Advances in Heisenberg categorification
X − = b b∨ b∈B
Alistair Savage University of Ottawa
Slides available online: alistairsavage.ca/talks Preprint: arXiv:1802.01626
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 1 / 34 Outline
Goals: 1 Describe a family of categories that categorify the Heisenberg algebra 2 Explain the relation to previous Heisenberg categories
Overview: 1 Strict monoidal categories and string diagrams 2 The Frobenius Heisenberg category 3 Actions on categories of modules 4 Work in progress: q-deformations
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 2 / 34 Strict monoidal categories
A strict monoidal category is a category C equipped with a bifunctor (the tensor product) ⊗: C × C → C, and a unit object 1, such that, for objects A, B, C and morphisms f, g, h, (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C), 1 ⊗ A = A = A ⊗ 1, (f ⊗ g) ⊗ h = f ⊗ (g ⊗ h),
11 ⊗ f = f = f ⊗ 11.
Remark: Non-strict monoidal categories In a (not necessarily strict) monoidal category, the equalities above are replaced by isomorphism, and we impose some coherence conditions.
Every monoidal category is monoidally equivalent to a strict one.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 3 / 34 k-linear monoidal categories
Fix a commutative ground ring k.
A strict k-linear monoidal category is a strict monoidal category such that each morphism space is a k-module, composition of morphisms is k-bilinear, tensor product of morphisms is k-bilinear.
The interchange law The axioms of a strict monoidal category imply the interchange law: For f g A1 −→ A2 and B1 −→ B2, the following diagram commutes:
1⊗g A1 ⊗ B1 / A1 ⊗ B2 f⊗g f⊗1 f⊗1 & A ⊗ B / A ⊗ B 2 1 1⊗g 2 2
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 4 / 34 Categorification via split Grothendieck group
Suppose C is an additive category (i.e. have ⊕).
IsoZ(C)= free abelian group generated by isom. classes of objects in C. The split Grothendieck group of C is
K0(C) = IsoZ(C)/h[X ⊕ Y ] = [X] + [Y ] | X,Y ∈ Ci.
If C is monoidal, then K0(C) is a ring:
[X] · [Y ] = [X ⊗ Y ].
Categorification For our purposes, to categorify a ring R is to find an additive monoidal category C such that ∼ K0(C) = R as rings.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 5 / 34 The Heisenberg algebra
Let h be the infinite-dimensional Heisenberg Lie algebra.
Thus, h is the complex Lie algebra with basis
± {c, qn : n ≥ 1}
and product
+ + − − ± + − [qm, qn ] = [qm, qn ] = [c, qn ] = 0, [qm, qn ] = δm,nnc.
The associative Heisenberg algebra at central charge ξ ∈ Z is
U(h)/hc − ξi.
We will describe categories that categorify these algebras.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 6 / 34 String diagrams
Fix a strict monoidal category C. We will denote a morphism f : A → B by:
B
f
A
The identity map idA : A → A is a string with no label:
A
A
We sometimes omit the object labels when they are clear or unimportant.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 7 / 34 String diagrams
Composition is vertical stacking and tensor product is horizontal juxtaposition:
f = fg f ⊗ g = f g g
The interchange law then becomes:
f g = f g = g f
A morphism f : A1 ⊗ A2 → B1 ⊗ B2 can be depicted:
B1 B2
f
A1 A2
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 8 / 34 Presentations of strict monoidal categories
One can give presentations of some strict k-linear monoidal categories, just as for monoids, groups, algebras, etc.
Objects: If the objects are generated by some collection Ai, i ∈ I, then we have all possible tensor products of these objects:
1,Ai,Ai ⊗ Aj ⊗ Ak ⊗ A`, etc.
Morphisms: If the morphisms are generated by some collection fj, j ∈ J, then we have all possible compositions and tensor products of these morphisms (whenever these make sense):
idAi , fj ⊗ (fifk) ⊗ (f`), etc.
