Categorifying quantum knot invariants

Transcription

1 Categorifying quantum knot invariants Ben Webster U. of Oregon November 26, 2010 Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

2 This talk is online at Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

3 The story thus far Let me just remind you what we ve done thus far. To a simple Lie algebra g, we ve associated a 2-category U. λ λ α j λ α j α i λ α i λ λ α i λ α i + α j λ α i i j j Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

4 The story thus far Let me just remind you what we ve done thus far. To a simple Lie algebra g, we ve associated a 2-category U. To a tensor product of simple representations V λ1 V λl, we ve associated a module category T λ -mod and its derived category V λ = D + (T λ -mod). λ 1 λ 2 λ 3 λ 1 λ 2 λ 3 Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

5 The story thus far Let me just remind you what we ve done thus far. To a simple Lie algebra g, we ve associated a 2-category U. To a tensor product of simple representations V λ1 V λl, we ve associated a module category T λ -mod and its derived category V λ = D + (T λ -mod). To a braid σ, we ve associated a braiding functor B σ : V λ V σλ. λ 1 λ 3 λ 2 λ 1 λ 2 λ 3 Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

6 The story thus far Let me just remind you what we ve done thus far. To a simple Lie algebra g, we ve associated a 2-category U. To a tensor product of simple representations V λ1 V λl, we ve associated a module category T λ -mod and its derived category V λ = D + (T λ -mod). To a braid σ, we ve associated a braiding functor B σ : V λ V σλ. λ 1 λ 3 λ 2 λ 1 λ 2 λ 3 All of these categorify the familiar functors from the the theory of quantum groups. Now, in order to get link invariants, we need to learn to turn around. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

7 Coevalution and quantum trace First, consider the case where we have two highest weights λ and w 0 λ = λ. We must first define an isomorphism between V λ and V λ. That is to say, a pairing V λ V λ C(q). We start with a chosen highest weight vector of both representations v λ, v λ (this comes from the irrep in T λ λ -mod = k -mod). So, a pairing is fixed by a choice of lowest weight vector. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

8 Coevalution and quantum trace First, consider the case where we have two highest weights λ and w 0 λ = λ. We must first define an isomorphism between V λ and V λ. That is to say, a pairing V λ V λ C(q). We start with a chosen highest weight vector of both representations v λ, v λ (this comes from the irrep in T λ λ -mod = k -mod). So, a pairing is fixed by a choice of lowest weight vector. Pick a reduced expression w 0 = s n s 1 with corresponding roots α 1,, α n. Then we have a lowest weight vector of the form v low = F (α n (s n 1 s 1 λ)) i n F (α 2 (s 1λ)) i 2 F (α 1 (λ)) i 1 v λ We will always choose this one. (Corresponds to self-dual simple in T λ w 0 λ ). Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

9 Invariants We should look for a categorification of the unique invariant vector c V λ V λ. We can actually guess quite easily what this should be. The space of invariants in any tensor product category is spanned by the classes of simples killed by all E i (there are the right number of these by the crystal structure on simples). Thus in V λ,λ, we need to find the one simple killed by all E i. This is easy to find; start with the unique simple/projective in T λ w 0 λ -mod; apply the red line labelled with λ. This is a projective P λ and its simple quotient L λ spans invariants. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

10 Invariants We should look for a categorification of the unique invariant vector c V λ V λ. We can actually guess quite easily what this should be. The space of invariants in any tensor product category is spanned by the classes of simples killed by all E i (there are the right number of these by the crystal structure on simples). Thus in V λ,λ, we need to find the one simple killed by all E i. This is easy to find; start with the unique simple/projective in T λ w 0 λ -mod; apply the red line labelled with λ. This is a projective P λ and its simple quotient L λ spans invariants. We can describe P λ explicitly; it corresponds to the sequence (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 (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

11 Coevalution and evaluation This shows us precisely how our functors should work in the V λ V λ case. The coevaluation functor is categorified by the functor V = Vect V λ,λ sending C L λ. The evaluation functor is categorified by T λ,λ L λ [2ρ (λ)](2 λ, ρ ): V λ,λ V = Dfd (Vect). Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

