On Weyl Groups and Artin Groups Associated to Orbifold Projective Lines

0.39 MB·

by  Yuuki Shiraishi

ON WEYL GROUPS AND ARTIN GROUPS ASSOCIATED TO
ORBIFOLD PROJECTIVE LINES
4
1 YUUKI SHIRAISHI, ATSUSHI TAKAHASHI, AND KENTARO WADA
0
2
Abstract. We associate a generalized root system in the sense of Kyoji Saito to an
n
a orbifold projective line via the derived category of finite dimensional representations of
J
a certainbound quiver algebra. We generalize results by Saito–Takebayshiand Yamada

9
forellipticWeylgroupsandellipticArtingroupstotheWeylgroupsandthefundamental
1
groups of the regular orbit spaces associated to the generalized root systems. Moreover
]
G we study the relation between this fundamental group and a certain subgroup of the
A autoequivalencegroupofatriangulatedsubcategoryofthederivedcategoryof2-Calabi–
.
h Yau completion of the bound quiver algebra.
t
a
m
[
1. Introduction

1
v
The purpose of the present paper is to associate a generalized root system in the
1
3 sense of Kyoji Saitotoanorbifoldprojective lineandtounderstand someproperties ofthe
6
4 Weyl group of the generalized root system. In particular, we want to know the description
.
1
of this Weyl group as the group with generators and relations and the connection between
0
4 the Weyl group and the fundamental group of the regular orbit space associated to the
1
: generalized root system.
v
i Saito–Takebayashi [32] introduces the generalized Coxeter relations in order to de-
X
r scribe by generators and relations an elliptic Weyl group, the Weyl group of an elliptic
a
rootsystem whichisaninvariantofthemirrordual object toanellipticorbifoldprojective
line. It isknown that thebase space of a universal unfolding of asimple elliptic singularity
is described as the orbit space associated to the corresponding elliptic root system and,
in particular, the complement of the discriminant of the base space is homeomorphic to
the regular orbit space (cf. Chapter 3 in [24]). Yamada [37] gives the generalized Coxeter
relations to the fundamental group of this regular orbit space and shows that there is a
natural surjective map from the fundamental group of it to the Weyl group.
On the other hand, one of the most important theme in mirror symmetry is to un-
derstand the relations among complex geometry, symplectic geometry and representation
theory via the isomorphisms of Frobenius structures constructed fromthese mathematical
areas. The Frobenius structure is defined by Dubrovin [9] in order to obtain the geometric
Date: January 21, 2014.
1
2 YUUKISHIRAISHI,ATSUSHITAKAHASHI,ANDKENTAROWADA
description of the WDVV equations on the topological field theory. On the other hand,
the Frobenius structure was first discovered by K. Saito as the flat structure on the base
space of a universal unfolding of an isolated hypersurface singularity in his study of a
primitive form [30] and on a complexified quotient space of a real Euclid space by the
finite reflection group on it [31, 33]. The notion of the generalized root system is intro-
duced in [30] in order to understand algebraically vanishing cycles in the Milnor fiber and
the period mapping of a primitive form. From the view point of mirror symmetry, the
generalized root system should exist on other mathematical areas. Moreover the general-
ized root system introduced in other areas should play an important role to understand
them as vanishing cycles do in the singularity theory. Let us give the plan of this paper
and then explain more precisely our motivation.
Let r ≥ 3 be a positive integer, A = (a ,...,a ) an r-tuple of positive integers
1 r
greater than one and Λ = (λ ,...,λ ) an r-tuple of pairwise distinct elements of P1(k)
1 r
normalized such that λ = ∞, λ = 0 and λ = 1. We consider an orbifold projective line
1 2 3
P1 of type (A,Λ) (see Definition 2.24), which is the smooth proper Deligne–Mumford
A,Λ
stack whose coarse moduli space is P1(k) with the orbifold points λ of orders a for
i i
i = 1,...,r.
