way of doing computations efficiently? What about Alexander polynomial? Could this prescription be modified to give Knot Floer homology? These categories seem to arise from perverse sheaves on/fukaya categories of quiver varieties. Can the knot homology be described in that picture? Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29

89 Thanks, y all. Ben Webster (MIT/Oregon) Categorification of quantum knot invariants March 17, / 29