K-theoretic Hall algebras for quivers with

potential

Tudor Padurariu

MIT

June 23, 2020

Table of Contents

Quivers

Hall algebras

K-theoretic Hall algebras for quivers with potential

Definition of quivers.

A quiver is a directed graph.Q = (V ,E , s, t : E ! V )Examples: Jordan quiver

For k a field, the path algebra kQ is the k-vector space with basispaths in Q and multiplication given by concatenation of paths.Example: kJ = k[x ].

Representations of quivers.

Let d 2 NV be a dimension vector. Representations of the path

algebra can be described as follows:

The stack of representations of Q of dimension d is

X (d) =Y

e2EAds(e)dt(e)/

Y

v2VGL(dv ).

Table of Contents

Quivers

Hall algebras

K-theoretic Hall algebras for quivers with potential

Classical Hall algebras

(Steinitz ⇠ 1905, Hall, Ringel ⇠ 1990)Let Q be a quiver, k a finite field with q elements, v = q

1/2.

The Hall algebra HQ is the C-vector space with basis isomorphismclasses of Q-reps over k and multiplication encoding extensions ofsuch representations.

Consider the subalgebra HsphQ ⇢ HQ generated by dimension

vectors (0, · · · , 0, 1, 0, · · · , 0).

Theorem (Ringel-Green)

Let Q be a quiver with no loops with corresponding Kac-Moody

algebra g. ThenH

sphQ = U

>v (g).

From classical Hall algebras to preprojective Hall algebras

The algebra HsphQ can be categorified (Lusztig) by a category C(d)

of constructible sheaves on the stacks X (d). Multiplication isdefined by m = p⇤q

⇤, where

X (d)⇥ X (e)q � X (d , e)

p�! X (d + e)

(A,B/A) (Ad⇢ B

d+e)! B .

Consider the singular support functor:

C(d)! DbC⇤Coh(T ⇤

X (d)).

Grojnowski proposed the study of algebras defined usingD

bC⇤Coh(T ⇤

X (d)).

Preprojective/ 2d Hall algebras

(Grojnowski 1995, Schi↵mann-Vasserot 2009, Yang-Zhao 2014)For a quiver Q, define

KHA(Q) =M

d2NI

KC⇤

0 (T ⇤X (d)).

Example: for J the Jordan quiver, T ⇤X (d) is the stack of sheaves

on A2 with support dimension 0 and length d .

Conjecture

Let Q be an arbitrary quiver. Then

KHA(Q) = U>q (cgQ),

where Uq(cgQ) is the Okounkov-Smirnov quantum group.

Hall algebras for CY 3d categories.

(Kontsevich-Soibelman 2010)Conjecturally, for any Calabi-Yau 3-category one can construct aHall algebra. When the category is DbCoh(X ) for X a Calabi-Yau3-fold, the number of generators is expected to be given byenumerative invariants of X .

Locally, these algebras are expected to be modelled by quivers withpotential. In this situation, one can construct the Hall algebra.

This construction generalizes the preprojective/ 2d KHA for aquiver Q.

classical HAsa�nization������! 2d KHAs ⇢ 3d KHAs.

Table of Contents

Quivers

Hall algebras

K-theoretic Hall algebras for quivers with potential

Quivers with potential.

A potential W is a linear combination of cycles in Q.Example:

Define the Jacobi algebra

Jac(Q,W ) = CQ/

✓@W

@e, e 2 E

◆.

Example: Jac(Q3,W3) = C[x , y , z ].

Let J (d) be the stack of reps of Jac(Q,W ) of dimension d . Wehave that

J (d) = crit (TrW : X (d)! A1).

Example: T ors (A3, d) = crit (TrW3 : gl(d)3/GL(d)! A1).

Definition of KHAs for quivers with potential.

Consider the NV -graded vector space

KHA(Q,W ) =M

d2NV

K0(Dsg (X (d)0)) =M

d2NV

Kcrit(J (d)).

Here Dsg (X (d)0) = DbCoh (X (d)0)/Perf (X (d)0) is the category

of singularities of X (d)0 = (TrW )�1(0) ⇢ X (d).

Theorem (P.)

KHA(Q,W ) is an associative algebra with multiplication

m = p⇤q⇤, where X (d)⇥ X (e)

q � X (d , e)

p�! X (d + e).

Relations between 2d and 3d KHAs.

For any quiver Q, there exists a pair ( eQ, fW ) such that

KHA2d(Q) = KHA

3d( eQ, fW ).

Example: For the Jordan quiver J, the pair is (Q3,W3) and the

algebra is the positive part of Uq

✓ccgl1

◆.

Representations of KHAs

Theorem (P.)

For any pair (Q,W ), KHA has natural representations constructed

using framings of Q.

Example: KHA(Q3,W3) acts on

M

d2NI

K0(Dsg (X (1, d)ss0 )) =M

d2NI

Kcrit(Hilb (A3, d)).

Theorem (P.)

For a pair ( eQ, fW ), KHA has natural representations constructed

using Nakajima quiver varieties.

Example: KHA(Q3,W3) acts on

M

d2NK0(Hilb (A

2, d)).

PBW theorem for KHAs.

Let Q be a symmetric quiver.

Theorem (P.)

There exists a filtration F•on KHA(Q,W ) such that

gr KHA(Q,W ) = dSym (F 1).

The first piece in the filtration is

F1 =

MK0(M(d)) ⇢ KHA(Q,W )

for some natural categories M(d) ⇢ Dsg (X (d)0).

Sketch of the proof of the PBW theorem.

The proof follows from semi-orthogonal decompositions in the zeropotential case

Db(X (d)) = hp⇤q

⇤ ⇥M(di ),M(d)i,

where the summands on the left are generated by multiplicationfrom non-trivial decompositions d = d1 + · · ·+ dk .

The category M(d) is the subcategory of Db(X (d)) generated byOX (d) ⌦ V (�), where � is a dominant weight of G (d) such that

�+ ⇢ 21

2sum [0,�] ⇢ MR.

The above semi-orthogonal decomposition induces correspondingsemi-orthogonal decompositions of Dsg (X (d)0).

Thank you for your attention!