Arithmetic Hall algebras Maxim Kontsevich I.H.E.S. 7 September - PowerPoint PPT Presentation

Arithmetic Hall algebras Maxim Kontsevich I.H.E.S. 7 September 2019 1/19

Hall algebras and dilogarithm identities q = p r : power of a prime , F q : finite field , Q : finite quiver Gives an abelian category A with finite sets of morphisms and extensions. Definition : Hall algebra of A is a free Z -module spanned by isomorphism classes of objects in A , with associative product given by � c E [ E 1 ] · [ E 2 ] = E 1 , E 2 [ E ] E c E E 1 , E 2 := number of subobjects F ⊂ E such that [ F ] = [ E 1 ] , [ E / F ] = [ E 2 ] Example Q = • , A = category of vector spaces. Set of isomorphism classes = Z ≥ 0 = { 0 , 1 , 2 , . . . } . Structure constants are q -binomial coefficients n = # Gr ( n 1 , n 1 + n 2 )( F q ) = [ n 1 + n 2 ] q ! c n 1 + n 2 � ( 1 + q + · · · + q k ‘ − 1 ) [ n 1 ] q ![ n 2 ] q ! , [ n ] q ! := n 1 , n 2 k = 1 2/19

Why people like Hall algebras? s ❄ ✲ ✲ ✛ ✲ ✲ Theorem (C.Ringel) If Q is Dynkin diagram then s s s s s s Hall algebra = U q n + Q i.e. quantum deformation of universal enveloping algebra of the upper-triangular part n + Q of the corresponding semi-simple Lie algebra g Q . Generalization (J.A.Green) For general acyclic quiver, certain subalgebra of its Hall algebra is isomorphic to U q n + for the corresponding Kac-Moody algebra. = ⇒ HUGE industry in representation theory. Non-Dynkin quivers: ∞ many indecomposable representations, Hall algebra is too big. 3/19

More recent development: interaction with Bridgeland stability , giving multiplicative identities in completed Hall algebras, and then in quantum tori (as certain quotient algebras of Hall algebras) Definition Central charge for quiver Q is a collection of complex numbers z v ∈ Upper half-plane := { z ∈ C | , ℑ z > 0 } , ∀ v ∈ Vertices ( Q ) For a representation E � = 0 its argument arg E ∈ ( 0 o , 180 o ) (or better ( 0 , π ) ) is the argument of non-zero complex number Z ( E ) := Z ( − → � dim ( E )) = z v · dim E v v ∈ Vertices ( E ) Representation E � = 0 is θ - semistable if ∀F � E , F � = 0 arg E = θ and arg F ≤ θ 4/19

Every representation E has canonical Harder-Narasimhan filtration 0 = E 0 � E 1 � · · · � E m = E , m ≥ 0 arg E 1 / E 0 > arg E 2 / E 1 > · · · > arg E m / E m − 1 E 1 / E 0 , E 2 / E 1 , . . . are all semistable Z ( E 2 / E 1 ) s ❑ ❆ · · · · · · ❆ Z ( E 1 / E 0 ) ❆ Z ( E m / E m − 1 ) ✐ P P ❆ s ✘✘✘✘✘✘✘✘ ✿ s P P P ❆ P P P ❆ 0 5/19

Existence and uniquenes of Harder-Narasimhan filtration ⇐ ⇒ Product formula : � � A Hall θ A Hall = Q Q , Z ,θ where ◮ A Hall := 1 + · · · = � [ E ] [ E ] , the formal sum of all objects, ◮ A Hall Q , Z ,θ = 1 + � θ − semistable [ E ] [ E ] ◮ the product is in the clockwise order, the l.h.s. does not depend on the choice of central charge. The product formula defines uniquely the r.h.s. from A Hall and Z . Q All this is quite abstract, we need to map Hall algebra to something more manageable. 6/19