We then often impose some relations on these morphism spaces. String diagrams: We can build complex diagrams out of our simple generating diagrams.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 9 / 34 Monoidally generated symmetric groups
Define a strict k-linear monoidal category S with one generating object ↑ and denote id↑ =
We have one generating morphism
: ↑ ⊗ ↑→↑ ⊗ ↑ .
We impose the relations:
= , = .
Then ⊗n EndS (↑ ) = kSn is the group algebra of the symmetric group on n letters. Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 10 / 34 The degenerate affine Hecke category
Start again with the strict k-linear monoidal category S, but add a morphism: : ↑→↑ We impose the additional relation:
− = .
Now End(↑⊗n) is the degenerate affine Hecke algebra (of type A).
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 11 / 34 The wreath product category
Fix an associative k-algebra F . We add an endomorphism of ↑ for each element of F .
More precisely, let W(F ) be the strict k-linear monoidal category obtained from S by adding morphisms such that we have an algebra homomorphism:
F → End ↑, f 7→ f
We impose the additional relations:
f = , f ∈ F f
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 12 / 34 The wreath product category
⊗n ⊗n EndW(F )(↑ ) = F o Sn is a wreath product algebra.
As a vector space, ⊗n ⊗n F o Sn = F ⊗k kSn. Multiplication is determined by
⊗n (f1 ⊗ π1)(f2 ⊗ π2) = f1(π1 · f2) ⊗ π1π2, f1, f2 ∈ F , π1, π2 ∈ Sn,
⊗n where π1 · f2 denotes the natural action of Sn on F by permutation of the factors.
Note: W(k) = S, the symmetric group category.
Want: An affine version of the wreath product category. F = k should recover the degenerate affine Hecke category. Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 13 / 34 Frobenius algebras: Definition
Frobenius algebra A Frobenius algebra is a f.d. associative algebra F together with a linear trace map tr: F → k such that the induced map
F → Homk(F, k), f 7→ (g 7→ tr(gf)) , is an isomorphism.
For simplicity, we assume that the trace is symmetric:
tr(fg) = tr(gf), for all f, g ∈ F.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 14 / 34 Frobenius algebras: Examples
Example (k)
k is a Frobenius algebra with tr = idk.
Example (Matrix algebra) Any matrix algebra over a field is a Frobenius algebra with the usual trace.
Example (k[x]/(xk)) k k[x]/(x ) is a Frobenius algebra with
` tr(x ) = δ`,k−1.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 15 / 34 Frobenius algebras: Examples
Example (Group algebra) Suppose G is a finite group.
The group algebra kG is a Frobenius algebra with
tr(g) = δg,1G , g ∈ G.
Example (Zigzag algebra) Associated to every quiver is a zigzag algebra. These are Frobenius algebras.
Example (Hopf algebras) Every f.d. Hopf algebra is a Frobenius algebra.
From now on: F is a Frobenius algebra with trace tr.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 16 / 34 Frobenius algebras: dual bases
Fix a basis B of F . The dual basis is
B∨ = {b∨ | b ∈ B}
defined by ∨ tr b c = δb,c, b, c ∈ B.
It is easy to check that X b ⊗ b∨ ∈ F ⊗ F b∈B is independent of the basis B.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 17 / 34 Affine wreath product category
Start with the wreath product category W(F ), but add a morphism:
: ↑→↑
We impose the additional relations:
X − = b b∨ , f = , f ∈ F f b∈B
Call the resulting category AW(F ) the affine wreath product category. Now ⊗n EndAW(F )(↑ ) is an affine wreath product algebra.
Note: AW(k) is the degenerate affine Hecke category.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 18 / 34 Adjunction
Suppose a strict monoidal category C has two objects ↑ and ↓, with
id↑ = , id↓ = .
A morphism 1 →↓ ⊗ ↑ would have string diagram
, where = id1 .
We typically omit the dotted line and draw:
: 1 →↓ ⊗ ↑ .
