Categorification of quantum groups and quantum knot invariantsmathserver.neu.edu/~bwebster/berkeley-cat.pdf · projection of the knot. They then use the theory of quantum groups to

Categorification of quantum groupsand quantum knot invariants

Ben Webster

MIT/Oregon

March 17, 2010

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

The big picture

quantum groups Uq(g)

ribbon category of Uq(g)-reps

quantum knot polynomials

Khovanov-Lauda/Rouquier2-categories U

HAVE

quantum knot homologies

WANT


HAVE


The big picture





HAVE


WANT


HAVE

???


The big picture





HAVE


WANT


HAVE

ribbon 2-category of U-reps?

??


The big picture





HAVE


WANT


HAVE

categorifications of tensorproducts of simples

!


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 holonomyaround the knot for a certain probability distribution on the space ofg-connections on S3.

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 iscontrolled by quantum groups.


Reshetikhin-Turaev invariants

More concretely, Reshetikhin and Turaev gave a mathematical construction ofthese invariants.

They label each component of the knot with a representation, and choose aprojection of the knot. They then use the theory of quantum groups to attachmaps to small diagrams like:

⊗

⊗

⊗

⊗

C[q, q−1]

C[q, q−1]

W

W

V

V V V∗

V V∗

These are called the braiding, the quantum trace and the coevaluation.

Composing these together for a given link results in a scalar: theReshetikhin-Turaev invariant for that labeling.





⊗

⊗

⊗

⊗

C[q, q−1]

C[q, q−1]

W

W

V

V V V∗

V V∗







⊗

⊗

⊗

⊗

C[q, q−1]

C[q, q−1]

W

W

V

V V V∗

V V∗




A historical interlude

Progress has been made on categorifying these in a piecemeal fashion for awhile

Khovanov (’99): Jones polynomial (C2 for sl2).

Oszvath-Szabo, Rasmussen (’02): Alexander polynomial (which isactually a gl(1|1) invariant, and doesn’t fit into our general picture).

Khovanov (’03): C3 for sl3.

Khovanov-Rozansky (’04): Cn for sln.

Stroppel-Mazorchuk, Sussan (’06-’07): ∧iCn for sln.

Cautis-Kamnitzer (’06): ∧iCn for sln.

Khovanov-Rozansky(’06): Cn for son.

What I want to show is a unified, pictorial construction that should include allof these.

p=proven, c=conjectured

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.













.




p Khovanov (’99): Jones polynomial (C2 for sl2).

? Oszvath-Szabo, Rasmussen (’02): Alexander polynomial (which isactually a gl(1|1) invariant, and doesn’t fit into our general picture).

p Khovanov (’03): C3 for sl3.

c Khovanov-Rozansky (’04): Cn for sln.

p Stroppel-Mazorchuk, Sussan (’06-’07): ∧iCn for sln.

c Cautis-Kamnitzer (’06): ∧iCn for sln.

c Khovanov-Rozansky(’06): Cn for son.

What I want to show is a unified, pictorial construction that should include allof these. p=proven, c=conjectured.


Decategorification?

For our purposes, decategorification means sending

a vector space V −→ its dimension dim V .

a graded vector space V −→ its q-dimension dimq V .

an abelian category C −→ its Grothendieck group K0(C)a graded abelian category C −→

its q−Grothendieck group Z[q, q−1]⊗Z K0(C).

An exact functor F : C → C′ −→the induced map [F] : K0(C)→ K0(C′).

So “F : C → C′ categorifies φ : V → V ′” means “there are isomorphismsK0(C) ∼= V and K0(C′) ∼= V ′, such that the map induced by F is φ.”


Decategorification?

For our purposes, decategorification means sending

a vector space V −→ its dimension dim V .

a graded vector space V −→ its q-dimension dimq V .

an abelian category C −→ its Grothendieck group K0(C)a graded abelian category C −→

its q−Grothendieck group Z[q, q−1]⊗Z K0(C).

An exact functor F : C → C′ −→the induced map [F] : K0(C)→ K0(C′).

So “F : C → C′ categorifies φ : V → V ′” means “there are isomorphismsK0(C) ∼= V and K0(C′) ∼= V ′, such that the map induced by F is φ.”



The idea now is to categorify everything in sight.

