Higher representation theory in algebra and geometry:Lecture VIII
Ben Webster
UVA
April 8, 2014
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 1 / 36
References
For this lecture, useful references include:
B.W., Knot invariants and higher representation theory
The slides for the talk are on my webpage at:http://people.virginia.edu/~btw4e/lecture-8.pdf
You can also find some proofs that I didn’t feel like going through in class at:https://pages.shanti.virginia.edu/Higher_Rep_Theory/
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 2 / 36
The future
So, there are 3 class meetings left. In what time is left, I want to try to covertwo interesting applications of the theory we’ve discussed.
the construction of knot invariants using this theory. We’ve alreadydiscussed one special case of this, using sl2-categorifications to obtainthe Jones polynomial. This generalizes to other types. There’s also a“dual” construction of these knot invariants for sln, which we’ll likelyget to in Lecture 9. This also includes some interesting connections toalgebraic geometry.
the perspective on the representation theory of Cherednik algebrasafforded by higher representation theory. This is is, of course, anenormous topic, but I think it’s an exciting application of the theory, andone worth discussing a bit. I anticipate that will be Lecture 10.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 3 / 36
Knot invariants
Roadmap
quantum groups Uq(g)
ribbon category of Uq(g)-reps
quantum knot polynomials(Jones polynomial, etc.)
Khovanov-Lauda/Rouquier2-categories U
HAVE
quantum knot homologies
WANT
quantum knot homologies
HAVE
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 4 / 36
Knot invariants
Roadmap
quantum groups Uq(g)
ribbon category of Uq(g)-reps
quantum knot polynomials(Jones polynomial, etc.)
Khovanov-Lauda/Rouquier2-categories U
HAVE
quantum knot homologies
WANT
quantum knot homologies
HAVE
???
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 4 / 36
Knot invariants
Roadmap
quantum groups Uq(g)
ribbon category of Uq(g)-reps
quantum knot polynomials(Jones polynomial, etc.)
Khovanov-Lauda/Rouquier2-categories U
HAVE
quantum knot homologies
WANT
quantum knot homologies
HAVE
ribbon 2-category of U-reps?
??
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 4 / 36
Knot invariants
Roadmap
quantum groups Uq(g)
ribbon category of Uq(g)-reps
quantum knot polynomials(Jones polynomial, etc.)
Khovanov-Lauda/Rouquier2-categories U
HAVE
quantum knot homologies
WANT
quantum knot homologies
HAVE
categorifications of tensorproducts of simples
!
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 4 / 36
Knot invariants Braiding
Reshetikhin-Turaev invariants
Let me briefly indicate how the left side of the diagram works.
Quantum groups are deformations of universal enveloping algebras. Perhapsthe most important thing about them is that they deform the tensor productof U(g) representations. Given two reps V , W, we still have a Uq(g)-action onV ⊗W.
However, in this new definition, the obvious map V ⊗W → W ⊗ V is not amap of representations. Luckily, this can be fixed by changing the map a littlebit, and multiplying by a formal sum R ∈ Uq(g)⊗ Uq(g) called the “universalR-matrix.”
bV,W : V ⊗W R·−→ V ⊗Wflip−→ W ⊗ V
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 5 / 36
Knot invariants Braiding
Reshetikhin-Turaev invariants
Let me briefly indicate how the left side of the diagram works.
Quantum groups are deformations of universal enveloping algebras. Perhapsthe most important thing about them is that they deform the tensor productof U(g) representations. Given two reps V , W, we still have a Uq(g)-action onV ⊗W.
However, in this new definition, the obvious map V ⊗W → W ⊗ V is not amap of representations. Luckily, this can be fixed by changing the map a littlebit, and multiplying by a formal sum R ∈ Uq(g)⊗ Uq(g) called the “universalR-matrix.”
bV,W : V ⊗W R·−→ V ⊗Wflip−→ W ⊗ V
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 5 / 36
Knot invariants Braiding
Reshetikhin-Turaev invariants
Proposition
The maps bV,W make Uq(g) into a braided monoidal category.
One way to think about this fact is that if you represent
bV,W 7→
Then the maps induced by switching factors of big tensor products satisfy thebraid relations.
(1W ⊗ bU,V)(bU,W ⊗ 1V)(1U ⊗ bV,W)
=
(bV,W ⊗ 1U)(1V ⊗ bU,W)(bU,V ⊗ 1W)
On the other hand bV,WbW,V 6= 1, as the picture above suggests.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 6 / 36
Knot invariants Braiding
Reshetikhin-Turaev invariants
Proposition
The maps bV,W make Uq(g) into a braided monoidal category.
One way to think about this fact is that if you represent
bV,W 7→
Then the maps induced by switching factors of big tensor products satisfy thebraid relations.
(1W ⊗ bU,V)(bU,W ⊗ 1V)(1U ⊗ bV,W)
=
(bV,W ⊗ 1U)(1V ⊗ bU,W)(bU,V ⊗ 1W)
On the other hand bV,WbW,V 6= 1, as the picture above suggests.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 6 / 36
Knot invariants Cups and caps
Reshetikhin-Turaev invariants
The other important structure on representations of a quantum group is takingdual of representations. As with switching tensor factors, we have to becareful about left and right. There is a contravariant functor V 7→ V∗ calledright dual (there’s also left dual which is the same vector space with adifferent Uq(g)-action).
The category of Uq(g)-representations has canonical maps
evaluation V∗ ⊗ V → C(q), represented by
coevaluation C(q)→ V ⊗ V∗, represented by
If you want the maps the other way, you need to take left dual.