12 Coevalution and evaluation This shows us precisely how our functors should work in the V λ V λ case. The coevaluation functor is categorified by the functor V = Vect V λ,λ sending C L λ. The evaluation functor is categorified by T λ,λ L λ [2ρ (λ)](2 λ, ρ ): V λ,λ V = Dfd (Vect). Here Ṁ is the functor that send a right module M to a left module where the algebra acts by the vertical reflection anti-automorphism. The infinite dimensionalness I mentioned earlier exactly arises from the fact that L λ does not have to have a finite length projective resolution, and even worse, P λ can appear infinitely many times in the resolution. However, this won t happen when λ s are all miniscule. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

13 Quantum trace and cotrace Now, we know that if we want quantum trace, we should compromise between L λ [2ρ (λ)](2 λ, ρ ) and L λ [ 2ρ (λ)]( 2 λ, ρ ) This tells us that the square of the ribbon twist has to act by [4ρ (λ)](4 λ, ρ ). Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

14 Quantum trace and cotrace Now, we know that if we want quantum trace, we should compromise between L λ [2ρ (λ)](2 λ, ρ ) and L λ [ 2ρ (λ)]( 2 λ, ρ ) This tells us that the square of the ribbon twist has to act by [4ρ (λ)](4 λ, ρ ). Definition The positive ribbon twist acts on the category by [2ρ (λ)](2 λ, ρ ). The quantum trace functor is categorified by the functor V = Vect V λ,λ sending C L λ. The quantum cotrace functor is categorified by T λ,λ L λ [2ρ (λ)](2 λ, ρ ): V λ,λ V = Dfd (Vect). Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

15 Ribbon structure Warning: 2ρ (λ) and 2 λ, ρ don t have to be even. Thus the resulting ribbon structure ( 1) 2ρ (λ) q 2 λ,ρ is not the usual one. For each ribbon element, there is a notion of quantum dimension, and in this picture, qdim V q=1 = ( 1) 2ρ (λ) dim V. For example, in sl 2, qdim V n = ( 1) n qn+1 q n 1 q q 1. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

16 Ribbon structure Warning: 2ρ (λ) and 2 λ, ρ don t have to be even. Thus the resulting ribbon structure ( 1) 2ρ (λ) q 2 λ,ρ is not the usual one. For each ribbon element, there is a notion of quantum dimension, and in this picture, qdim V q=1 = ( 1) 2ρ (λ) dim V. For example, in sl 2, Proposition qdim V n = ( 1) n qn+1 q n 1 q q 1. The quantum knot invariants for the Snyder-Tingley ribbon structure differ from those of the usual ribbon structure by L i ( 1) (wr(li) 1)2ρ (λi) where the L i are the components of the link, and λ i are their labels. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

17 Ribbon structure From now on, all my knots are ribbon knots (in the blackboard framing), and I ll really get invariants of ribbon knots (but twists just give grading shifts). = Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

18 The unknot 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 Euler characteristic given by the quantum dimension of V λ. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

19 The unknot 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 Euler characteristic given by the quantum dimension of V λ. If V λ is miniscule, the dimension of A λ is really the dimension of V λ. In particular, if λ = ω i for g = sl n, then A λ = H (Grass(i, n)). Conjecture If λ is miniscule, A λ = H (Gr λ ). Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

20 The unknot 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 Euler characteristic given by the quantum dimension of V λ. If V λ is miniscule, the dimension of A λ is really the dimension of V λ. In particular, if λ = ω i for g = sl n, then A λ = H (Grass(i, n)). Conjecture If λ is miniscule, A λ = H (Gr λ ). On the other hand, if λ is not miniscule, things blow up. For example, if g = sl 2 and λ = 2, then i,j ( t)j dim q A j λ q 2 t q 2 t 2 Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

21 The unknot 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 Euler characteristic given by the quantum dimension of V λ. If V λ is miniscule, the dimension of A λ is really the dimension of V λ. In particular, if λ = ω i for g = sl n, then A λ = H (Grass(i, n)). Conjecture If λ is miniscule, A λ = H (Gr λ ). On the other hand, if λ is not miniscule, things blow up. For example, if g = sl 2 and λ = 2, then i,j ( t)j dim q A j λ = q 2 t q 2 t 2 + q2 q 2 t 1 t 2 q 4 Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