Starting from an orbifold projective line P1 and passing through the bounded
A,Λ
derived category Dbcoh(P1 ) of coherent sheaves on P1 , we shall associate to it a gen-
A,Λ A,Λ
eralized root system R , which is independent from Λ, based on Proposition 2.9 which
A
says that a generalized root system is an invariant for a triangulated category satis-
e
fying some conditions which is expected to hold for triangulated categories of a good
algebro-geometric origin [3, 7, 22, 26, 28]. One of key facts is the triangulated equivalence
between Dbcoh(P1 ) and the bounded derived category Db(kT ) of finite dimensional
A,Λ A,Λ
representations of the bound quiver algebra kT (see Definition 2.21), which is a di-
A,Λ
e
rect consequence of the result by Geigle–Lenzing [11]. The reason why we consider the
e
bound quiver T is that we can obtain the generalized Coxeter relations introduced by
A,Λ
Saito–Takebayashi [32] and Yamada [37]. At this moment, we have not succeeded yet to
e
obtain a generalization of Coxeter relations for Ringel’s canonical algebra [27] which is
also derived equivalent to Dbcoh(P1 ).
A,Λ
The results for elliptic Weyl groups in [32] are naturally extended. We describe
the cuspidal Weyl group, the Weyl group W(R ) of R , as a semi-direct product of the
A A
Weyl group W(R ) of R and the Grothendieck group K (R ) (see Theorem 3.5 and
A A 0 A
e e
Definition 3.6), where R is a generalized root system associated to the bounded derived
A
category Db(kT ) of finite dimensional representations of the path algebra kT for the
A A
subquiver T of T (see Definition 2.19). Recall that, for elliptic cases, the hyperbolic
A A
e
ON WEYL GROUPS AND ARTIN GROUPS ASSOCIATED TO ORBIFOLD PROJECTIVE LINES 3
extension of W(R ) is isomorphic to W(R ) ⋉ K (R ). We check in Theorem 4.5 the
A A 0 A
generalized Coxeter relations in [32], a generalization of Coxeter relations for the bound
e
components in Coxeter–Dynkin diagram, is also valid for the Weyl group W(R ).
A
The results for elliptic Artin groups in [37] are also naturally extended. We define
e
a group G(T ) (see Definition 5.1) for the Coxeter–Dynkin diagram T , which is called
A A
the cuspidal Artin group of type A. Since the group W(R ) ⋉ K (R ) naturally acts
A 0 A
e e
on the complexified Tits cone E(R ) associated to R in a properly discontinuous way
A A
and its action is free on the complement set E(R)reg of the reflection hyperplanes, we
can consider the regular orbit space E(R)reg/(W(R )⋉K (R )). We show in Theorem
A 0 A
5.8 that G(T ) is isomorphic to the fundamental group G(R ) of the regular orbit space
A A
E(R)reg/(W(R )⋉K (R )).
A 0 A
e e
From now on we shall return to our motivation in mirror symmetry. In particular,
the generalized root system R will be related to the space of stability conditions for
A
the derived category. The second named author expects the following conjecture. For
e
simplicity, we restrict ourselves to the non-elliptic case:
r
Conjecture 1.1. Set χ := 2+ (1/a −1) and assume that χ 6= 0.
A i A
i=1
X
(i) The complex manifold
(E(R )×C)/W(R )⋉K (R ) if χ > 0,
A A 0 A A
M := (1.1)
A

(E(R )×H)/W(R )⋉K (R ) if χ < 0
 A A 0 A A
should be isomorphic to the space of stability conditions Stab(Dbcoh(P1 )) for
 A,Λ
Dbcoh(P1 ). Here the group W(R )⋉K (R ) acts on E(R ) as reflections and
A,Λ A 0 A A
translations, and on C or H trivially.
(ii) There should exist a Frobenius structure on M which is locally isomorphic to the
A
one constructed from the Gromov–Witten theory of P1 .