χ : Z I ⊗ Z I → Z , � � χ ( d ′ , d ′′ ) := d ′ v d ′′ d ′ v d ′′ Euler form : v − u arrows vertices v → u v Meaning : if d ′ , d ′′ are dimension vectors of E ′ , E ′′ then χ ( d ′ , d ′′ ) = dim Hom ( E ′ , E ′′ ) − dim Ext ( E ′ , E ′′ ) Define quantum torus (associated with quiver Q ) as associative algebra over Q with linear basis { e d } where d ∈ Z I ≥ 0 (the set of all possible dimension vectors), with multiplication given by e d ′ · e d ′′ = q − χ ( d ′′ , d ′ ) e d ′ + d ′′ , e 0 = 1 Now we have a Homomorphism: e − → dim E Hall algebra → Quantum Torus : [ E ] �→ # Aut ( E ) 7/19

Apply this homomorphism the product formula:   1 � � � A Q := Image of [ E ] =  · e d =   # Aut ( E )  − → [ E ] d ∈ Z I ≥ 0 [ E ]: dim E = d � v → u d v d u � v → u d v d u q q � � = v # GL ( d v , F q ) e d = e d � dv ( dv − 1 ) ( q − 1 ) d v [ d v ] q ! � v q 2 d ∈ Z I d ∈ Z I ≥ 0 ≥ 0 Define for any angle θ ∈ ( 0 , π ) generating series for θ -semistable representations (plus trivial one): 1 A Q , Z ,θ := Image of A Hall � = 1 + # Aut ( E ) · e − → Q dim E [ E ]: θ − semistable � Product Formula in quantum torus : A Q = � θ A Q , Z ,θ The product is in the clockwise order, the l.h.s. does not depend on the choice of central charge. This decomposition defines uniquely the r.h.s. from A Q and Z . 8/19

1 Example Dynkin quiver for sl 2 : • The generating series is the quantum exponent: n ( n − 1 ) ( e 1 / ( q − 1 )) n q 2 � � # GL ( n , F q ) e n =: E q ( e 1 ) 1 = [ n ] q ! n ≥ 0 n ≥ 0 9/19

1 2 Example : Dynkin quiver for sl 3 : •− → • Indecomposable representations R 01 , R 11 , R 10 : id → F q F q → F q F q − 0 − − → 0 Short exact sequence: 0 → R 01 → R 11 → R 10 → 0. Case 1 : arg z 1 ≥ arg z 2 : all three R 01 , R 11 , R 10 are semistable, Case 2 : arg z 1 < arg z 2 : only R 01 and R 10 are semistable. z 1 + z 2 s ✻ s s s s z 1 z 2 z 2 z 1 ■ ❅ ✒ � ❅ ■ ✒ � ❅ � ❅ � ❅ � ❅ � 0 0 10/19

1 2 Example : Dynkin quiver for sl 3 : •− → • (continuation) We get two product decompositions for A Q , hence an identity E q ( e 1 ) · E q ( e 1 e 2 ) · E q ( e 2 ) = E q ( e 2 ) · E q ( e 1 ) , e 2 · e 1 = q e 1 · e 2 This is called quantum dilogarithm identity, as in the formal limit q → 1 it converges to the classical 5-term identity for the dilogarithm x n � Li 2 ( x ) := n 2 n ≥ 1 11/19

From quivers to curves Representations of quivers over a finite field form an abelian category of cohomological dimension 1 , similar to the category of coherent sheaves on a smooth compact algebraic curve C over F q . The central charge in this case is a homomorphism of abelian groups √ Z : K 0 ( Coh C ) → C , Z ([ E ]) := − deg E + − 1 · rk E Semistable objects are 1) torsion sheaves, 2) usual semistable vector bundles. Example : charges of some semistable objects for C = P 1 : O ( 2 ) O ( 1 ) O O ( − 1 ) O ( − 2 ) M.Kapranov, O.Schiffmann and s s s s s E.Vasserot studied Hall algebras i − 2 i − 1 i + 1 i + 2 i for Coh ( C ) , the structure is controlled O / m 2 O / m x x by cuspidal automorphic forms. s s ❜ − 2 − 1 0 12/19