Not all is lost! After all, we have a map which switches tensor factors. Butshould we take
or ?
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 7 / 36
Knot invariants Cups and caps
Reshetikhin-Turaev invariants
The other important structure on representations of a quantum group is takingdual of representations. As with switching tensor factors, we have to becareful about left and right. There is a contravariant functor V 7→ V∗ calledright dual (there’s also left dual which is the same vector space with adifferent Uq(g)-action).
The category of Uq(g)-representations has canonical maps
evaluation V∗ ⊗ V → C(q), represented by
coevaluation C(q)→ V ⊗ V∗, represented by
If you want the maps the other way, you need to take left dual.
Not all is lost! After all, we have a map which switches tensor factors. Butshould we take
or ?
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 7 / 36
Knot invariants Cups and caps
Reshetikhin-Turaev invariants
The other important structure on representations of a quantum group is takingdual of representations. As with switching tensor factors, we have to becareful about left and right. There is a contravariant functor V 7→ V∗ calledright dual (there’s also left dual which is the same vector space with adifferent Uq(g)-action).
The category of Uq(g)-representations has canonical maps
evaluation V∗ ⊗ V → C(q), represented by
coevaluation C(q)→ V ⊗ V∗, represented by
If you want the maps the other way, you need to take left dual.
Not all is lost! After all, we have a map which switches tensor factors. Butshould we take
or ?
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 7 / 36
Knot invariants Cups and caps
Reshetikhin-Turaev invariants
Of course, we can’t play favorites. Instead we should take the geometricmean.
If V is irreducible, there’s a unique constant aV ∈ C(q) (actually a power of q)such that
1√
aV=
√aV .
A natural choice of√
aV (I really mean functorial) is called a ribbonstructure. The reason for the name is that if we interpret the diagrams asdrawn with ribbon, then they are with a left and right twist added,respectively.
DefinitionThis map is called quantum trace and its vertical flip is called quantumcotrace.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 8 / 36
Knot invariants Cups and caps
Reshetikhin-Turaev invariants
This allows us to associate a map for any oriented tangle labeled withrepresentations, by associating the braiding to a crossing and appropriate traceor evaluation to cups:
⊗
⊗
⊗
⊗
C[q, q−1]
C[q, q−1]
W
W
V
V V V∗
V V∗
Composing these together for a given ribbon link results in a scalar: theReshetikhin-Turaev invariant for that labeling.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 9 / 36
Knot invariants Cups and caps
Reshetikhin-Turaev invariants
This allows us to associate a map for any oriented tangle labeled withrepresentations, by associating the braiding to a crossing and appropriate traceor evaluation to cups:
⊗
⊗
⊗
⊗
C[q, q−1]
C[q, q−1]
W
W
V
V V V∗
V V∗
Composing these together for a given ribbon link results in a scalar: theReshetikhin-Turaev invariant for that labeling.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 9 / 36
Knot invariants Cups and caps
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’ll give you is a unified, pictorial construction that should include all ofthese. For that, we need tensor products. p=proven, c=conjectured.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 10 / 36
Knot invariants Cups and caps
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’ll give you is a unified, pictorial construction that should include all ofthese. For that, we need tensor products. p=proven, c=conjectured.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 10 / 36
Knot invariants Cups and caps
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’ll give you is a unified, pictorial construction that should include all ofthese. For that, we need tensor products. p=proven, c=conjectured.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 10 / 36
Knot invariants Cups and caps
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’ll give you is a unified, pictorial construction that should include all ofthese. For that, we need tensor products. p=proven, c=conjectured.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 10 / 36
Knot invariants Cups and caps
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’ll give you is a unified, pictorial construction that should include all ofthese. For that, we need tensor products. p=proven, c=conjectured.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 10 / 36
Knot invariants Cups and caps
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’ll give you is a unified, pictorial construction that should include all ofthese. For that, we need tensor products. p=proven, c=conjectured.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 10 / 36
Knot invariants Cups and caps
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’ll give you is a unified, pictorial construction that should include all ofthese. For that, we need tensor products. p=proven, c=conjectured.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 10 / 36
Knot invariants Cups and caps
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’ll give you is a unified, pictorial construction that should include all ofthese. For that, we need tensor products. p=proven, c=conjectured.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 10 / 36
Knot invariants Cups and caps
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’ll give you is a unified, pictorial construction that should include all ofthese. For that, we need tensor products. p=proven, c=conjectured.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 10 / 36
Knot invariants Cups and caps
A historical interlude
Progress has been made on categorifying these in a piecemeal fashion for awhile
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.
p Khovanov-Rozansky (’04): Cn for sln.
p Stroppel-Mazorchuk, Sussan (’06-’07): ∧iCn for sln.
p Cautis-Kamnitzer (’06): ∧iCn for sln.
c Khovanov-Rozansky(’06): Cn for son.
What I’ll give you is a unified, pictorial construction that should include all ofthese. For that, we need tensor products. p=proven, c=conjectured.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 10 / 36
Knot invariants Cups and caps
A historical interlude
Progress has been made on categorifying these in a piecemeal fashion for awhile
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.
p Khovanov-Rozansky (’04): Cn for sln.
p Stroppel-Mazorchuk, Sussan (’06-’07): ∧iCn for sln.
p Cautis-Kamnitzer (’06): ∧iCn for sln.
c Khovanov-Rozansky(’06): Cn for son.