A,Λ
(iii) Set the complex manifold
(E(R )reg ×C)/W(R )⋉K (R ) if χ > 0,
Mreg := A A 0 A A (1.2)
A (E(R )reg ×H)/W(R )⋉K (R ) if χ < 0.
 A A 0 A A
Then the universal covering Mreg of Mreg should be the space of stability condi-
A A
tions Stab(Dˇ ) for Dˇ where Dˇ is the smallest full triangulated subcategory
A,Λ A,Λ A,Λ
f
of the derived category of the 2-Calabi–Yau completion Π (kT ) of kT con-
2 A,Λ A,Λ
taining kT , closed under isomorphisms and taking direct summand.
A,Λ
e e
(iv) Assume that r = 3. By [14, 35] and (ii), the Frobenius structure on M in (ii)
A
e
can be identified with the one on the Z-covering of the base space of a universal
4 YUUKISHIRAISHI,ATSUSHITAKAHASHI,ANDKENTAROWADA
unfolding of a cusp polynomial f := xa1+xa2+xa3−s−1x x x with the primitive
A 1 2 3 µA 1 2 3
form ζ(0) := ζ obtained in [14] for χ > 0 and in [35] for χ < 0. Under this
A A A
identification, the stability function
Z : Stab(Dˇ ) −→ (K (Dˇ )⊗ C)∗ (1.3)
A,Λ 0 A,Λ Z
should be given by the period mapping:
ζˇ(−1) : Mreg −→ (K (Dˇ )⊗ C)∗, (1.4)
A 0 A,Λ Z
Z
where the element ζˇ(−1) isfthe formal Fourier–Laplace transform (the Gelfand–
Leray form) of the primitive form ζ(−1) twisted by −1 (cf. Section 5 of [29]).
We shall explain some known results concerning this conjecture. Looijenga shows
in Chapter 3 of [24] that the base space of a universal unfolding of f for r = 3 and
A
χ ≤ 0 can be described as the orbit space in Conjecture 1.1 (i) as complex manifolds.
A
In particular, the complement of the discriminant of this universal unfolding of f is
A
homeomorphic to the regular orbit space in Conjecture 1.1 (iii) as complex manifolds.
If χ > 0, Dubrovin–Zhang [10] constructed a Frobenius structure on M . It
A A
is shown in Ishibashi–Shiraishi–Takahashi [13, 14] and Shiraishi–Takahashi [35] that this
Frobenius structure is isomorphic to the one constructed from the Gromov–Witten theory
of P1 and the one constructed from the universal unfolding of the cusp polynomial f
A,Λ A
with the primitive form ζ .
A
ThegroupG(R )isrelatedtoasubgroupoftheautoequivalencegroupAuteq(Dˇ ).
A A,Λ
ˇ ˇ
Let Br(D ) be the subgroup of the autoequivalence group Auteq(D ) generated by
A,Λ A,Λ
e
spherical twist functors, defined in Seidel–Thomas [34], of simple kT -modules regarded
A,Λ
as Π (kT )-modules. Then we have the natural surjective group homomorphism from
2 A,Λ
e
G(R ) to Br(Dˇ ) in Theorem 6.12. Under Conjecture 1.1 (iii) and (iv), this surjectivity
A A,Λ
e
can be viewed as the property of the covering map and the fundamental group since
e
Br(Dˇ ) is regarded as the group of covering transformations for Stab(Dˇ ). Similar to
A,Λ A,Λ
the results by Bridgeland for K3 surfaces in [4] and Kleinian singularities in [5], we expect
the following conjecture:
Conjecture 1.2. The group homomorphism G(T ) ։ Br(Dˇ ) in Theorem 6.12 should
A A,Λ
also be injective, and hence isomorphism. In other words, the space of stability condition
e
Stab(Dˇ ) should be simply connected.
A,Λ
Similar known results for the injectivity of the group homomorphism in Conjecture
1.2 are obtained by Brav–Thomas [6], Ishii–Ueda–Uehara [15] and Seidel–Thomas [34].
