Weyl Groups and Artin Groups Associated to Weighted Projective Lines (joint work with Yuuki Shiraishi and Kentaro Wada) Atsushi TAKAHASHI OSAKA UNIVERSITY November 15, 2013 at NAGOYA 1 / 29
Aim Want to understand interesting correspondences among 1. Complex (Algebraic) Geometry. 2. Symplectic Geometry. 3. Representation Theory. McKay correspondence, strange duality,... 2 / 29
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 Lagrangian submanifolds). 3. Derived category of finite dimesional modules over a finite dimensional algebra. Equivalences among them = Homological Mirror Symmetry 3 / 29
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 Frobenius algebras over a complex manifold 4 / 29
These Mirror Symmetries should be related via 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 groups associated to weighted projective lines. 5 / 29
Weighted Projective Lines A = (a 1,..., a r ): a tuple of positive integers (r 3). Set r r ( ) 1 µ A := 2 + (a r 1), χ A := 2 + 1. a r k=1 k=1 Λ = (λ 4,..., λ r ): a tuple of pairwise distinct points on P 1 (C) \ {, 0, 1}. 6 / 29
Set R A,Λ := C[X 1,..., X r ] /I Λ where I Λ is an ideal generated by X a i i X a 2 2 + λ ix a 1 1, i = 3,..., r (λ 3 := 1). Denote by L A an abelian group defined as the quotient ( r ) /( L A := ZX ) i a ixi a jxj ; 1 i < j r. i=1 Definition 1 Let r, A and Λ be as above. Define a stack P 1 A,Λ by P 1 A,Λ := [(Spec(R A,Λ)\{0}) /Spec(CL A )], which is called the weighted projective line of type (A, Λ). 7 / 29
Theorem 2 (Geigle Lenzing 87) The category D b coh(p 1 A,Λ ) admits a full strongly exceptional collection. In particular, there are triangulated equivalences D b coh(p 1 A,Λ ) = D b (CC A,Λ ) = D b (C T A,Λ ) where C A,Λ is the Ringel s canonical algebra of type (A, Λ) and T A,Λ is a bound quiver in the next slide. Theorem 3 (T 08) Assume that A = (a 1, a 2, 2). We have a triangulated equivalence D b Fuk (f A ) = D b (CC A,Λ ) = D b (C T A ), where f A := x a 1 1 + x a 2 2 + x 3 2 cx 1x 2 x 3 for some c C \ {0}. 8 / 29
The bound quiver T A,Λ (1,a 1 1) 1 22222222 2 1 (1.1) 2 22222222 (r,1) 11111111 (2,1) (r 1,1) 1...... (r,a r 1) 3 3333333 (2,a2 1)... (r 1,a r 1 1) There are two relations from 1 to 1 depending on Λ. 9 / 29
The star quiver T A (1,a 1 1) 1 (1.1) 2 22222222 (r,1) (r,a r 1) 11111111 (2,1) (r 1,1) 1...... 3 3333333 (2,a2 1)... (r 1,a r 1 1) 10 / 29
Root lattice Denote by K 0 ( T A,Λ ) the Grothendieck group of D b (C T A,Λ ); K 0 ( T A,Λ ) = v T A,Λ Z α v. The Euler form χ : K 0 ( T A,Λ ) K 0 ( T A,Λ ) Z is defined by χ([m], [N ]) := p Z( 1) p dim C Hom D (M, N [p]). b (C T A,Λ ) The Cartan form I : K 0 ( T A,Λ ) K 0 ( T A,Λ ) Z is defined by Define similarly for T A. I (γ, γ ) := χ(γ, γ ) + χ(γ, γ). 11 / 29
Coxeter Dynkin diagram T A I ( α v, α v ) = 0 v v I ( α v, α v ) = 1 v v I ( α v, α v ) = +2 v v (1,a 1 1) 1 3 3333333 1 (1.1) 3 3333333 (r,1) (r 1,1)... 2 22222222 (2,1)... (r,ar 1) 4 4444444 (2,a 2 1)... (r 1,a r 1 1) 12 / 29
Coxeter Dynkin diagram T A I (α v, α v ) = 0 v v I (α v, α v ) = 1 v v (1,a 1 1) 1 (1.1) 3 3333333... 2 22222222 (2,1) (r,1) (r 1,1)... (r,ar 1) 4 4444444 (2,a 2 1)... (r 1,a r 1 1) χ A 0 the Cartan form I on K 0 (T A ) is non-degenerate. 13 / 29