What I’ll give you is a unified, pictorial construction that should include all ofthese. For that, we need tensor products. p=proven, c=conjectured.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 10 / 36
Tensor products Definition
Tensor products
In the case of sl2, we introduced a graphical calculus for elements ofVλ = Vλ1 ⊗ · · · ⊗ Vλ` .
A downward black line on the left means acting by Fi.
A red line at the left labeled by λ corresponds to vλ ⊗−, where vλ is thehighest weight vector of Vλ.
So, we obtain a spanning set of Vλ consisting of vectors like
Fi(vλ1 ⊗ Fjvλ2)↔λ1 + λ2−αj + αi
λ1 + λ2−αjλ2 λ2 − αj
iλ1jλ2
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 11 / 36
Tensor products Definition
Tensor products
In the case of sl2, we introduced a graphical calculus for elements ofVλ = Vλ1 ⊗ · · · ⊗ Vλ` .
A downward black line on the left means acting by Fi.
A red line at the left labeled by λ corresponds to vλ ⊗−, where vλ is thehighest weight vector of Vλ.
So, we obtain a spanning set of Vλ consisting of vectors like
Fi(vλ1 ⊗ Fjvλ2)↔λ1 + λ2−αj + αi
λ1 + λ2−αjλ2 λ2 − αj
iλ1jλ2
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 11 / 36
Tensor products Definition
Tensor products
Let Tλ be the algebra whose elements are k-linear combinations of immersed1-manifolds with
black components oriented, dotted and labeled with i ∈ Γ andred components have no intersections, and are labeled with the weights λin order modulo the relations
i λ
=
λi
λi
λ i
=
iλ
λi ii λ
=
ii λ
+∑
a+b=λi−1b
iλ
a
i
ij λ
=
ij λ
= =
any diagram witha black line at
the far left is 0.
and. . . . . .Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 12 / 36
Tensor products Definition
Diagrams
i j
=
i j
unless i = j
i i
=
i i
+
i i
i i
=
i i
+
i i
i j
=
ji
Qij(y1, y2)
ki j
=
ki j
unless i = k = j± 1
i i
= 0
ii j
=
ii j
−
ii j
Qij(y3, y2)− Qij(y1, y2)
y3 − y1
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 13 / 36
Tensor products Definition
Diagrams
i j
=
i j
unless i = j
i i
=
i i
+
i i
i i
=
i i
+
i i
i j
=
ji
Qij(y1, y2)
ki j
=
ki j
unless i = k = j± 1
i i
= 0
ii j
=
ii j
−
ii j
Qij(y3, y2)− Qij(y1, y2)
y3 − y1
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 13 / 36
Tensor products Definition
Categorical action
Recall, last time, we defined the notion of a categorical action of g. For this,we need functors Fi and Ei.
These are induction and restriction functors, which can think of as tensorproduct with the bimodules:
Fi = · · ·
i
· · ·
right action
left action
E = · · ·
i
· · ·
right action
left action
The action of Rm on the power Fm is by attaching pictures at the bottom.Adjunction is essentially automatic.
The tricky part is checking the sl2 relations. This is hard.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 14 / 36
Tensor products Definition
Grothendieck groups
Theorem
The GG of Tλ -pmod is the Lusztig integral form of Vλ, sending the functor Fi
to the action of Fi, and the functor λ (adding a red line) to the inclusion
V−⊗vhigh↪→ V ⊗ Vλ.
But we’d like to talk about the category Tλ -mod, which doesn’t have thesame Grothendieck group: the map
K0(Tλ -pmod)→ K0(Tλ -mod)
is injective, but not surjective, since not all simple modules have finiteprojective resolutions. (Think about k[x]/(x2)).
However, this map is an isomorphism after tensoring with C(q), so everyfinite dimensional Tλ-module defines a class in Vλ.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 15 / 36
Tensor products Definition
Grothendieck groups
Theorem
The GG of Tλ -pmod is the Lusztig integral form of Vλ, sending the functor Fi
to the action of Fi, and the functor λ (adding a red line) to the inclusion
V−⊗vhigh↪→ V ⊗ Vλ.
But we’d like to talk about the category Tλ -mod, which doesn’t have thesame Grothendieck group: the map
K0(Tλ -pmod)→ K0(Tλ -mod)
is injective, but not surjective, since not all simple modules have finiteprojective resolutions. (Think about k[x]/(x2)).
However, this map is an isomorphism after tensoring with C(q), so everyfinite dimensional Tλ-module defines a class in Vλ.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 15 / 36
Tensor products Definition
Bases
What does the representation theory of this algebra look like?
Projectives are just summands of the modules Pκi = Tλe(i, κ) wheree(i, κ) is the sequence corresponding to a particular ordering of red andblack dots. The indecomposables give you a “canonical basis” (Lusztig’sif g is symmetric type).
Simple modules are endowed with a crystal structure (exactly as inLauda and Vazirani), which is the tensor product of the crystals for Vλi .These give you a “dual canonical basis.”
These objects both give bases of the Grothendieck group which are not verycompatible with the tensor product structure. If we’re ever going to do anycalculations, we’re going to need objects that correspond to pure tensors.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 16 / 36
Tensor products Definition
Bases
What does the representation theory of this algebra look like?
Projectives are just summands of the modules Pκi = Tλe(i, κ) wheree(i, κ) is the sequence corresponding to a particular ordering of red andblack dots. The indecomposables give you a “canonical basis” (Lusztig’sif g is symmetric type).
Simple modules are endowed with a crystal structure (exactly as inLauda and Vazirani), which is the tensor product of the crystals for Vλi .These give you a “dual canonical basis.”