The above conjecture is the further theme to be worked on with Conjecture 1.1.
ON WEYL GROUPS AND ARTIN GROUPS ASSOCIATED TO ORBIFOLD PROJECTIVE LINES 5
Acknowledgement
ThesecondnamedauthorissupportedbyJSPSKAKENHIGrantNumber24684005.
2. Notations and terminologies
Throughout this paper, k denotes an algebraically closed field of characteristic zero.
2.1. Generalized root systems. Inthissubsection, werecallthedefinitionofthesimply
laced generalized root system introduced by K. Saito [30].
Definition 2.1. A simply-laced generalized root system R consists of
• a free Z-module K (R) of finite rank (=: µ) called the root lattice,
0
• a symmetric bi-linear form I : K (R)×K (R) −→ Z,
R 0 0
• a subset ∆ (R) of K (R) called the set of real roots such that
re 0
(i) K (R) = Z∆ (R),
0 re
(ii) for all α ∈ ∆ (R), I(α,α) = 2,
re
(iii) for all α ∈ ∆ (R), the element r of Aut(K (R),I ), the group of automor-
re α 0 R
phisms of K (R) respecting I , defined by
0 R
r (λ) := λ−I (λ,α)α, λ ∈ K (R), (2.1)
α R 0
makes ∆ (R) invariant, namely, r (∆ (R)) = ∆ (R),
re α re re
(iv) thereexistsasubsetB = {α ,...,α }of∆ (R)calledaroot basisofRwhich
1 µ re
µ
satisfies K (R) = Zα , W(R) = hr ,...,r i and ∆ (R) = W(R)B
0 i α1 αµ re
i=1
M
where W(R) is the Weyl group of R defined by
W(R) := hr | α ∈ ∆ (R)i ⊂ Aut(K (R),I ), (2.2)
α re 0 R
• an element c of W(R) called the Coxeter transformation which has the presen-
R
tation c = r ···r with respect to a root basis B.
R α1 αµ
An element of ∆ (R) is called a real root and an element of B is called a real simple root.
re
For a real simple root α ∈ B, the reflection r is called a simple reflection.
α
One can define a notion of isomorphism of simply-laced generalized root systems in
the obvious way.
Definition 2.2. Let R = (K (R),I ,∆ (R),c ) be a simply-laced generalized root sys-
0 R re R
tem with a root basis B = {α ,...,α } of R. The Coxeter–Dynkin diagram Γ is a finite
1 µ B
graph defined as follows:
• the set of vertices is B = {α ,...,α },
1 µ
6 YUUKISHIRAISHI,ATSUSHITAKAHASHI,ANDKENTAROWADA
• the edge between vertices α and α of Γ is given by the following rule:
i j B
◦ ◦ if I (α ,α ) = 0, (2.3a)
αi αj R i j
◦ ◦ if I (α ,α ) = −1, (2.3b)
αi αj R i j
◦ ◦ if I (α ,α ) = −t, (t ≥ 2), (2.3c)
αi t αj R i j
◦ ◦ if I (α ,α ) = +1, (2.3d)
αi αj R i j
◦ ◦ if I (α ,α ) = +2, (2.3e)
αi αj R i j
◦ ◦ if I (α ,α ) = +t, (t ≥ 3). (2.3f)
αi t αj R i j
2.2. Generalized root systems from triangulated categories. In this subsection,
we deduce a simply laced generalized root system from a certain algebraic triangulated
category which satisfies plausible conditions.
Definition 2.3. Let D be a k-linear triangulated category with the translation functor
[1]. Consider a free abelian group F with generators {[X] | X ∈ D} and a subgroup F
0
of F generated by [X]−[Y]+[Z] for all exact triangles X −→ Y −→ Z −→ X[1] in D.
The Grothendieck group K (D) of D is a quotient group F/F .
