Hall algebras and Fukaya categories

Transcription

1 Hall algebras and Fukaya categories Peter Samuelson July 4, 2017 Geometric Representation Theory, Glasgow (based on work with H. Morton and work in progress with B. Cooper) Peter Samuelson Hall algebras and Fukaya categories July 4, / 13

2 Outline Plan: The Hall algebra Motivation: the Hall algebra of an elliptic curve The Fukaya category of a surface Some computations in Hall(Fuk(S)) Peter Samuelson Hall algebras and Fukaya categories July 4, / 13

3 Hall algebras An abelian category A is finitary if Hom, Ext 1 are always finite sets. Ex: A = kq fdmod, with k a finite field, Q a quiver w/o cycles. Hall algebra: Ha(A) = Q{iso classes in A}, with product [M][N] := [E] c E M,N [E], ce M,N := # { E E E = M E/E = N } Rmk: Can rephrase as a convolution product using Ob(A) Ob(A) Exact(A) Ob(A) Thm: [Ringel, special case] If Q is the A n quiver, then Ha(A) = U q (sl n+1 ) +, where q = #k and simple E i. Thm: [Toën, Xiao-Xu] There exists a definition of the Hall algebra DHa(C) for a (finitary) triangulated category C. Peter Samuelson Hall algebras and Fukaya categories July 4, / 13

4 The elliptic Hall algebra, I X a smooth elliptic curve over F p, E X the Hall algebra of Coh(X ), E q,t is the elliptic Hall algebra of Burban and Schiffmann Theorem (Burban, Schiffmann 12) For any X, there exist σ, σ C such that E + q=σ,t= σ is isomorphic to E X. E q,t is generated by u x for x Z 2, modulo { 0 if x, y collinear [u y, u x ] = ±θ x+y /α 1 if ** holds (**): x primitive, the triangle 0, x, x + y has no interior lattice points. θ kx0 s k := exp α i u ix0 s i, α k := (1 q k )(1 t k )(1 (qt) k ) k i 1 k 0 Peter Samuelson Hall algebras and Fukaya categories July 4, / 13

5 The elliptic Hall algebra, II E q,t has been related to many objects by many people: Equivariant K-theory of Hilbert schemes, [Schiffmann-Vasserot], [Feigin-Tsymbaliuk], [Negut],... Positivity conjectures in algebraic combinatorics, [Bergeron-Garsia-Leven-Xin], [Mellit], [Haiman],... m/n interpolation between Macdonald/Schur symmetric functions, [Gorsky-Negut], [Maulik-Okunkov] triply graded homology of torus knots, [Gorsky-Negut], [G-Oblomkov-Rasmussen-Shende], [Mellit], [Elias-Hogancamp], [G-N-R], [O-Rozansky]... the q-heisenberg category [Cautis-Lauda-Licata-S-Sussan] the skein algebra of the torus, [Morton-S] First case of homological mirror symmetry: if X an elliptic curve/c, then Question: What is the Ha(Fuk(S))? D b (Coh(X )) = Fukaya(T 2 ) Peter Samuelson Hall algebras and Fukaya categories July 4, / 13

6 The Homflypt skein algebra of the torus The Homflypt skein module of a 3-manifold M is Sk q (M) = C(q){oriented links in M} modulo the skein relations: = (q q 1 ) Rmk: Sk q (F [0, 1]) is an algebra, by stacking (where F is a surface). [P 1,1, P 0,1 ] = = (q q 1 ) = P 1,2 Theorem (Morton, S) There exist generators {P x x Z 2 } of Sk q (T 2 ), with relations [P x, P y ] = (q d q d )P x+y d = det[x y] Corollary Sk q (T 2 ) is isomorphic to E q,q, the elliptic Hall algebra at t = q. Peter Samuelson Hall algebras and Fukaya categories July 4, / 13

7 Mirror symmetry The iso Sk q (T 2 ) = E q,q seems roughly analogous to Kontsevich s homological mirror symmetry for an elliptic curve: X a complex variety, Y its mirror (a symplectic manifold) D b (Coh(X )) = Fukaya(Y ) Example: [Polishchuk, Zaslow] X an elliptic curve (over C), Y the topological torus (with a symplectic form), Ob(Fuk(Y )) are lines decorated by a rep of U(n), A vector bundle E X maps to a line of slope slope(e). Question: Does this analogy make any sense? Is Ha(Fuk(S)) related to the algebra Sk q (S)? Peter Samuelson Hall algebras and Fukaya categories July 4, / 13

