Advances in Heisenberg categorification X − = b b_ b2B 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 |-linear monoidal categories Fix a commutative ground ring |. A strict |-linear monoidal category is a strict monoidal category such that each morphism space is a |-module, composition of morphisms is |-bilinear, tensor product of morphisms is |-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 ] j X; Y 2 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 ± fc; qn : n ≥ 1g and product + + − − ± + − [qm; qn ] = [qm; qn ] = [c; qn ] = 0; [qm; qn ] = δm;nnc: The associative Heisenberg algebra at central charge ξ 2 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 |-linear monoidal categories, just as for monoids, groups, algebras, etc. Objects: If the objects are generated by some collection Ai, i 2 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 2 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 |-linear monoidal category S with one generating object " and denote id" = We have one generating morphism : " ⊗ "!" ⊗ " : We impose the relations: = ; = : Then ⊗n EndS (" ) = |Sn 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 |-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 |-algebra F . We add an endomorphism of " for each element of F . More precisely, let W(F ) be the strict |-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 2 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 ⊗| |Sn: Multiplication is determined by ⊗n (f1 ⊗ π1)(f2 ⊗ π2) = f1(π1 · f2) ⊗ π1π2; f1; f2 2 F ; π1; π2 2 Sn; ⊗n where π1 · f2 denotes the natural action of Sn on F by permutation of the factors. Note: W(|) = S, the symmetric group category. Want: An affine version of the wreath product category. F = | 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 ! | such that the induced map F ! Hom|(F; |); 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 2 F: Alistair Savage (Ottawa) Heisenberg categorification June 8, 2018 14 / 34 Frobenius algebras: Examples Example (|) | is a Frobenius algebra with tr = id|. Example (Matrix algebra) Any matrix algebra over a field is a Frobenius algebra with the usual trace. Example (|[x]=(xk)) 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 |G is a Frobenius algebra with tr(g) = δg;1G ; g 2 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_ = fb_ j b 2 Bg defined by _ tr b c = δb;c; b; c 2 B: It is easy to check that X b ⊗ b_ 2 F ⊗ F b2B 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 2 F f b2B Call the resulting category AW(F ) the affine wreath product category. Now ⊗n EndAW(F )(" ) is an affine wreath product algebra. Note: AW(|) 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 |-linear monoidal category with: Objects: Generated by object ". Morphisms: Generated by : " ⊗ "!" ⊗ "; : "!"; f : "!"; f 2 F; with relations f = ; = ; = f ; f 2 F; X − = b b_ ; f = f ; f 2 F: b2B For n 2 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 2 Z, ξ ≤ 0. (Actually, we can take any ξ 2 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 2 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.