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Weyl Groups and Artin GroupsAssociated to Weighted Projective Lines

(joint work with Yuuki Shiraishi and Kentaro Wada)

Atsushi TAKAHASHI

OSAKA UNIVERSITY

November 15, 2013 at NAGOYA

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Aim

Want to understand interesting correspondences among

1. Complex (Algebraic) Geometry.

2. Symplectic Geometry.

3. Representation Theory.

McKay correspondence, strange duality, ...

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Three kinds of triangulated categories from different origins:

1. Derived category of coherent sheaves on an algebraic stack.

2. Derived category of Fukaya category (of Lagrangiansubmanifolds).

3. Derived category of finite dimesional modules over a finitedimensional algebra.

Equivalences among them = Homological Mirror Symmetry

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On the other hand,

there are three different constructions of Frobenius structures:

1. Gromov–Witten theory.

2. Deformation theory.

3. Invariant theory of Weyl groups.

Isomorphisms among these = Classical Mirror Symmetry

Frobenius structure: a “flat family” of commutative Frobeniusalgebras over a complex manifold

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These Mirror Symmetries should be relatedvia space of Bridgeland’s stability conditions.

In order to make this idea some precise statements,we first study basic properties of Weyl groups and Artin groupsassociated to weighted projective lines.

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Weighted Projective Lines

A = (a1, . . . , ar ): a tuple of positive integers (r ≥ 3).Set

µA := 2 +r∑

k=1

(ar − 1) , χA := 2 +r∑

k=1

(1

ar− 1

).

Λ = (λ4, . . . , λr ): a tuple of pairwise distinct points onP1(C) \ {∞, 0, 1}.

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Set RA,Λ := C[X1, . . . ,Xr ] /IΛ where IΛ is an ideal generated by

X aii − X a2

2 + λiXa11 , i = 3, . . . , r (λ3 := 1).

Denote by LA an abelian group defined as the quotient

LA :=

(r⊕

i=1

ZXi

)/(ai Xi − aj Xj ; 1 ≤ i < j ≤ r

).

Definition 1Let r , A and Λ be as above. Define a stack P1

A,Λ by

P1A,Λ := [(Spec(RA,Λ)\{0}) /Spec(CLA)] ,

which is called the weighted projective line of type (A,Λ).

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Theorem 2 (Geigle–Lenzing ’87)

The category Dbcoh(P1A,Λ) admits a full strongly exceptional

collection. In particular, there are triangulated equivalencesDbcoh(P1

A,Λ)∼= Db(CCA,Λ) ∼= Db(CTA,Λ) where CA,Λ is the

Ringel’s canonical algebra of type (A,Λ) and TA,Λ is a boundquiver in the next slide.

Theorem 3 (T ’08)

Assume that A = (a1, a2, 2). We have a triangulated equivalenceDbFuk→(fA) ∼= Db(CCA,Λ) ∼= Db(CTA), wherefA := xa11 + xa22 + x23 − cx1x2x3 for some c ∈ C \ {0}.

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The bound quiver TA,Λ

• bb

EEEE

EEEE

EEE

��������������

YY

3333

3333

3333

331∗

• oo

(1,a1−1)

· · · oo • oo

<<yyyyyyyyy

(1.1)

•

||yyyyyyyyy

//1

"EEE

EEEE

EE • //

(r ,1)

· · · // •(r ,ar−1)

•

xxxxxxxxx (2,1)

•

FFFF

FFFF

F(r−1,1)

. . .

}}zzzzzzzz

. . .

""DDD

DDDD

D

•(2,a2−1)

. . . . . . . . . •(r−1,ar−1−1)

There are two relations from 1 to 1∗ depending on Λ.

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. . . . . .

The star quiver TA

• oo

(1,a1−1)

· · · oo • oo

(1.1)

•

||yyyyyyyyy

//1

"EEE

EEEE

EE • //

(r ,1)

· · · // •(r ,ar−1)

•

xxxxxxxxx (2,1)

•

FFFF

FFFF

F(r−1,1)

. . .

}}zzzzzzzz

. . .

""DDD

DDDD

D

•(2,a2−1)

. . . . . . . . . •(r−1,ar−1−1)

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Root lattice

Denote by K0(TA,Λ) the Grothendieck group of Db(CTA,Λ);

K0(TA,Λ) =⊕

v∈TA,Λ

Zαv .

The Euler form χ : K0(TA,Λ)× K0(TA,Λ) −→ Z is defined by

χ([M], [N ]) :=∑p∈Z

(−1)p dimCHomDb(CTA,Λ)

(M,N [p]).

The Cartan form I : K0(TA,Λ)× K0(TA,Λ) −→ Z is defined by

I (γ, γ′) := χ(γ, γ′) + χ(γ′, γ).