Similarly, we can have : ↑ ⊗ ↓→ 1.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 19 / 34 Adjunction
We say that ↓ is right adjoint to ↑ (and ↑ is left adjoint to ↓ ) if there exist morphisms : 1 →↓ ⊗ ↑ . and : ↑ ⊗ ↓→ 1. such that = and = .
(This is analogous to the unit-counit formulation of adjunction of functors.) We say ↑ and ↓ are biadjoint if they are both left and right adjoint to each other. So we also have : 1 →↑ ⊗ ↓ and : ↓ ⊗ ↑→ 1 such that = and = .
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 20 / 34 The Frobenius Heisenberg category
Recall the affine wreath product category AW(F ). It is the strict k-linear monoidal category with: Objects: Generated by object ↑. Morphisms: Generated by : ↑ ⊗ ↑→↑ ⊗ ↑,
: ↑→↑, f : ↑→↑, f ∈ F, with relations f = , = , = f , f ∈ F, X − = b b∨ , f = f , f ∈ F. b∈B For n ∈ N, define o n = n dots.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 21 / 34 The Frobenius Heisenberg category
Fix a central charge ξ ∈ Z, ξ ≤ 0.
(Actually, we can take any ξ ∈ Z, but we choose ξ ≤ 0 for simplicity of exposition.)
To AW(F ) we add another object ↓ that is right adjoint to ↑:
= and = .
We can then define right crossings:
:= : ↑↓→↓↑ .
(We start denoting tensor product by juxtaposition: ↑↓:=↑ ⊗ ↓.)
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 22 / 34 The Frobenius Heisenberg category
We then impose the crucial inversion relation:
The following matrix of morphisms is an isomorphism in the additive envelope: " # ∨ b , 0 ≤ r ≤ −ξ − 1, b ∈ B :(↑ ⊗ ↓) ⊕ 1⊕(−ξ dim F ) →↓ ⊗ ↑ . r
More precisely, we add in some other morphisms that are the matrix components of an inverse to the above morphism.
We call the resulting category HeisF,ξ the Frobenius Heisenberg category.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 23 / 34 The Frobenius Heisenberg category
Theorem (S. 2018) There are unique morphisms
: 1 →↑↓, : ↓↑→ 1 (1)
such that the following relations hold:
k ∨ X X b = , = + −k−s−2 a∨b k,s≥0 a,b∈B a s
= δξ,0 , f r = δr,−ξ−1 tr(f) if 0 ≤ r < −ξ.
In addition HeisF,ξ can be presented equivalently by replacing the inversion relation with the existence of morphisms (1) and above relations.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 24 / 34 The Frobenius Heisenberg category
The previous theorem involves left crossings := and negatively dotted bubbles: for r ≤ −ξ, r r+1 X ∨ r+ξ−1 f := (−1) det bj−1bj i−j−ξ . i,j=1 b1,...,br−1∈B
Theorem (S. 2018)
1 The objects ↓ and ↑ are biadjoint.
2 The category HeisF,ξ is strictly pivotal (isotopy invariance for morphisms). 3 One can compute an infinite grassmannian relation, curl relations, bubble slide relations, and an alternating braid relation (omitted here).
4 Under a mild assumption on F , the category HeisF,ξ categorifies the Heisenberg algebra at central charge ξ dim F .
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 25 / 34 Heisenberg categorification and actions (ξ ≤ −1)
Suppose ξ ≤ −1.
The category HeisF,ξ acts naturally on modules for cyclotomic wreath product algebras. We have a chain of algebras
k = A0 ⊆ A1 ⊆ A2 ⊆ · · · .
Then
↑ acts by induction from An-mod to An+1-mod,
↓ acts by restriction from An-mod to An−1-mod. The morphisms (diagrams) act by certain natural transformations.
Fact that ↑ and ↓ are biadjoint corresponds to fact that induction and restriction are biadjoint.