0 0
Any triangulated category of our interest in this paper is equipped with an enhance-
ment. We briefly recall some terminologies.
Definition 2.4 (Keller [19]). Let D be a k-linear triangulated category. We say that D
is algebraic if it is equivalent as a triangulated category to the stable category of some
k-linear Frobenius category.
It is important to note that for an algebraic k-linear triangulated category D, we
have functorial cones and RHom-complexes once we fix an enhancement, a differential
graded category which yields D (see Theorem 3.8 in [20] for precise statements).
Definition 2.5. Let D be an algebraic k-linear triangulated category with the translation
functor [1] with a fixed enhancement.
(i) For X,Y ∈ D, denote by RHom• (X,Y) ∈ D(k) the RHom-complex such that
D
Hom (X,Y[p]) = Hp(RHom• (X,Y)) for all p ∈ Z, where D(k) is the derived
D D
category of complexes of k-modules.
(ii) Ak-lineartriangulatedcategoryD issaidtobeof finite type ifthetotaldimension
•
of the graded k-module Hom (X,Y) := Hom (X,Y[p])[−p] is finite for all
D D
p∈Z
M
X,Y ∈ D.
ON WEYL GROUPS AND ARTIN GROUPS ASSOCIATED TO ORBIFOLD PROJECTIVE LINES 7
Definition 2.6. Let D be an algebraic k-linear triangulated category of finite type with
a fixed enhancement.
(i) An object E in D is called an exceptional object (or is called exceptional) if
RHom• (E,E) ∼= k ·id in D(k).
D E
(ii) An exceptional collection E = (E ,...,E ) in D is a finite ordered set of excep-
1 n
tional objects satisfying the condition that RHom• (E ,E ) ∼= 0 in D(k) for all
D i j
i > j. An exceptional collection consisting of two objects is an exceptional pair.
(iii) An exceptional collection E = (E ,...,E ) in D is said to be isomorphic to
1 n
another exceptional collection E′ = (E′,...,E′) in D if E ∼= E′ in D for all
1 n i i
i = 1,...,µ.
(iv) An exceptional collection E = (E ,...,E ) in D is called a strongly excep-
1 n
tional collection if, for all i,j = 1,...,n, the complex RHom• (E ,E ) is iso-
D i j
morphic in D(k) to a complex concentrated in degree zero, equivalently, we have
Hom (E ,E [p]) = 0 for p 6= 0.
D i j
(v) An exceptional collection E in D is called full if the smallest full triangulated
subcategory of D containing all elements in E is equivalent to D.
(vi) For an exceptional pair (X,Y), one has new exceptional pairs (L Y,Y) called the
X
left mutation of (X,Y) and (Y,R X) called the right mutation of (X,Y). Here
Y
the object L Y[1] is defined as the cone of the evaluation morphism ev
X
RHom• (X,Y)⊗L X −e→v Y, (2.4a)
D
where (−) ⊗L X is the left adjoint of the functor RHom (X,−) : D −→ D(k).
D
Similarly, the object R X is defined as the cone of the evaluation morphism ev∗
Y
X −e→v∗ RHom• (X,Y)∗ ⊗L Y. (2.4b)
D
where (−)∗ denotes the duality Hom (−,k).
k
Here we recall the braid group action on the set of isomorphism classes of full
exceptional collections.
Definition 2.7. The Artin’s braid group B on µ-stands is a group presented by the
µ
following generators and relations:
Generators: {b | i = 1,...,µ−1}
i
Relations:
b b = b b for |i−j| ≥ 2, (2.5a)
i j j i
b b b = b b b for i = 1,...,µ−2. (2.5b)
i i+1 i i+1 i i+1
8 YUUKISHIRAISHI,ATSUSHITAKAHASHI,ANDKENTAROWADA
Consider the group G := B ⋉Zµ, the semi-direct product of the braid group B
µ µ µ
and the free abelian group of rank µ, defined by the group homomorphism B → S →
µ µ
Aut Zµ, where the first homomorphism is b 7→ (i,i+ 1) and the second one is induced
Z i
by the natural actions of the symmetric group S on Zµ.