These objects both give bases of the Grothendieck group which are not verycompatible with the tensor product structure. If we’re ever going to do anycalculations, we’re going to need objects that correspond to pure tensors.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 16 / 36
Tensor products Definition
Bases
What does the representation theory of this algebra look like?
Projectives are just summands of the modules Pκi = Tλe(i, κ) wheree(i, κ) is the sequence corresponding to a particular ordering of red andblack dots. The indecomposables give you a “canonical basis” (Lusztig’sif g is symmetric type).
Simple modules are endowed with a crystal structure (exactly as inLauda and Vazirani), which is the tensor product of the crystals for Vλi .These give you a “dual canonical basis.”
These objects both give bases of the Grothendieck group which are not verycompatible with the tensor product structure. If we’re ever going to do anycalculations, we’re going to need objects that correspond to pure tensors.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 16 / 36
Tensor products Definition
Bases
What does the representation theory of this algebra look like?
Projectives are just summands of the modules Pκi = Tλe(i, κ) wheree(i, κ) is the sequence corresponding to a particular ordering of red andblack dots. The indecomposables give you a “canonical basis” (Lusztig’sif g is symmetric type).
Simple modules are endowed with a crystal structure (exactly as inLauda and Vazirani), which is the tensor product of the crystals for Vλi .These give you a “dual canonical basis.”
These objects both give bases of the Grothendieck group which are not verycompatible with the tensor product structure. If we’re ever going to do anycalculations, we’re going to need objects that correspond to pure tensors.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 16 / 36
Tensor products Standard modules
Standard modules
Well, how would we construct the pure tensor v1 ⊗ Fiv2? We have modulescorresponding to
λ1 λ2 i
Fi(v1 ⊗ v2) = v1 ⊗ Fiv2 + qλiFiv1 ⊗ v2
and
λ1 λ2i
Fiv1 ⊗ v2
So we’d like to subtract the former from the latter. Of course, in categoriesyou can’t subtract, but you can look for submodules. As it happens, the mapgiven by is injective, so modding out by its image gives a module withthe right class in the Grothendieck group.
Can this phenomenon be generalized?
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 17 / 36
Tensor products Standard modules
Standard modules
Well, how would we construct the pure tensor v1 ⊗ Fiv2? We have modulescorresponding to
λ1 λ2 i
Fi(v1 ⊗ v2) = v1 ⊗ Fiv2 + qλiFiv1 ⊗ v2
and
λ1 λ2i
Fiv1 ⊗ v2
So we’d like to subtract the former from the latter. Of course, in categoriesyou can’t subtract, but you can look for submodules. As it happens, the mapgiven by is injective, so modding out by its image gives a module withthe right class in the Grothendieck group.
Can this phenomenon be generalized?
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 17 / 36
Tensor products Standard modules
Standard modules
a “left” crossing a “right” crossing
DefinitionThe standard module Sκλ is the quotient of Pκλ by the submodule generated byall diagrams with at least one “left” crossing as above, and no “right”crossings.
Put another way, we can associate a composition to the module Pκi bycounting the number of black strands between each pair of reds, and we modout by the images of all maps from projectives strictly higher in dominanceorder.
In the example of the last slide, we just use that (1, 0) > (0, 1).
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 18 / 36
Tensor products Standard modules
Standard modules
a “left” crossing a “right” crossing
DefinitionThe standard module Sκλ is the quotient of Pκλ by the submodule generated byall diagrams with at least one “left” crossing as above, and no “right”crossings.
Put another way, we can associate a composition to the module Pκi bycounting the number of black strands between each pair of reds, and we modout by the images of all maps from projectives strictly higher in dominanceorder.
In the example of the last slide, we just use that (1, 0) > (0, 1).
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 18 / 36
Tensor products Standard modules
Standard modules
As you may have guessed
Proposition
[Sκi ] = Fiκ(1)−1 · · ·Fi1v1 ⊗ · · · ⊗ Fin · · ·Fκ(`)vn
This makes standard modules invaluable as “test objects” for functors to seethat they behave correctly on the Grothendieck group.
For example, FiSκi has a filtration which categorifies the usual formula
∆(`)(Fi) = Fi ⊗ K̃i ⊗ · · · ⊗ K̃i + · · ·+ 1⊗ · · · ⊗ 1⊗ Fi
and similarly for EiSκi .
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 19 / 36
Tensor products Standard modules
Standard modules
As you may have guessed
Proposition
[Sκi ] = Fiκ(1)−1 · · ·Fi1v1 ⊗ · · · ⊗ Fin · · ·Fκ(`)vn
This makes standard modules invaluable as “test objects” for functors to seethat they behave correctly on the Grothendieck group.
For example, FiSκi has a filtration which categorifies the usual formula
∆(`)(Fi) = Fi ⊗ K̃i ⊗ · · · ⊗ K̃i + · · ·+ 1⊗ · · · ⊗ 1⊗ Fi
and similarly for EiSκi .
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 19 / 36
Braiding functors Definition
Derived category
What functors? Well, we had a whole lot of maps earlier, corresponding toany tangle (though it was enough to define them for small pictures).
Unfortunately, if we want to categorify these using the yoga we’ve used thusfar, we run into a problem: the coefficients aren’t positive.
If you want to have a “direct minus” in a category, you have to use some kindof category of complexes. We let Vλ be the bounded-above derived categoryof Tλ -mod.