In particular, what we’d like to find is

graded categories Vλ1,··· ,λn such that

K0q(Vλ1,··· ,λn) ∼= Vλ = Vλ1 ⊗ · · · ⊗ Vλn

where Vλi is the representation of Uq(g) of highest weight λ.Graded functors categorifying:

the Chevalley generators Ei,Fi of Uq(g).the braiding maps relating different orderings of the highest weights.the coevalution and quantum trace maps.






K0q(Vλ1,··· ,λn) ∼= Vλ = Vλ1 ⊗ · · · ⊗ Vλn

where Vλi is the representation of Uq(g) of highest weight λ.

Graded functors categorifying:







K0q(Vλ1,··· ,λn) ∼= Vλ = Vλ1 ⊗ · · · ⊗ Vλn








K0q(Vλ1,··· ,λn) ∼= Vλ = Vλ1 ⊗ · · · ⊗ Vλn


the Chevalley generators Ei,Fi of Uq(g).

the braiding maps relating different orderings of the highest weights.the coevalution and quantum trace maps.






K0q(Vλ1,··· ,λn) ∼= Vλ = Vλ1 ⊗ · · · ⊗ Vλn


the Chevalley generators Ei,Fi of Uq(g).the braiding maps relating different orderings of the highest weights.

the coevalution and quantum trace maps.






K0q(Vλ1,··· ,λn) ∼= Vλ = Vλ1 ⊗ · · · ⊗ Vλn




Pictorial categorification

One of the most successful programs of categorification has been theunderstanding of quantum groups, which goes back to Lusztig. Someremarkable progress on this story was made in the 80’s and 90’s. Usingcategorifications:

Kazhdan and Lusztig defined a basis of the Hecke algebra.

Lusztig defined the canonical basis of Uq(g) and its representations.

but this work at its heart was all geometric; there were a lot of perversesheaves involved.

While geometry has a lot of power, it’s also kinda hard. Luckily, in the pastfew years Rouquier and Khovanov-Lauda were able to redigest this wholestory combinatorially, and so I can tell you an entirely pictorial story (thoughthere’s a still a little geometry tucked away in corners).



One of the most successful programs of categorification has been theunderstanding of quantum groups, which goes back to Lusztig. Someremarkable progress on this story was made in the 80’s and 90’s. Usingcategorifications:

Kazhdan and Lusztig defined a basis of the Hecke algebra.

Lusztig defined the canonical basis of Uq(g) and its representations.

but this work at its heart was all geometric; there were a lot of perversesheaves involved.

While geometry has a lot of power, it’s also kinda hard. Luckily, in the pastfew years Rouquier and Khovanov-Lauda were able to redigest this wholestory combinatorially, and so I can tell you an entirely pictorial story (thoughthere’s a still a little geometry tucked away in corners).



Now we define an algebra R(g) generated by pictures consisting of strandseach colored with a simple root, each labeled with any number of dots withthe 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 ∼= sl3 with simple roots α1 = (1,−1, 0) and α2 = (0, 1,−1).


The Khovanov-Lauda relations

The relations for g ∼= sl3 are given by (keeping in mind there is anautomorphism interchanging blue and green).

== +

= 0 = +

=

=

= +




== +

= 0 = +

=

=

= +




== +

= 0 = +

=

=

= +




== +

= 0 = +

=

=

= +



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 amonoidal 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 derivedequivalent to Lusztig’s categorification of U+(g) and the canonical basis iscategorified by the indecomposible projective modules.


















Irreducible representations

Fix a highest weight λ of g. We want a module category over R(g) -modgenerated by a “highest weight object” which I draw as a red line. Thus, if Iact “horizontally” with R(g), I’ll get pictures like

λ

But I need to impose relations to get a finite dimensional Grothendieck group:

λ

=

λ

α∨1 (λ) · · · = 0

λ

=

λ

α∨2 (λ) · · · = 0

Theorem (Lauda-Vazirani)

The category Vλ with its V∞ action categorifies the irreduciblerepresentation Vλ. The canonical basis is categorified by the indecomposibleprojective modules.


Irreducible representations

Fix a highest weight λ of g. We want a module category over R(g) -modgenerated by a “highest weight object” which I draw as a red line. Thus, if Iact “horizontally” with R(g), I’ll get pictures like

λ

But I need to impose relations to get a finite dimensional Grothendieck group:

λ

=

λ

α∨1 (λ) · · · = 0

λ

=

λ

α∨2 (λ) · · · = 0

Theorem (Lauda-Vazirani)

The category Vλ with its V∞ action categorifies the irreduciblerepresentation Vλ. The canonical basis is categorified by the indecomposibleprojective modules.