In other words An is a Frobenius extension of An−1.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 26 / 34 Heisenberg categorification and actions (ξ = 0)
F = k case Heisk,0 is the affine oriented Brauer category of Brundan–Comes–Nash–Reynolds.
Heisk,0 acts naturally on gln(k)-mod: If V is the natural rep, then ↑ 7→ V ⊗ − ↓ 7→ V ∗ ⊗ −
General case: open problem
What does HeisF,0 act naturally on for a general Frobenius algebra F ?
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 27 / 34 Historical remarks
Original Heisenberg category (Khovanov) Morphisms were planar diagrams up to isotopy, so strictly pivotal property was part of the definition. Central charge ξ = −1 and F = k.
Frobenius modification (central charge −1) For F the zigzag algebra, defined by Cautis–Licata and studied in relation to geometry of the Hilbert scheme. General definition given in joint work with Rosso. Still have central charge ξ = −1.
Higher central charge (Mackaay–S.) Generalized to higher central charge (with F = k). Again, pivotal property part of the definition.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 28 / 34 Historical remarks
Inversion relation approach, F = k (Brundan) New approach to the definition of higher charge category (Mackaay-S.) using the inversion relation. Now, pivotal property is a consequence of the definition. Advantage: proof that category acts on modules over degenerate (cyclotomic) affine Hecke algebras is much easier. Uses a well-known Mackey-type theorem.
Current work Follows inversion relation approach of Brundan. Defines a Frobenius algebra version of higher charge category (Mackaay–S.). Defines a higher charge version of previous Frobenius Heisenberg category (Rosso–S.).
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 29 / 34 Historical remarks
Summarizing the relationship between the Heisenberg categories appearing in the literature, we have:
HeisF,ξ
ξ=−1 F =k
Rosso–S. Mackaay–S. Cautis–Licata (F = zigzag) Brundan (inversion)
F =k ξ=−1 Khovanov
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 30 / 34 Some remarks
One can actually work in a more general setting than the one described here:
1 F can be a graded Frobenius superalgebra. Then HeisF,ξ is a strict k-linear graded monoidal supercategory. 2 The trace need not be symmetric. In general, there exists a Nakayama automorphism ψ : F → F such that
¯ tr(fg) = (−1)fg¯ tr(gψ(f)) for all f, g ∈ F.
Then, for instance, f = ψ(f) , f ∈ F,
3 Above remarks mean we can take F to be the Clifford superalgebra. Then HeisF,ξ acts on modules for affine Sergeev algebras (a.k.a. degenerate affine Hecke–Clifford algebras).
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 31 / 34 Work in progress (with J. Brundan)
One can q-deform the Frobenius Heisenberg category. When F = k, this corresponds to deg. affine Hecke algebra affine Hecke algebra. Generating objects: ↑ and ↓ Generating morphisms: , , .
Relations: Fix z ∈ k.
= , = , = ,
− = z , = ,
+ inversion relation.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 32 / 34 Work in progress (with J. Brundan)
Case: ξ = 0 When ξ = 0, one obtains the affine oriented skein category, an affinization of the HOMFLY-PT skein category.
This category acts on modules for Uq(gln).
Certain closed diagrams correspond to the Casimir elements in Uq(gln).
Relation to previous constructions When ξ = −1, the category contains the previously defined q-deformed Heisenberg category (Licata–S. 2013).
Main difference between two constructions is that, in the previous q-deformed Heisenberg category, the dot was not invertible.
Case ξ 6= 0: Action on modules for cyclotomic Hecke algebras.
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 33 / 34 Work in progress (with J. Brundan)
Generally, one can again incorporate a graded Frobenius superalgebra to get a more general quantum Frobenius Heisenberg category.
When ξ 6= 0, category should act on cyclotomic quotients of quantum affine wreath product algebras. The theory of these algebras is yet to be developed.
When ξ = 0, the natural action is an open question for general F . Should be some F -deformation of Uq(gln).
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 34 / 34 Happy 60th Birthday!
Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 34 / 34