I bet lots of you are happier with the homotopy category, but that doesn’twork so well for me. Working in that category would require me knowingsome projective resolutions that are very hard to write down.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 20 / 36
Braiding functors Definition
Derived category
What functors? Well, we had a whole lot of maps earlier, corresponding toany tangle (though it was enough to define them for small pictures).
Unfortunately, if we want to categorify these using the yoga we’ve used thusfar, we run into a problem: the coefficients aren’t positive.
If you want to have a “direct minus” in a category, you have to use some kindof category of complexes. We let Vλ be the bounded-above derived categoryof Tλ -mod.
I bet lots of you are happier with the homotopy category, but that doesn’twork so well for me. Working in that category would require me knowingsome projective resolutions that are very hard to write down.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 20 / 36
Braiding functors Definition
Braiding and duals
TheoremGiven any sequence λ:
For any `-strand braid σ, we have a functor Vλ → Vσλ which inducesthe usual braided structure on the GG.
For any λ, and λ+ given by adding an adjacent pair of dual highestweights, we have functors Vλ+ → Vλ inducing evaluation and quantumtrace on GG, and dually for coevaluation and quantum cotrace (but for afunny ribbon structure!).
My goal for the rest of this talk is to describe these functors, and how theygive knot invariants.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 21 / 36
Braiding functors Definition
Braiding and duals
TheoremGiven any sequence λ:
For any `-strand braid σ, we have a functor Vλ → Vσλ which inducesthe usual braided structure on the GG.
For any λ, and λ+ given by adding an adjacent pair of dual highestweights, we have functors Vλ+ → Vλ inducing evaluation and quantumtrace on GG, and dually for coevaluation and quantum cotrace (but for afunny ribbon structure!).
My goal for the rest of this talk is to describe these functors, and how theygive knot invariants.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 21 / 36
Braiding functors Definition
Braiding and duals
TheoremGiven any sequence λ:
For any `-strand braid σ, we have a functor Vλ → Vσλ which inducesthe usual braided structure on the GG.
For any λ, and λ+ given by adding an adjacent pair of dual highestweights, we have functors Vλ+ → Vλ inducing evaluation and quantumtrace on GG, and dually for coevaluation and quantum cotrace (but for afunny ribbon structure!).
My goal for the rest of this talk is to describe these functors, and how theygive knot invariants.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 21 / 36
Braiding functors Definition
Braiding and duals
TheoremGiven any sequence λ:
For any `-strand braid σ, we have a functor Vλ → Vσλ which inducesthe usual braided structure on the GG.
For any λ, and λ+ given by adding an adjacent pair of dual highestweights, we have functors Vλ+ → Vλ inducing evaluation and quantumtrace on GG, and dually for coevaluation and quantum cotrace (but for afunny ribbon structure!).
My goal for the rest of this talk is to describe these functors, and how theygive knot invariants.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 21 / 36
Braiding functors Definition
Braiding
So, now we need to look for braiding functors.
Consider the bimodule Bi over Tλ and T(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⊗Tλ Bi : Vλ → V(i,i+1)·λ categorifies the
braiding map Ri : Vλ → V(i,i+1)·λ. The inverse functor is given byRHom(Bi,−).
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 22 / 36
Braiding functors Definition
Braiding
So, now we need to look for braiding functors.
Consider the bimodule Bi over Tλ and T(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⊗Tλ Bi : Vλ → V(i,i+1)·λ categorifies the
braiding map Ri : Vλ → V(i,i+1)·λ. The inverse functor is given byRHom(Bi,−).
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 22 / 36
Braiding functors Definition
Braiding
So, now we need to look for braiding functors.
Consider the bimodule Bi over Tλ and T(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⊗Tλ Bi : Vλ → V(i,i+1)·λ categorifies the
braiding map Ri : Vλ → V(i,i+1)·λ. The inverse functor is given byRHom(Bi,−).
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 22 / 36
Braiding functors Definition
Braiding
So, firstly, what does derived tensor product mean? It means, amongst otherthings, that we could take a projective resolution of Bi as a bimodule. Thiswill be a complex in the category Tλ ⊗ Tλ -pmod which is unique up tohomotopy.
Unfortunately, I don’t understand at the moment how to write down thiscomplex explicitly. In most cases, it must have infinite length and is quitecomplex, but it would facilitate computation quite a bit.
On the other hand, part of the magic of homological algebra is that you canfigure some things out without knowing this.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 23 / 36
Braiding functors Checking properties
Braiding
In particular, how does one check that it actually acts as the braiding? Bylooking at test objects.
Note that Vλ1 ⊗ Vλ2 is generated over Uq(g) by vectors of the form v⊗ vhigh
and under the braiding, these are sent to q?vhigh ⊗ v. As we know, thesevectors are categorified by standard modules of the form S0,n
i .
Proposition
B1L⊗ S(0,n)i
∼= S(0,0)i (?)
Proof: · · · · · ·
λj+1
λj+1
λj
λj
· · ·
· · ·
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 24 / 36
Braiding functors Checking properties
Braiding
In particular, how does one check that it actually acts as the braiding? Bylooking at test objects.
Note that Vλ1 ⊗ Vλ2 is generated over Uq(g) by vectors of the form v⊗ vhigh
and under the braiding, these are sent to q?vhigh ⊗ v. As we know, thesevectors are categorified by standard modules of the form S0,n
i .
Proposition
B1L⊗ S(0,n)i
∼= S(0,0)i (?)
Proof: · · · · · ·
λj+1
λj+1
λj
λj
· · ·
· · ·
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 24 / 36
Braiding functors Checking properties
Braiding
In particular, how does one check that it actually gives a braid groupoidaction?