22 The unknot 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 Euler characteristic given by the quantum dimension of V λ. If V λ is miniscule, the dimension of A λ is really the dimension of V λ. In particular, if λ = ω i for g = sl n, then A λ = H (Grass(i, n)). Conjecture If λ is miniscule, A λ = H (Gr λ ). On the other hand, if λ is not miniscule, things blow up. For example, if g = sl 2 and λ = 2, then i,j ( 1)j dim q A j λ = q q 2 Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

23 The unknot For example, if g = sl 2 and λ = 1, then L λ is the head of the projective for. Its projective resolution looks like Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

24 The unknot For example, if g = sl 2 and λ = 1, then L λ is the head of the projective for. Its projective resolution looks like If λ = 2 then there are 3 indecomposable projectives, and the Cartan matrix is [L λ ] = 3[P λ ] 3/2[P 2 ] + [P 3 ] Therefore, no finite resolution! Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

25 Coevalution and quantum trace To give these functors in general, you can construct natural bimodules K µ. This is given by the picture. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

26 Coevalution and quantum trace To give these functors in general, you can construct natural bimodules K µ. This is given by the picture. λ 1 αi 1 (µ) αi n (s in 1 s i1 µ) µ i i i µ 1 i 1 n n λ l λ 1 L µ λ l Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

27 Coevalution and quantum trace There s exactly one interesting relation here, which says that µ µ i 1 i n µ µ i 1 i n = L µ L µ F i v c λ = F i (v c λ ). Theorem Tensor product with this bimodule categorifies evaluation/quantum trace, and Hom with it categorifies coevaluation/quantum cotrace. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

28 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 (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

29 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(Vect) V = D(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, and finite-dimensional in each homological and each graded degree. The graded Euler characteristic of this complex is J V,L (q). As usual, we can take a generating series of F L (C). This will not be a polynomial, but it will be a rational function in q, t. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

30 Knot invariants Start with C. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

31 Knot invariants V V A 1 = C K 1,2 V Replace with projective resolution B 1 Start with C. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

32 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 (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

33 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 (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

34 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 (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

35 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 (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

36 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 (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

37 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 (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

38 Comparison to other knot homologies So, one of the things I had promised you is that this homology would be a generalization of previous known ones. How does one check a thing like that? Well, it helps that this whole construction is actually exactly the same as the one done by Stroppel to interpret Khovanov homology in terms of parabolic category O. Thus, we just have to make sure that the categories and functors we feed into this construction are the same. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

39 Comparison to classical representation theory Theorem If g = sl n and λ = (ω 1,..., ω 1 ), then T λ -mod is isomorphic is a direct sum of blocks of category O for sl l (which block one takes depends on n). If λ is a sequence of fundamental weights then one must take parabolic category O for a parabolic whose block sizes are given by the indices of the fundamental weights. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

40 Comparison to classical representation theory Theorem If g = sl n and λ = (ω 1,..., ω 1 ), then T λ -mod is isomorphic is a direct sum of blocks of category O for sl l (which block one takes depends on n). If λ is a sequence of fundamental weights then one must take parabolic category O for a parabolic whose block sizes are given by the indices of the fundamental weights. For λ = (ω 3, ω 2, ω 4 ), we take the category generated by the simples for column strict tableaux with entries in [1, n] on the Young pyramid Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

41 Comparison to classical representation theory Theorem If g = sl n and λ = (ω 1,..., ω 1 ), then T λ -mod is isomorphic is a direct sum of blocks of category O for sl l (which block one takes depends on n). If λ is a sequence of fundamental weights then one must take parabolic category O for a parabolic whose block sizes are given by the indices of the fundamental weights. For λ = (ω 3, ω 2, ω 4 ), we take the category generated by the simples for column strict tableaux with entries in [1, n] on the Young pyramid Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

42 Comparison to classical representation theory If you want non-fundamental weights, you need to start taking subcategories generated by particular projective modules. Projectives correspond to column strict Young tableaux, so let me try to tell you which ones to take. First write each weight λ i = a n 1 ω n 1 + a n 2 ω n 2 + and get a sequence of fundamental weights, with demarcated groups within which the indices decrease such that the groups sum to our desired weights. The desired projectives are those that correspond to column strict tableaux on the Young pyramid which are semi-standard on each group. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