From curves over finite field to number fields Category Coh ( C ) is similar to proto-abelian category of Arakelov coherent sheaves for a number field case. In the case of � Spec Z we get pairs (Γ , h ) where Γ is a finitely generated abelian group (=usual coherent sheaf on Spec Z ) and h is a positive quadratic form on real vector space Γ R := Γ ⊗ R (=extension to the archimedean infinity). Γ is without torsion ⇐ ⇒ an Euclidean lattice. Central charge is given by √ � � Z (Γ , h ) := − log #Γ tors + log vol (Γ R / Γ) + − 1 · r k Γ Possible values of central charge of semistable objects are √ − 1 · y ∈ C | x ∈ R , y ∈ Z > 0 } {− log 2 , − log 3 , . . . } ⊔ { x + 13/19

(Oriented) semstable euclidean lattices of covolume = 1 and of rank n ≥ 1 are those for which covolume if any sublattice is ≥ 1, they form a closed real-algebraic subset of SL ( n , Z ) \ SL ( n , R ) / SO ( n , R ) . The case of covolume � = 1 reduces to the case of covolume 1 by the action of 1-parameter group of automorphisms of the category of Arakelov coherent sheaves: (Γ , h ) �→ (Γ , e t · h ) , t ∈ R ≃ Pic ( � Spec Z ) Analog of countings of representations and of semistable representations: calculations of volumes of SL ( n , Z ) \ SL ( n , R ) / SO ( n , R ) and its semistable part: V ( n ) = vol ( SL ( n , Z ) \ SL ( n , R )) ζ ( 2 ) ζ ( 3 ) . . . ζ ( n ) vol ( S 0 ) . . . vol ( S n − 1 ) ≥ V ss ( n ) > 0 = 2 vol ( SO ( n , R )) 14/19

Relation between ( V ( n )) n ≥ 1 and ( V ss ( n )) n ≥ 1 One can write a recursive formula which in principle allows to write numbers ( V ss ( n )) n ≥ 1 as complicated polynomial expressions in explicitly known volumes ( V ( n )) n ≥ 1 (L.Weng). Alternatively, one can use quantum torus with continuous grading by Z × R , and after some manipulations obtain a functional relation. Introduce generating series: V ( n ) t n ∈ t R > 0 [[ t ]] , V ss ( n ) t n ∈ t R > 0 [[ t ]] � F ss ( t ) := � F ( t ) := n ≥ 1 n ≥ 1 Theorem ∃ ! series Φ( x , y ) ∈ y R [[ x , y ]] such that � � x ∂ x ∂ ∂ x + y ∂ Φ + x ∂ ∂ x ( F ss ( x )Φ) = 0 ◮ ∂ x ∂ y ◮ Φ( t , t ) = F ss ( t ) ◮ Φ( 0 , t ) = F ( t ) 15/19

From functions to constructible sheaves Q : finite quiver, k : any field (not necessarily finite). For any dimension vector d ∈ Z I ≥ 0 ( I =set of vertices) the stack of representations of dimension d is a smooth Artin stack M d over k , of virtual dimension = − χ ( d , d ) . For any d ′ , d ′′ consider stack M d ′ , d ′′ parametrizing short exact sequences dim E ′ = d , − − → → 0 → E ′ → E → E ′′ → 0 , dim E ′′ = d ′′ We have universal diagram M d ′ , d ′′ π 1 π 2 � ❅ � ✠ ❘ ❅ M d ′ × M d ′′ M d ′ + d ′′ If k is finite field, Hall algebra product is ( π 2 ) ∗ ◦ π ∗ 1 for functions on k -points. For general fields: functor ( π 2 ) ! ◦ π ∗ 1 = ( π 2 ) ∗ ◦ π ∗ 1 on l -adic constructible sheaves. 16/19