8 Fuk(S), following [Haiden, Katzarkov, Kontsevich] A grading on S is a foliation η Γ(P(TS)), A grading on a curve c(t) is (for each t) a path from η to c (t) The intersection index of α, β (intersecting at p) is ι p (α, β) Z, η α (t) β (t) η, (middle is shortest counterclockwise path ) (S, M) a marked surface has its boundary components divided into marked / unmarked intervals M i, U i (alternating), M = M i, An arc is an embedded graded curve in S with endpoints in M, An arc system A is a set of disjoint, nonisotopic arcs. A boundary path is a curve in M between arcs, with S to the right. The Fukaya category of (S, M, η, A) is an A category with Objects: arcs in the arc system A Morphisms: boundary paths between arcs in A, (and identity) composition is concatenation of boundary paths, or 0 if undefined Higher compositions µ i determined by discs Peter Samuelson Hall algebras and Fukaya categories July 4, / 13

9 Fuk(S), continued Let D π (Fuk(S, M, A)) be the homotopy category of twisted complexes over Fuk(S, M, A). This is triangulated. Rough summary: D π (Fuk(S, M)): Objects are arcs, morphisms are intersection points, composition comes from disks. Example: The disk M is n intervals M i, A is n 1 arcs E i, with E i from M i 1 to M i, (suitably graded), For this A, higher compositions vanish, Fuk an honest category. functor Fuk(D n, M, A) A n 1 quiver mod, E i simple[i] D π (Fuk(D n, M, A)) D b (A n 1 ) is an equivalence, DHa(D b (A n 1 )) computed explicitly in [Hernandez, Leclerc] Peter Samuelson Hall algebras and Fukaya categories July 4, / 13

10 DHa(D b (A n )) Theorem (Hernandez-Leclerc) The derived Hall algebra DHa(Q) is generated by the elements z i,m, for i I = {1,, n} and m Z, subject to the relations z i,m z j,m z j,m z i,m = 0 ifi j = 0, z 2 i,mz j,m (q + q 1 )z i,m z j,m z i,m + z j,m z 2 i,m = 0 z i,m z j,m+1 q i j z j,m+1 z i,m = δ ij q 2 1 if i j = 1 q 1 z i,m z j,m+k q ( 1)k i j z j,m+k z i,m = 0 for k > 1 Peter Samuelson Hall algebras and Fukaya categories July 4, / 13

11 Skein relations in DHa(Fuk(S)) Rmk: If ι p (X, Y ) = 0, 1, one resolution X # p Y has a canonical grading. Suppose: α and β are simple graded curves in (S, M) α and β intersect once (transversally) at p the endpoints of α and β lie on four distinct interval components of M Theorem (Cooper, S) The following relation holds in the Hall algebra of Fuk(S, M): { ±(q q [α, β] = 1 ) α # p β if ι p (X, Y ) {0, 1} 0 otherwise This is a graded analogue of the HOMFLY skein relation. Proof sketch: Compute explicitly for S the disk with 4 marked points, using [HL]. Assumptions imply Fuk(D, 4) Fuk(S, M) is fully faithful, and therefore induces a map between Hall algebras. Peter Samuelson Hall algebras and Fukaya categories July 4, / 13

12 Concrete corollary Let U + = U q (sl n ) +, generated by E i with 1 i < n Define E i,j := [E i+1,j, E i ] q for i < j, where E i,i+1 := E i. Brundan showed these are Lusztig s PBW generators. Corollary The E i,j map to arcs in Fuk(D, n) and satisfy [E a,c, E b,d ] = (q q 1 )E a,d E b,c [E a,b, E b,c ] q = E a,c [E a,c, E b,c ] q = [E a,c, E a,b ] q = 0 [E a,b, E c,d ] = [E a,d, E b,c ] = 0 for a < b < c < d. This is a PBW-presentation of U + (i.e. unordered words in E i,j are a basis). (Relations were known, e.g. to Rosso.) Any decomposition {1,, n} = {1,, a 1 }... {a k,, n} produces an embedding U q (sl k ) + U q (sl n ) + Peter Samuelson Hall algebras and Fukaya categories July 4, / 13

13 Thanks Thanks for listening! Peter Samuelson Hall algebras and Fukaya categories July 4, / 13