43 Comparison to classical representation theory If you want non-fundamental weights, you need to start taking subcategories generated by particular projective modules. Projectives correspond to column strict Young tableaux, so let me try to tell you which ones to take. First write each weight λ i = a n 1 ω n 1 + a n 2 ω n 2 + and get a sequence of fundamental weights, with demarcated groups within which the indices decrease such that the groups sum to our desired weights. The desired projectives are those that correspond to column strict tableaux on the Young pyramid which are semi-standard on each group. For λ = (ω 3 + ω 2, ω 4 ), we get the sequence (ω 3, ω 2, ω 4 ) and take the category generated by the projectives for column strict tableaux with entries in [1, n] on the Young pyramid which are semi-standard on the first two rows: Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

44 Comparison to classical representation theory If you want non-fundamental weights, you need to start taking subcategories generated by particular projective modules. Projectives correspond to column strict Young tableaux, so let me try to tell you which ones to take. First write each weight λ i = a n 1 ω n 1 + a n 2 ω n 2 + and get a sequence of fundamental weights, with demarcated groups within which the indices decrease such that the groups sum to our desired weights. The desired projectives are those that correspond to column strict tableaux on the Young pyramid which are semi-standard on each group. For λ = (ω 3 + ω 2, ω 4 ), we get the sequence (ω 3, ω 2, ω 4 ) and take the category generated by the projectives for column strict 5 tableaux with entries in [1, n] on the Young pyramid which are 3 semi-standard on the first two rows: Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

45 Comparison to other knot homologies Theorem Assume now that g = sl n and all weights λ i = ω pi are fundamental T λ µ -mod is a block of parabolic category O for sl P p i for (p 1,..., p l )-block upper triangular matrices. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

46 Comparison to other knot homologies Theorem Assume now that g = sl n and all weights λ i = ω pi are fundamental T λ µ -mod is a block of parabolic category O for sl P p i for (p 1,..., p l )-block upper triangular matrices. F i and E i are translation functors, given by different eigenspaces of the Casimir on C N and (C N ). Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

47 Comparison to other knot homologies Theorem Assume now that g = sl n and all weights λ i = ω pi are fundamental T λ µ -mod is a block of parabolic category O for sl P p i for (p 1,..., p l )-block upper triangular matrices. F i and E i are translation functors, given by different eigenspaces of the Casimir on C N and (C N ). the (co)evaluation/quantum (co)trace functors are given by Zuckerman functors composed with Enright-Shelton equivalences. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

48 Comparison to other knot homologies Theorem Assume now that g = sl n and all weights λ i = ω pi are fundamental T λ µ -mod is a block of parabolic category O for sl P p i for (p 1,..., p l )-block upper triangular matrices. F i and E i are translation functors, given by different eigenspaces of the Casimir on C N and (C N ). the (co)evaluation/quantum (co)trace functors are given by Zuckerman functors composed with Enright-Shelton equivalences. if p i = 1, then the braiding functors are twisting functors. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

49 Comparison to other knot homologies Theorem Assume now that g = sl n and all weights λ i = ω pi are fundamental T λ µ -mod is a block of parabolic category O for sl P p i for (p 1,..., p l )-block upper triangular matrices. F i and E i are translation functors, given by different eigenspaces of the Casimir on C N and (C N ). the (co)evaluation/quantum (co)trace functors are given by Zuckerman functors composed with Enright-Shelton equivalences. if p i = 1, then the braiding functors are twisting functors. For the standard representation of sl n, this invariant is the same as the homology given by Mazorchuk-Stroppel and Sussan. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

50 Comparison to other knot homologies Theorem Assume now that g = sl n and all weights λ i = ω pi are fundamental T λ µ -mod is a block of parabolic category O for sl P p i for (p 1,..., p l )-block upper triangular matrices. F i and E i are translation functors, given by different eigenspaces of the Casimir on C N and (C N ). the (co)evaluation/quantum (co)trace functors are given by Zuckerman functors composed with Enright-Shelton equivalences. if p i = 1, then the braiding functors are twisting functors. For the standard representation of sl n, this invariant is the same as the homology given by Mazorchuk-Stroppel and Sussan. For n = 2, 3, it s known that this is Khovanov s original homology. For n > 3, it s conjectured to be Khovanov-Rozansky, but it isn t proven. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