The positive and negative twists are inverse because they are adjoint andderived equivalences. (Not easy! Must show that half twist sendsprojectives to tiltings.)Homological algebra song and dance: for reduced expression in thesymmetric group, its positive lift to a braid sends projectives to modules.So we just have to check that as modulesBi ⊗T Bi+1 ⊗T Bi ∼= Bi+1 ⊗T Bi ⊗T Bi+1
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 25 / 36
Braiding functors Checking properties
Braiding
In particular, how does one check that it actually gives a braid groupoidaction?
The positive and negative twists are inverse because they are adjoint andderived equivalences. (Not easy! Must show that half twist sendsprojectives to tiltings.)Homological algebra song and dance: for reduced expression in thesymmetric group, its positive lift to a braid sends projectives to modules.So we just have to check that as modulesBi ⊗T Bi+1 ⊗T Bi ∼= Bi+1 ⊗T Bi ⊗T Bi+1
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 25 / 36
Braiding functors Checking properties
Braiding
In particular, how does one check that it actually gives a braid groupoidaction?
The positive and negative twists are inverse because they are adjoint andderived equivalences. (Not easy! Must show that half twist sendsprojectives to tiltings.)Homological algebra song and dance: for reduced expression in thesymmetric group, its positive lift to a braid sends projectives to modules.So we just have to check that as modulesBi ⊗T Bi+1 ⊗T Bi ∼= Bi+1 ⊗T Bi ⊗T Bi+1
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 25 / 36
Braiding functors Checking properties
Braiding
In particular, how does one check that it actually gives a braid groupoidaction?
The positive and negative twists are inverse because they are adjoint andderived equivalences. (Not easy! Must show that half twist sendsprojectives to tiltings.)Homological algebra song and dance: for reduced expression in thesymmetric group, its positive lift to a braid sends projectives to modules.So we just have to check that as modulesBi ⊗T Bi+1 ⊗T Bi ∼= Bi+1 ⊗T Bi ⊗T Bi+1
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 25 / 36
Braiding functors Checking properties
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∗. We must first define an isomorphism between Vλ∗ and V∗λ. Thatis to say, a pairing Vλ × Vλ∗ → C(q).
We start with a chosen highest weight vector of both representations vλ, vλ∗(this comes from the irrep in Tλλ -mod ∼= k -mod). So, a pairing is fixed by achoice of lowest weight vector.
Pick a reduced expression
w0 = s1 · · · sn with corresponding roots α1, · · · , αn.
Then we have a lowest weight vector of the form
vlow = F(α∨n (sn−1···s1λ))in · · ·F(α∨2 (s1λ))
i2 F(α∨1 (λ))i1 vλ
We will always choose this one.Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 26 / 36
Braiding functors Checking properties
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∗. We must first define an isomorphism between Vλ∗ and V∗λ. Thatis to say, a pairing Vλ × Vλ∗ → C(q).
We start with a chosen highest weight vector of both representations vλ, vλ∗(this comes from the irrep in Tλλ -mod ∼= k -mod). So, a pairing is fixed by achoice of lowest weight vector.
Pick a reduced expression
w0 = s1 · · · sn with corresponding roots α1, · · · , αn.
Then we have a lowest weight vector of the form
vlow = F(α∨n (sn−1···s1λ))in · · ·F(α∨2 (s1λ))
i2 F(α∨1 (λ))i1 vλ
We will always choose this one.Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 26 / 36
Braiding functors Checking properties
Invariants
We should look for a categorification of the unique invariant vectorc ∈ Vλ ⊗ Vλ∗ . We can actually guess quite easily what this should be.
The space of invariants is orthogonal under the Euler form to all projectives ofthe form FiM for any i. We know by counting arguments that all but oneindecomposable projective is a summand of a FiM.
We actually know exactly what this remaining projective Pλ is; it correspondsto the sequence of weights and roots
(λ, α(α∨1 (λ))1 , α
(α∨2 (s1λ))2 , . . . , α
(α∨n (sn−1···s1λ))n , λ∗).
So, an element of invariants is given by the simple quotient of Pλ. Denote thisLλ.
It’s pretty easy to check by hand that Lλ is killed by all Ei.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 27 / 36
Braiding functors Checking properties
Invariants
We should look for a categorification of the unique invariant vectorc ∈ Vλ ⊗ Vλ∗ . We can actually guess quite easily what this should be.
The space of invariants is orthogonal under the Euler form to all projectives ofthe form FiM for any i. We know by counting arguments that all but oneindecomposable projective is a summand of a FiM.
We actually know exactly what this remaining projective Pλ is; it correspondsto the sequence of weights and roots
(λ, α(α∨1 (λ))1 , α
(α∨2 (s1λ))2 , . . . , α
(α∨n (sn−1···s1λ))n , λ∗).
So, an element of invariants is given by the simple quotient of Pλ. Denote thisLλ.
It’s pretty easy to check by hand that Lλ is killed by all Ei.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 27 / 36
Braiding functors Checking properties
Coevalution and evaluation
The coevaluation functor is categorified by the functorV∅ ∼= Vect→ Vλ,λ∗ sending C→ Lλ.
The evaluation functor is categorified by
RHom(Lλ,−)[2ρ∨(λ)](2〈λ, ρ〉) : Vλ,λ∗ → V∅ ∼= Dfd(Vect).
Now, we know that if we want quantum trace, we should compromise between
Lλ[2ρ∨(λ)](2〈λ, ρ〉) and Lλ[−2ρ∨(λ)](−2〈λ, ρ〉)
DefinitionThe positive ribbon twist acts on the category by [2ρ∨(λ)](2〈λ, ρ〉).