Define similarly for TA.

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Coxeter–Dynkin diagram TA

I (αv , αv ′) = 0 ⇐⇒ ◦v ◦v ′

I (αv , αv ′) = −1 ⇐⇒ ◦v ◦v ′

I (αv , αv ′) = +2 ⇐⇒ ◦v ◦v ′

•

DDDD

DDDD

��������������

22222222222222

1∗

•

(1,a1−1)

· · · •

zzzzzzzz

(1.1)

•

zzzzzzzz 1

DDDD

DDDD

•

(r,1)

· · · •

(r,ar−1)

•

yyyyyyyyy (2,1)

•

EEEE

EEEE

E(r−1,1)

. . .

{{{{{{{{

. . .

CCCC

CCCC

•(2,a2−1)

. . . . . . . . . •(r−1,ar−1−1)

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Coxeter–Dynkin diagram TA

I (αv , αv ′) = 0 ⇐⇒ ◦v ◦v ′

I (αv , αv ′) = −1 ⇐⇒ ◦v ◦v ′

•

(1,a1−1)

· · · •

(1.1)

•

zzzzzzzz 1

DDDD

DDDD

•

(r,1)

· · · •

(r,ar−1)

•

yyyyyyyyy (2,1)

•

EEEE

EEEE

E(r−1,1)

. . .

{{{{{{{{

. . .

CCCC

CCCC

•(2,a2−1)

. . . . . . . . . •(r−1,ar−1−1)

χA = 0 ⇐⇒ the Cartan form I on K0(TA) is non-degenerate.

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Weyl groups

Define the simple reflection rv on K0(TA,Λ)Q by

rv (λ) := λ− I (λ, αv )αv , λ ∈ K0(TA,Λ)Q.

Definition 4The subgroup of GL(K0(TA,Λ)Q) generated by simple reflections is

called the Weyl group associated to TA,Λ and is denoted by W (TA)

(since it depends only on the Coxeter–Dynkin diagram TA).

Remark 5Actually, W (TA) depends only on the (generalized) root systemassociated to Db(CTA,Λ) (∼= Dbcoh(P1

A,Λ)).

Define similarly the Weyl group W (TA) associated to TA.

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Definition 6

1. If χA = 0, then W (TA) is called the elliptic Weyl group.

2. If χA < 0, then W (TA) is called the cuspidal Weyl group.

If χA > 0, then W (TA) is isomorphic to an affine Weyl group.Indeed, we have the following:

Theorem 7 (Shiraishi–T–Wada, in preparation)

There is a split-exact sequence

{1} −→ K0(TA)/rad(I )t−→ W (TA)

p−→ W (TA) −→ {1},

where t and p are defined by

t(αv )(λ) := tv (λ) := λ− I (λ, αv )(α1∗ − α1),

p(r1) = p(r1∗) = r1, p(rv ) = rv , v ∈ TA.

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Remark 8The element α1∗ − α1 ∈ K0(TA,Λ) belongs to the radical of theCartan form I .

Lemma 9 (Key Lemma)

1. We have t1 = r1r1∗ , t(i ,1) = r(i ,1)t1r(i ,1)t−11 and

t(i ,j) = r(i ,j)t(i ,j−1)r(i ,j)t−1(i ,j−1).

2. The elements tv , v ∈ TA commute with each other.

3. Let N be the smallest normal subgroup of W (TA) containingt1. We have N = Ker(p) and t(i ,j) ∈ N for all i , j .

If χA = 0, then the group W (TA)⋉ K0(TA) is a central extensionof the elliptic Weyl group W (TA), which is called the hyperbolicextension of W (TA) (Saito–Takebayashi).

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Weyl groups as generalized Coxeter groups

Generalizing a result by Saito–Takebayashi for χA = 0, we have

Theorem 10 (STW)

Let W ′(TA) be a group described by the generators {wv | v ∈ TA}and the generalized Coxeter relations. Then we haveW ′(TA) ∼= W (TA)⋉ K0(TA).

w2v = 1 for all v ∈ TA, (W0)

(wv wv ′)2 = 1 if I (αv , αv ′) = 0, (W1.0)

(wv wv ′)3 = 1 if I (αv , αv ′) = −1, (W1.1)

w(i ,1)u1w(i ,1)u1 = u1w(i ,1)u1w(i ,1), (W2)

w(i ,1)u(j ,1) = u(j ,1)w(i ,1), w(j ,1)u(i ,1) = u(i ,1)w(j ,1) if i = j . (W3)

where u1 := w1w1∗ and u(i ,1) := w(i ,1)u1w(i ,1)u−11 .

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The correspondence (wv , uv ) ⇐⇒ (rv , tv ) gives the isomorphism.