Tensor products

Now, some of you might think: “Wait, why is looking at the category of2-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 thetensor 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 categorificationsof things, and the canonical basis of the tensor product is not the tensorproduct of canonical bases.

In fact, we’ll see that there are objects categorifying the tensor product of thecanonical bases, but they are not projective.


Tensor products








Tensor products








Tensor products








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 anynumber 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


λ1 λ2 λ3










λ1 λ2 λ3










λ1 λ2 λ3



First, we impose the Khovanov-Lauda relations from before. Also, we alsoneed some relations involving red lines.

= +∑

a+b=α∨1 (λ)−1ba =

= =

λ

=

λ

α∨1 (λ)

λ

=

λ

α∨1 (λ)

λ

=

λ

α∨2 (λ)

λ

=

λ

α∨2 (λ)Any diagram where ablue or green strandis to the left of allred strands is 0.




= +∑

a+b=α∨1 (λ)−1ba =

= =

λ

=

λ

α∨1 (λ)

λ

=

λ

α∨1 (λ)

λ

=

λ

α∨2 (λ)

λ

=

λ





= +∑

a+b=α∨1 (λ)−1ba =

= =

λ

=

λ

α∨1 (λ)

λ

=

λ

α∨1 (λ)

λ

=

λ

α∨2 (λ)

λ

=

λ

α∨2 (λ)

Any diagram where ablue or green strandis to the left of allred strands is 0.




= +∑

a+b=α∨1 (λ)−1ba =

= =

λ

=

λ

α∨1 (λ)

λ

=

λ

α∨1 (λ)

λ

=

λ

α∨2 (λ)

λ

=

λ



Comparison to Lauda-Vazirani

If there’s only red line, then we only get one new interesting relation:

λ

=

λ

α∨2 (λ) · · · · · · = 0

If there’s only one red line labeled with λ, then we just get back the categoryfor a simple representation.



For a sequence of representations λ = (λ1, . . . , λ`), let Eλ be the subalgebrawhere the red lines are labeled with λ in order.

TheoremK(Eλ -mod) ∼= Vλ1 ⊗ · · · ⊗ Vλ` = Vλ.

We can also describe the action of U−q (g) on Vλ using this categorification:

TheoremThe “horizontal” action of R(g) -mod induces the usual action of U−q (g) onVλ.

This is the first sanity check: the grading shifts necessary to get to quantumcoproduct are “built in.”



For a sequence of representations λ = (λ1, . . . , λ`), let Eλ be the subalgebrawhere the red lines are labeled with λ in order.

TheoremK(Eλ -mod) ∼= Vλ1 ⊗ · · · ⊗ Vλ` = Vλ.

We can also describe the action of U−q (g) on Vλ using this categorification:

TheoremThe “horizontal” action of R(g) -mod induces the usual action of U−q (g) onVλ.

This is the first sanity check: the grading shifts necessary to get to quantumcoproduct are “built in.”


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 allpictures where all red/black crossings are “negative.”

positive crossingnegative crossing

Let Sλ denote the right Eλ-module given by quotient by this ideal.

Proposition

End(Sλ) ∼= Eλ1 ⊗ · · · ⊗ Eλ` .

So, the failure of Sλ to be projective is exactly what encodes the differencebetween our tensor product category, and the naive tensor product.


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 allpictures where all red/black crossings are “negative.”

positive crossingnegative crossing

Let Sλ denote the right Eλ-module given by quotient by this ideal.

Proposition

End(Sλ) ∼= Eλ1 ⊗ · · · ⊗ Eλ` .

So, the failure of Sλ to be projective is exactly what encodes the differencebetween our tensor product category, and the naive tensor product.


Standard modules

We also obtain a functor from the naive product category to our category

given by −L⊗Eλ1⊗···⊗Eλ` Sλ : Vλ1;...;λ` → Vλ.

Proposition

This functor induces Vλ∼= K(Vλ1)⊗ · · · ⊗ K(Vλ`) ∼= K(Vλ).

To see why this is so, consider Fi(Sλ). This has a filtration given by elements

λ1

λ1

· · ·

λm

λm

· · ·

i

i

whose successive quotients match the terms of the coproduct

∆(`)(F) = 1⊗ · · · ⊗ Fi + · · ·+ Fi ⊗ K−1i ⊗ · · · ⊗ K−1

i .