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 28 / 36
Braiding functors Checking properties
Coevalution and evaluation
The coevaluation functor is categorified by the functorV∅ ∼= Vect→ Vλ,λ∗ sending C→ Lλ.
The evaluation functor is categorified by
RHom(Lλ,−)[2ρ∨(λ)](2〈λ, ρ〉) : Vλ,λ∗ → V∅ ∼= Dfd(Vect).
Now, we know that if we want quantum trace, we should compromise between
Lλ[2ρ∨(λ)](2〈λ, ρ〉) and Lλ[−2ρ∨(λ)](−2〈λ, ρ〉)
DefinitionThe positive ribbon twist acts on the category by [2ρ∨(λ)](2〈λ, ρ〉).
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 28 / 36
Braiding functors Checking properties
Ribbon structure
So this decategorifies to (−1)2ρ∨(λ)q2〈λ,ρ〉. Note: this is a strange ribbonelement! (It appeared in work of Snyder and Tingley on half-twist elements.)
For each ribbon element, there is a notion of “quantum dimension,” and in thispicture, qdimV|q=1 = (−1)2ρ∨(λ) dim V . For example, in sl2,
qdimVn = (−1)n qn+1 − q−n−1
q− q−1 .
From now on, all my knots are ribbon knots (in the blackboard framing), andI’ll really get invariants of ribbon knots (but twists just give grading shifts).
=
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 29 / 36
Braiding functors Checking properties
Coevalution and quantum trace
In particular, the algebra (which is the invariant of the circle)
Aλ = Ext•(Lλ,Lλ)[2ρ∨(λ)](2〈λ, ρ〉)
has graded Euler characteristic given by the quantum dimension of Vλ.
If Vλ is miniscule, then everything works beautifully. The dimension of Aλ isreally the dimension of Vλ. In particular, if λ = ωi for g = sln, thenAλ ∼= H∗(Grass(i, n)).
Conjecture
If λ is miniscule, Aλ ∼= H∗(Grλ).
On the other hand, if λ is not miniscule, things blow up. For example, ifg = sl2 and λ = 2, then
∑i,j(−t)j dimq Aj
λ 6= q−2t2 + 1 + q2t−2∑i,j(−t)j dimq Aj
λ = q−2t2 + 1 + q2t−2 + q2−q2t1−t2q4
∑i,j(−1)j dimq Aj
λ = q−2 + 1 + q2
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 30 / 36
Braiding functors Checking properties
Coevalution and quantum trace
In particular, the algebra (which is the invariant of the circle)
Aλ = Ext•(Lλ,Lλ)[2ρ∨(λ)](2〈λ, ρ〉)
has graded Euler characteristic given by the quantum dimension of Vλ.
If Vλ is miniscule, then everything works beautifully. The dimension of Aλ isreally the dimension of Vλ. In particular, if λ = ωi for g = sln, thenAλ ∼= H∗(Grass(i, n)).
Conjecture
If λ is miniscule, Aλ ∼= H∗(Grλ).
On the other hand, if λ is not miniscule, things blow up. For example, ifg = sl2 and λ = 2, then
∑i,j(−t)j dimq Aj
λ 6= q−2t2 + 1 + q2t−2∑i,j(−t)j dimq Aj
λ = q−2t2 + 1 + q2t−2 + q2−q2t1−t2q4
∑i,j(−1)j dimq Aj
λ = q−2 + 1 + q2
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 30 / 36
Braiding functors Checking properties
Coevalution and quantum trace
In particular, the algebra (which is the invariant of the circle)
Aλ = Ext•(Lλ,Lλ)[2ρ∨(λ)](2〈λ, ρ〉)
has graded Euler characteristic given by the quantum dimension of Vλ.
If Vλ is miniscule, then everything works beautifully. The dimension of Aλ isreally the dimension of Vλ. In particular, if λ = ωi for g = sln, thenAλ ∼= H∗(Grass(i, n)).
Conjecture
If λ is miniscule, Aλ ∼= H∗(Grλ).
On the other hand, if λ is not miniscule, things blow up. For example, ifg = sl2 and λ = 2, then
∑i,j(−t)j dimq Aj
λ 6= q−2t2 + 1 + q2t−2
∑i,j(−t)j dimq Aj
λ = q−2t2 + 1 + q2t−2 + q2−q2t1−t2q4
∑i,j(−1)j dimq Aj
λ = q−2 + 1 + q2
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 30 / 36
Braiding functors Checking properties
Coevalution and quantum trace
In particular, the algebra (which is the invariant of the circle)
Aλ = Ext•(Lλ,Lλ)[2ρ∨(λ)](2〈λ, ρ〉)
has graded Euler characteristic given by the quantum dimension of Vλ.
If Vλ is miniscule, then everything works beautifully. The dimension of Aλ isreally the dimension of Vλ. In particular, if λ = ωi for g = sln, thenAλ ∼= H∗(Grass(i, n)).
Conjecture
If λ is miniscule, Aλ ∼= H∗(Grλ).
On the other hand, if λ is not miniscule, things blow up. For example, ifg = sl2 and λ = 2, then
∑i,j(−t)j dimq Aj
λ 6= q−2t2 + 1 + q2t−2
∑i,j(−t)j dimq Aj
λ = q−2t2 + 1 + q2t−2 + q2−q2t1−t2q4
∑i,j(−1)j dimq Aj
λ = q−2 + 1 + q2
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 30 / 36
Braiding functors Checking properties
Coevalution and quantum trace
In particular, the algebra (which is the invariant of the circle)
Aλ = Ext•(Lλ,Lλ)[2ρ∨(λ)](2〈λ, ρ〉)
has graded Euler characteristic given by the quantum dimension of Vλ.