Remark 11 (cf. Yamada ’00)

Under W0, the relations W2 and W3 are equivalent to thoseintroduced by Saito–Takebayashi (’97) for χA = 0.

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Orbit space

From now on, we assume that χA = 0 to simplify some definitions.(χA = 0 ⇐⇒ the Cartan matrix for TA is non-degenerate.)

To TA, one can associate a Kac–Moody Lie algebra and hence onecan define a set of roots in a standard way.

Consider (the interior of) the complexified Tits cone

E(TA) := {h ∈ h(TA) | ⟨α, Im(h)⟩ > 0 for α ∈ ∆(TA)+im},

where ∆(TA)+im is the subset of ∆(TA)

+ consisting of positiveimaginary roots.

The group W (TA) ∼= W (TA)⋉ K0(TA) acts properlydiscontinuously on E(TA).

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Consider the complex manifold of dimension µA:

MA := E(TA)/W (TA)×

{C χA > 0

H χA < 0

where H is the complex upper half plane.

Conjecture 12

The space of Bridgeland’s stability conditions on Dbcoh(P1A,Λ) is

isomorphic to MA.

Conjecture 13

There is a Frobenius structure on MA isomorphic to the oneconstructed from the Gromov–Witten theory for P1

A,Λ.

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Theorem 14 (Satake–T ’08)

“Conjecture 13 for χA = 0” is true.

Theorem 15 (Ishibashi–Shiraishi–T ’12)

Conjecture 13 is true if χA > 0.

Theorem 16 (Shiraishi–T, in preparation)

Conjecture 13 is true if the “property (P)” holds.

A PDF file is available athttp://frompde.sissa.it/workshop2013/talks/16Mon/Takahashi.pdf

Remark 17One can check the “property (P)” easily for χA ≥ 0.(Saito ’90 for χA = 0 , Dubrovin–Zhang ’98 for χA > 0)

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Fundamental groups of regular orbit spaces

Set

M regA := E(TA)

reg/W (TA)×

{C χA > 0

H χA < 0

where

E(TA)reg := E(TA) \ {reflection hyperplanes}.

Want to understand the universal covering of M regA .

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The universal covering of M regA should be the space of Bridgeland’s

stability conditions on some triangulated category associated tothe derived preprojective algebra (2-CY completion) of CTA,Λ.

The fundamental group of M regA should determine the

autoequivalence group of the derived category.

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Artin groups

Definition 18 (cf. Yamada ’00 when χA = 0)

The Artin group G (TA) is a group defined by the generators{gv | v ∈ TA} and relations:

gv gv ′ = gv ′ gv if I (αv , αv ′) = 0, (A1.0)

gv gv ′ gv = gv ′ gv gv ′ if I (αv , αv ′) = −1, (A1.1)

g(i ,1)s1g(i ,1)s1 = s1g(i ,1)s1g(i ,1), (A2)

g(i ,1)s(j ,1) = s(j ,1)g(i ,1), g(j ,1)s(i ,1) = s(i ,1)g(j ,1) if i = j . (A3)

where s1 := g1g1∗ and s(i ,1) := g(i ,1)s1g(i ,1)s−11 .

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Proposition 19

The correspondence gv 7→ wv for v ∈ TA induces an isomorphism

G (TA)/⟨g2

v | v ∈ TA⟩ ∼= W (TA).

Remark 20The elements s(i ,j) defined inductively by

s(i ,j) := g(i ,j)s(i ,j−1)g(i ,j)s−1(i ,j−1),

are mapped to t(i ,j). (tv (λ) := λ− I (λ, αv )(α1∗ − α1))

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Artin groups as fundamental groups

Generalizing Yamada’s result for χA = 0, we have

Theorem 21 (STW)

If χA = 0, then there is an isomorphism

G (TA) ∼= π1

(M reg

A , ∗)∼= π1

(E(TA)

reg/W (TA), ∗).

Key: Van der Lek’s description of π1(E(TA)

reg/W (TA), ∗).

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Lemma 22 (Van der Lek ’83)

The group π1

(E(TA)

reg/W (TA), ∗)can be described by the

generators {gv , sv | v ∈ TA} and the following relations:

gvgv ′ = gv ′gv if I (αv , αv ′) = 0, (A’1.0)

gvgv ′gv = gv ′gvgv ′ if I (αv , αv ′) = −1, (A’1.1)

gv sv ′ = sv ′gv if I (αv , αv ′) = 0, (A’2)

gv sv ′gv = sv ′sv if I (αv , αv ′) = −1. (A’3)

Roughly speaking, the correspondence (gv , sv ) ⇐⇒ (gv , sv ) gives

the isomorphism G (TA) ∼= π1

(E(TA)

reg/W (TA), ∗).

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Thank you very much!

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Happy 65th birthday Prof. Yamagata!

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