µ
Proposition 2.8 (cf. Proposition 2.1 in [3]). Let D be an algebraic k-linear triangu-
lated category of finite type with a fixed enhancement.. The group G acts on the set of
µ
isomorphism classes of full exceptional collections in D by mutations and translations:
b (E ,...,E ) := (E ,...,E ,E ,R E ,E ,...,E ), (2.6a)
i 1 µ 1 i−1 i+1 Ei+1 i i+2 µ
b−1(E ,...,E ) := (E ,...,E ,L E ,E ,E ,...,E ), (2.6b)
i 1 µ 1 i−1 Ei i+1 i i+2 µ
e (E ,...,E ) := (E ,...,E ,E [1],E ,...,E ), (2.6c)
i 1 µ 1 i−1 i i+1 µ
where we denote by e the i-th standard basis of Zµ. (cid:3)
i
Proposition 2.9. Let D be an algebraic k-linear triangulated category of finite type with
the translation functor [1] and a fixed enhancement. Assume that D satisfies the following
conditions:
(i) There exists a full strongly exceptional collection E = (E ,...,E ) in D.
1 µ
(ii) The action of the group G on the set of isomorphism classes of full exceptional
µ
collections in D is transitive.
(iii) For any exceptional object E′ ∈ D, there exists a full exceptional collection E′ in
D such that E′ ∈ E′.
Then the following quadruple
• the Grothendieck group K (D) of D,
0
• the Cartan form I : K (D)×K (D) −→ Z;
D 0 0
I ([X],[Y]) := χ ([X],[Y])+χ ([Y],[X]), X,Y ∈ D, (2.7)
D D D
where χ : K (D)×K (D) −→ Z is the Euler form defined by
D 0 0
χ ([X],[Y]) := (−1)pdim Hom (X,Y[p]), (2.8)
D k D
p∈Z
X
• the subset ∆ (D) of K (D) defines by
re 0
∆ (D) := W(B)B, B := {[E ],...,[E ]}, (2.9)
re 1 µ
where W(B) is a subgroup of Aut(K (D),I ) generated by reflections
0 D
r (λ) := λ−I (λ,[E ])[E ], λ ∈ K (D), i = 1,...,µ, (2.10)
[Ei] D i i 0
ON WEYL GROUPS AND ARTIN GROUPS ASSOCIATED TO ORBIFOLD PROJECTIVE LINES 9
• the automorphism c on K (D) induced by the Coxeter functor C := S [−1] on
D 0 D D
D where S is the Serre functor on D,
D
forms a simply-laced generalized root system R , which does not depend on the choice of
D
the full exceptional collection E.
Proof. Since E = (E ,...,E ) is a full strongly exceptional collection in D, it follows that
1 µ
µ
K (D) = Z[E ] on which the Euler form χ is well-defined and there exists the Serre
0 i D
i=1
M
functor S on D.
D
Lemma 2.10. We have
c = r ···r . (2.11)
D [E1] [Eµ]
Proof. The statement follows from a relation between the Serre functor S on D and the
D
helix generated by the full exceptional collection E. See p. 223 in [3], for example. (cid:3)
We only have to show that ∆ (D) satisfies the desired properties and it does not
re
depend on the particular choice of E since the objects K (D), I and c are invariants of
0 D D
the triangulated category D.
It is obvious from the definition of exceptional object that I (α,α) = 2 for all
D
α ∈ ∆ (D). Set
re
W(D) := {r ∈ Aut(K (D),I ) | α ∈ ∆ (D)}. (2.12)
α 0 D re
Lemma 2.11. For any α ∈ ∆ (D), we have
re
r r = r r . (2.13)
[Ei] α r[Ei](α) [Ei]
Proof. A direct calculation yield the statement. (cid:3)
Note that Lemma 2.11 implies that W(D) = W(B).