Standard modules

We also obtain a functor from the naive product category to our category

given by −L⊗Eλ1⊗···⊗Eλ` Sλ : Vλ1;...;λ` → Vλ.

Proposition

This functor induces Vλ∼= K(Vλ1)⊗ · · · ⊗ K(Vλ`) ∼= K(Vλ).

To see why this is so, consider Fi(Sλ). This has a filtration given by elements

λ1

λ1

· · ·

λm

λm

· · ·

i

i

whose successive quotients match the terms of the coproduct

∆(`)(F) = 1⊗ · · · ⊗ Fi + · · ·+ Fi ⊗ K−1i ⊗ · · · ⊗ K−1

i .


Braiding

So, now we need to look for braiding functors.

Consider the bimodule Bi over Eλ and E(i,i+1)·λ given by exactly the samesort of diagrams, but with a single crossing inserted between the ith andi + 1st crossings.

λ1

λ1

λ3

λ3

λ2

λ2

Theorem

The derived tensor product−L⊗Bi : Vλ → V(i,i+1)·λ categorifies the braiding

map Ri : Vλ → V(i,i+1)·λ. The inverse functor is given by RHom(Bi,−).


Braiding



λ1

λ1

λ3

λ3

λ2

λ2

Theorem




Braiding



λ1

λ1

λ3

λ3

λ2

λ2

Theorem




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−w0λ = λ∗. In this case, pick a reduced expression

w0 = s1 · · · sn with corresponding roots α1, · · · , αn.

There’s a unique simple module Lλ not killed by the idempotent for thesequence of weights and roots λ, α(α∨1 (λ))

1 , α(α∨2 (s1λ))2 , . . . , α

(α∨n (sn−1···s1λ))n , λ∗.

The coevaluation functor is categorified by the functorV∅ ∼= Vect→ Vλ,λ∗ sending C→ Lλ.

The quantum trace functor is categorified by

RHom(Lλ,−)[2ρ∨(λ)](2〈λ, ρ〉) : Vλ,λ∗ → V∅ ∼= Dfd(Vect).






1 , α(α∨2 (s1λ))2 , . . . , α

(α∨n (sn−1···s1λ))n , λ∗.









1 , α(α∨2 (s1λ))2 , . . . , α

(α∨n (sn−1···s1λ))n , λ∗.






In particular, the algebra (which is the invariant of the circle)

Aλ = Ext•(Lλ,Lλ)[2ρ∨(λ)](2〈λ, ρ〉)

has graded dimension given by the quantum dimension of Vλ.

Has anyone seen this algebra before?

One interesting candidate is the algebra structure that Feigin, Frenkel andRybnikov put on Vλ using the “quantum shift of argument algebra” at aprincipal nilpotent.

Another tantalizing possibility is that it is related to the geometry of Grλ.Perhaps a ring structure on intersection cohomology?




Aλ = Ext•(Lλ,Lλ)[2ρ∨(λ)](2〈λ, ρ〉)








Aλ = Ext•(Lλ,Lλ)[2ρ∨(λ)](2〈λ, ρ〉)







To do this in general, you can construct natural bimodules Kλ. Rather thangive a definition, let me just draw the picture.

λ1

λ1

· · ·

µ µ∗

· · ·

λ`

λ`α∨2 (µ) α∨1 (µ)α∨1 + α∨2 (µ)

Lµ

TheoremTensor product with this bimodule categorifies coevaluation, and Hom with itcategorifies quantum trace.




λ1

λ1

· · ·

µ µ∗

· · ·

λ`

λ`α∨2 (µ) α∨1 (µ)α∨1 + α∨2 (µ)

Lµ





λ1

λ1

· · ·

µ µ∗

· · ·

λ`

λ`α∨2 (µ) α∨1 (µ)α∨1 + α∨2 (µ)

Lµ



Ribbon structure

Now, I should have drawn all these pictures as ribbon knots, since framingmatters in our picture. Moreover, I need to associate an actual functor to theribbon twist.

=

The ribbon functor is just M 7→ M(〈2λ, ρ〉)[2ρ∨(λ)].

Note: this is a strange ribbon element! (It appeared in work of Snyder andTingley on half-twist elements.) But that won’t change things very much.


Ribbon structure


=




Ribbon structure


=




Knot invariants

Now, we start with a picture of our knot (in red), cut it up into theseelementary pieces, and compose these functors in the order the elementarypieces fit together.