If Vλ is miniscule, then everything works beautifully. The dimension of Aλ isreally the dimension of Vλ. In particular, if λ = ωi for g = sln, thenAλ ∼= H∗(Grass(i, n)).
Conjecture
If λ is miniscule, Aλ ∼= H∗(Grλ).
On the other hand, if λ is not miniscule, things blow up. For example, ifg = sl2 and λ = 2, then
∑i,j(−t)j dimq Aj
λ 6= q−2t2 + 1 + q2t−2∑i,j(−t)j dimq Aj
λ = q−2t2 + 1 + q2t−2 + q2−q2t1−t2q4
∑i,j(−1)j dimq Aj
λ = q−2 + 1 + q2
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 30 / 36
Braiding functors Checking properties
Coevalution and quantum trace
To do this in general, you can construct natural bimodules Kµ. This is givenby the picture.
λ1
λ1
· · ·µµ∗
· · ·
λ`
λ`
inini1i1
α∨i1(µ) α∨in(sin−1 · · · si1µ)
Lµ
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 31 / 36
Braiding functors Checking properties
Coevalution and quantum trace
To do this in general, you can construct natural bimodules Kµ. This is givenby the picture.
λ1
λ1
· · ·µµ∗
· · ·
λ`
λ`
inini1i1
α∨i1(µ) α∨in(sin−1 · · · si1µ)
Lµ
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 31 / 36
Braiding functors Checking properties
Coevalution and quantum trace
There’s exactly one interesting relation here, which says that
· · ·
µ µ∗
· · ·
ini1
Lµ
= · · ·
µ µ∗
· · ·
ini1
Lµ
Fiv⊗ cλ = Fi(v⊗ cλ).
TheoremTensor product with this bimodule categorifies coevaluation/quantum cotrace,and Hom with it categorifies evaluation/quantum trace.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 32 / 36
Braiding functors Checking properties
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∅ ∼= D(Vect)→ V∅ ∼= D(Vect). SoFL(C) is a complex of vector spaces (actually graded vector spaces).
TheoremThe cohomology of FL(C) is a knot invariant, and finite-dimensional in eachhomological and each graded degree. The graded Euler characteristic of thiscomplex is JV,L(q).
As usual, we can take a generating series of FL(C). This will not be apolynomial, but it should be a rational function.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 33 / 36
Braiding functors Checking properties
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∅ ∼= D(Vect)→ V∅ ∼= D(Vect). SoFL(C) is a complex of vector spaces (actually graded vector spaces).
TheoremThe cohomology of FL(C) is a knot invariant, and finite-dimensional in eachhomological and each graded degree. The graded Euler characteristic of thiscomplex is JV,L(q).
As usual, we can take a generating series of FL(C). This will not be apolynomial, but it should be a rational function.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 33 / 36
Braiding functors Checking properties
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)
Replace with injectiveresolution B6
A7 = RHom(K1,2V ,B6) Knot homology!
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 34 / 36
Braiding functors Checking properties
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)
Replace with injectiveresolution B6
A7 = RHom(K1,2V ,B6) Knot homology!
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 34 / 36
Braiding functors Checking properties
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)
Replace with injectiveresolution B6
A7 = RHom(K1,2V ,B6) Knot homology!
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 34 / 36
Braiding functors Checking properties
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)
Replace with injectiveresolution B6
A7 = RHom(K1,2V ,B6) Knot homology!
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 34 / 36
Braiding functors Checking properties
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)
Replace with injectiveresolution B6
A7 = RHom(K1,2V ,B6) Knot homology!
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 34 / 36
Braiding functors Checking properties
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)
Replace with injectiveresolution B6
A7 = RHom(K1,2V ,B6) Knot homology!
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 34 / 36
Braiding functors Checking properties
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)
Replace with injectiveresolution B6
A7 = RHom(K1,2V ,B6) Knot homology!
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 34 / 36
Braiding functors Checking properties
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)
Replace with injectiveresolution B6
A7 = RHom(K1,2V ,B6) Knot homology!
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 34 / 36
Braiding functors Checking properties
4d TQFT
One of the inspirations for studying categorifications is the connectionsbetween 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 Uq(g)representation to a circle and building the 2-and 3-dimensional layers fromthat.
Can one make a 4-dimensional TQFT of some kind out the category of2-representations of this categorified quantum group?
Gukov and other physicists have done work on this, but as far as I know,nothing mathematically rigorous has appeared.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 35 / 36
Braiding functors Checking properties
4d TQFT
One of the inspirations for studying categorifications is the connectionsbetween 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 Uq(g)representation to a circle and building the 2-and 3-dimensional layers fromthat.
Can one make a 4-dimensional TQFT of some kind out the category of2-representations of this categorified quantum group?
Gukov and other physicists have done work on this, but as far as I know,nothing mathematically rigorous has appeared.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 35 / 36
Braiding functors Checking properties
Next time
Next time I’ll talk about how to relate this construction to the other ones I’vementioned, especially those of Khovanov-Rozansky and Cautis-Kamnitzer.
Doing that will also require some discussion of connections to the geometryof quiver varieties.
Ben Webster (UVA) HRT : Lecture VIII April 8, 2014 36 / 36