Lemma 2.12. For an exceptional object E′ ∈ D, the class [E′] ∈ K (D) belongs to ∆ (D).
0 re
Proof. Choose a full exceptional collection E′ in D such that E′ ∈ E′. There exists an
element g ∈ G such that gE′ ∼= E. Since we have −[X] = r ([X]) and [R Y] =
µ [X] X
−r [X] = r r ([X]) in K (D) for any exceptional pair (X,Y), it turns out from
[Y] [Y] [X] 0
Lemme 2.11 that [E′] ∈ ∆ (D). (cid:3)
re
Set B′ := {[E′],....[E′]} for any full exceptional collection E′ = (E′,...,E′) in
1 µ 1 µ
D. Lemma 2.12 implies that W(D)B′ ⊂ W(D)W(D)B ⊂ W(D)B and hence W(D)B′ =
W(D)B. Therefore the set ∆ (D) does not depend on the particular choice of the full
re
exceptional collection E. Thus we have completed the proof of the proposition. (cid:3)
10 YUUKISHIRAISHI,ATSUSHITAKAHASHI,ANDKENTAROWADA
Remark 2.13. We assumed in Proposition 2.9 the existence of a full strongly exceptional
collection E in D in order to ensure that D has a unique enhancement in a suitable sense.
We refer [18] and [23] for some results on the uniqueness of enhancements for triangulated
categories and do not discuss this matter more in detail.
Definition 2.14. The generalized root system R in Proposition 2.9 is called the simply-
D
laced generalized root system associated to D.
From various points of view, which we do not discuss in detail in this paper, it is
natural to expect the assumptions of Proposition 2.9. Indeed, they are proven for derived
categories of hereditary Artin algebras by Crawley-Boevey [7] and Ringel [26] and for
derived categories of coherent sheaves on an orbifold projective line P1 (we shall recall
A,Λ
the definition later) by Meltzer [25]. The transitivity of the action of G is conjectured
µ
by Bondal–Polishchuk (Conjecture 2.2 in [3]), and is known for the derived categories of
coherent sheaves on P2 and P1 ×P1 by Rudakov [28], by arbitrary del Pezzo surfaces by
Kuleshov and Orlov [22], for example.
Remark 2.15. One can also consider the subset ∆s (D) of K (D) defined by
re 0
∆s (D) := {[E] ∈ K (D) | E is an exceptional object in D}, (2.14)
re 0
which is known as the set of Schur roots. Under the assumptions of Proposition 2.9, we
always have ∆s (D) ⊂ ∆ (D), however, ∆s (D) 6= ∆ (D) in general. A criteria to have
re re re re
∆s (D) in terms of the Weyl group W(D) is recently given by Hubery–Krause [12] for
re
derived categories of hereditary Artin algebras.
2.3. Generalized root systems associated to star quivers. We recall the definition
of quivers and their path algebras.
Definition 2.16. A quiver Q is a quadruple (Q ,Q ;s,t) where Q is a set called the set
0 1 0
of vertices, Q is a set called the set of arrows and s,t are maps from Q to Q which
1 1 0
associate the source vertex and the target vertex for each arrow. An arrow f with the
f
source s(f) and the target t(f) is often written as s(f) −→ t(f).
Definition 2.17. Let Q = (Q ,Q ;s,t) be a quiver.
0 1
(i) A path of length 0 is a symbol (v|v) defined for each vertex v ∈ Q .
0
(ii) A path of length l ≥ 1 from the vertex v to the vertex v′ in a quiver Q is a symbol
(v|f ···f |v′) with arrows f , i = 1,...,l such that s(f ) = v, t(f ) = v′ and
1 l i 1 l
s(f ) = t(f ), i = 1,...,l−1.
i+1 i
(iii) For a path p = (v|f ···f |v′), set s(p) := v and t(p) := v′.
1 l