For a link L, we get a functor FL : V∅ ∼= Dfd(Vect)→ V∅ ∼= Dfd(Vect). SoFL(C) is a complex of vector spaces (actually graded vector spaces).

TheoremThe cohomology of FL(C) is a knot invariant. The graded Euler characteristicof this complex is JV,L(q).


Knot invariants

Now, we start with a picture of our knot (in red), cut it up into theseelementary pieces, and compose these functors in the order the elementarypieces fit together.

For a link L, we get a functor FL : V∅ ∼= Dfd(Vect)→ V∅ ∼= Dfd(Vect). SoFL(C) is a complex of vector spaces (actually graded vector spaces).

TheoremThe cohomology of FL(C) is a knot invariant. The graded Euler characteristicof this complex is JV,L(q).


Knot invariants

V V∗

V V∗ V V∗

V V V∗ V∗

V V V∗ V∗

V V V∗ V∗

V V∗

Start with C.

A1 = C⊗ K1,2V

Replace with projectiveresolution B1

A2 = B1 ⊗ K1,2V

Replace with injectiveresolution B2

A3 = RHom(Bi,B2)Replace with projectiveresolution B3

A4 = B3 ⊗B1Replace with projectiveresolution B4

A5 = B4 ⊗B3Replace with injectiveresolution B5

A6 = RHom(K2,3V ,B5)


A7 = RHom(K1,2V ,B6) Knot homology!


Knot invariants

V V∗

V V∗ V V∗

V V V∗ V∗

V V V∗ V∗

V V V∗ V∗

V V∗

Start with C.

A1 = C⊗ K1,2V


A2 = B1 ⊗ K1,2V









Knot invariants

V V∗

V V∗ V V∗

V V V∗ V∗

V V V∗ V∗

V V V∗ V∗

V V∗

Start with C.

A1 = C⊗ K1,2V


A2 = B1 ⊗ K1,2V









Knot invariants

V V∗

V V∗ V V∗

V V V∗ V∗

V V V∗ V∗

V V V∗ V∗

V V∗

Start with C.

A1 = C⊗ K1,2V


A2 = B1 ⊗ K1,2V









Knot invariants

V V∗

V V∗ V V∗

V V V∗ V∗

V V V∗ V∗

V V V∗ V∗

V V∗

Start with C.

A1 = C⊗ K1,2V


A2 = B1 ⊗ K1,2V









Knot invariants

V V∗

V V∗ V V∗

V V V∗ V∗

V V V∗ V∗

V V V∗ V∗

V V∗

Start with C.

A1 = C⊗ K1,2V


A2 = B1 ⊗ K1,2V









Knot invariants

V V∗

V V∗ V V∗

V V V∗ V∗

V V V∗ V∗

V V V∗ V∗

V V∗

Start with C.

A1 = C⊗ K1,2V


A2 = B1 ⊗ K1,2V









Knot invariants

V V∗

V V∗ V V∗

V V V∗ V∗

V V V∗ V∗

V V V∗ V∗

V V∗

Start with C.

A1 = C⊗ K1,2V


A2 = B1 ⊗ K1,2V









Functoriality?

It’s not known at the moment if this is functorial in cobordisms betweenknots. How would one construct a functoriality map?

Cobordisms of knots can be cut (using a Morse function) into the moves of

∅

circle destruction

∅

circle creationsaddle move

Being able to define these maps requires that the cap and cup functors arebiadjoint (they’re clearly adjoint one way).

However, one has to prove that this map does not depend on the handledecomposition. Not easy!


Functoriality?

It’s not known at the moment if this is functorial in cobordisms betweenknots. How would one construct a functoriality map?

Cobordisms of knots can be cut (using a Morse function) into the moves of

∅

circle destruction

∅

circle creationsaddle move

Being able to define these maps requires that the cap and cup functors arebiadjoint (they’re clearly adjoint one way).

However, one has to prove that this map does not depend on the handledecomposition. Not easy!


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 modifiedto give Knot Floer homology?

These categories seem to arise from perverse sheaves on/Fukayacategories of quiver varieties. Can the knot homology be described inthat picture?


Open questions






Open questions






Open questions






Thanks, y’all.


Categorification of quantum groups and quantum knot invariantsmathserver.neu.edu/~bwebster/berkeley-cat.pdf · projection of the knot. They then use the theory of quantum groups to

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