weyl groups and artin groups associated to weighted ...iyama/yamagata65/atakahashi.pdfweyl groups...
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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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