51 Functoriality? It s not known at the moment if this is functorial in cobordisms between knots. What sort of structure would we expect to see if it were? Well, for one thing, the invariant of a circle would be a Frobenius algebra, with the Frobenius action coming from cobordisms between flat tangles. Theorem For λ miniscule, A λ is a finite-dimensional homogeneous Frobenius algebra. I conjectured earlier that this is cohomology of a smooth finite dimensional manifold. You can regard this as some evidence for that. What if λ isn t miniscule? Wildly false; has a really nice fake proof. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

52 Functoriality? In non-miniscule weights, can Humpty-Dumpty be put back together again? Well, maybe. You ll recall that a couple of days ago, I showed you λ j λ j λ j 0 = = λ j + λ j 1 + λ j λ λ λ λ and got rid of the bubbles with negative labels. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

53 Functoriality? What if I had kept them? I get a deformation on my original algebras, which looks like I took equivariant cohomology instead of cohomology. Then the algebra would have finite global dimension, so one could replay the past 3 talks with these more bubbly algebras, and get a finite dimensional answer! Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

54 Functoriality? What if I had kept them? I get a deformation on my original algebras, which looks like I took equivariant cohomology instead of cohomology. Then the algebra would have finite global dimension, so one could replay the past 3 talks with these more bubbly algebras, and get a finite dimensional answer! Except maybe there was a good reason to kill the bubbles: if you do this in the most naive way, what you get isn t a knot invariant. Reidemeister I is all wrong now. On the other hand, the invariant of a circle is a finite dimensional Frobenius algebra. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

55 Infinite-dimensional algebras On the other hand, maybe it s not so bad to be free from the tyranny of finite-dimensionality. Work in progress: Just as you can define highest weight representations, you can also define lowest weight representations. Of course, if g is finite-dimensional, these are equivalent, but if g is a Kac-Moody algebra, they are different. You can define tensor products of these modules, and braidings as before. Now it s easier to define cups and caps, since you should only define them for pairs of highest and lowest weight reps. You can now (probably) define knot homologies for Kac-Moody algebras, and they will be finite in each homological degree (but the Euler characteristic will not converge). Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

56 Roots of unity? Many applications in topology (for example, 3-manifold invariants) use quantum groups at roots of unity, rather than those at generic points of q. Is there any hope of categorifying them? Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

57 Roots of unity? Many applications in topology (for example, 3-manifold invariants) use quantum groups at roots of unity, rather than those at generic points of q. Is there any hope of categorifying them? Well, there is hope, but not a lot else at the moment... Theorem (Khovanov) If A is an algebra over F p, and : A A is a derivation such that p = 0, then the quotient of A -mod by -free modules has Grothendieck group which is naturally a module over Z[e 2πi /p ]. The ring Z[e 2πi /p ] itself arises from A = F p [X]/(X p ) and = d/dx. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

58 Roots of unity? Many applications in topology (for example, 3-manifold invariants) use quantum groups at roots of unity, rather than those at generic points of q. Is there any hope of categorifying them? Well, there is hope, but not a lot else at the moment... Theorem (Khovanov) If A is an algebra over F p, and : A A is a derivation such that p = 0, then the quotient of A -mod by -free modules has Grothendieck group which is naturally a module over Z[e 2πi /p ]. The ring Z[e 2πi /p ] itself arises from A = F p [X]/(X p ) and = d/dx. Obviously, it s an appealing idea to construct some algebra close to the ones we considered before with the right derivation, but at the moment I ve had no luck. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

59 4d TQFT? One of the inspirations for studying categorifications is the connections between higher categories and quantum field theory. The quantum knot invariants arise from a 3-d TQFT: Chern-Simons theory. You can think of this as built up from attaching the category of U q (g) representation to a circle and building the 2-and 3-dimensional layers from that. Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

60 4d TQFT? One of the inspirations for studying categorifications is the connections between higher categories and quantum field theory. The quantum knot invariants arise from a 3-d TQFT: Chern-Simons theory. You can think of this as built up from attaching the category of U q (g) representation to a circle and building the 2-and 3-dimensional layers from that. Can one make a 4-dimensional TQFT of some kind out the category of 2-representations of this categorified quantum group? Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26

61 That s all, folks! Ben Webster (U. of Oregon) Categorifying quantum knot invariants November 26, / 26