Weyl groups Define the simple reflection r v on K 0 ( T A,Λ ) Q by r v ( λ) := λ I ( λ, α v ) α v, λ K0 ( T A,Λ ) Q. Definition 4 The subgroup of GL(K 0 ( T A,Λ ) Q ) generated by simple reflections is called the Weyl group associated to T A,Λ and is denoted by W ( T A ) (since it depends only on the Coxeter Dynkin diagram T A ). Remark 5 Actually, W ( T A ) depends only on the (generalized) root system associated to D b (C T A,Λ ) ( = D b coh(p 1 A,Λ )). Define similarly the Weyl group W (T A ) associated to T A. 14 / 29
Definition 6 1. If χ A = 0, then W ( T A ) is called the elliptic Weyl group. 2. If χ A < 0, then W ( T A ) is called the cuspidal Weyl group. If χ A > 0, then W ( T A ) 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} K 0 (T A )/rad(i ) where t and p are defined by t W ( T p A ) W (T A ) {1}, t( α v )( λ) := t v ( λ) := λ I ( λ, α v )( α 1 α 1 ), p( r 1 ) = p( r 1 ) = r 1, p( r v ) = r v, v T A. 15 / 29
Remark 8 The element α 1 α 1 K 0 ( T A,Λ ) belongs to the radical of the Cartan form I. Lemma 9 (Key Lemma) 1. We have t 1 = r 1 r 1, t (i,1) = r (i,1) t 1 r (i,1) t 1 1 and t (i,j) = r (i,j) t (i,j 1) r (i,j) t 1 (i,j 1). 2. The elements t v, v T A commute with each other. 3. Let N be the smallest normal subgroup of W ( T A ) containing t 1. We have N = Ker(p) and t (i,j) N for all i, j. If χ A = 0, then the group W (T A ) K 0 (T A ) is a central extension of the elliptic Weyl group W ( T A ), which is called the hyperbolic extension of W ( T A ) (Saito Takebayashi). 16 / 29
Weyl groups as generalized Coxeter groups Generalizing a result by Saito Takebayashi for χ A = 0, we have Theorem 10 (STW) Let W ( T A ) be a group described by the generators { w v v T A } and the generalized Coxeter relations. Then we have W ( T A ) = W (T A ) K 0 (T A ). w 2 v = 1 for all v T A, (W0) ( w v w v ) 2 = 1 if I ( α v, α v ) = 0, (W1.0) ( w v w v ) 3 = 1 if I ( α v, α v ) = 1, (W1.1) w (i,1) ũ 1 w (i,1) ũ 1 = ũ 1 w (i,1) ũ 1 w (i,1), w (i,1) ũ (j,1) = ũ (j,1) w (i,1), w (j,1) ũ (i,1) = ũ (i,1) w (j,1) if i j. (W2) (W3) where ũ 1 := w 1 w 1 and ũ (i,1) := w (i,1) ũ 1 w (i,1) ũ 1 1. 17 / 29
The correspondence ( w v, ũ v ) ( r v, t v ) gives the isomorphism. Remark 11 (cf. Yamada 00) Under W0, the relations W2 and W3 are equivalent to those introduced by Saito Takebayashi ( 97) for χ A = 0. 18 / 29
Orbit space From now on, we assume that χ A 0 to simplify some definitions. (χ A 0 the Cartan matrix for T A is non-degenerate.) To T A, one can associate a Kac Moody Lie algebra and hence one can define a set of roots in a standard way. Consider (the interior of) the complexified Tits cone E(T A ) := {h h(t A ) α, Im(h) > 0 for α (T A ) + im }, where (T A ) + im is the subset of (T A) + consisting of positive imaginary roots. The group W ( T A ) = W (T A ) K 0 (T A ) acts properly discontinuously on E(T A ). 19 / 29
Consider the complex manifold of dimension µ A : { M A := E(T A )/W ( T C χ A > 0 A ) H χ A < 0 where H is the complex upper half plane. Conjecture 12 The space of Bridgeland s stability conditions on D b coh(p 1 A,Λ ) is isomorphic to M A. Conjecture 13 There is a Frobenius structure on M A isomorphic to the one constructed from the Gromov Witten theory for P 1 A,Λ. 20 / 29
Theorem 14 (Satake T 08) Conjecture 13 for χ A = 0 is true. Theorem 15 (Ishibashi Shiraishi T 12) Conjecture 13 is true if χ A > 0. Theorem 16 (Shiraishi T, in preparation) Conjecture 13 is true if the property (P) holds. A PDF file is available at http://frompde.sissa.it/workshop2013/talks/16mon/takahashi.pdf Remark 17 One can check the property (P) easily for χ A 0. (Saito 90 for χ A = 0, Dubrovin Zhang 98 for χ A > 0) 21 / 29
Fundamental groups of regular orbit spaces Set where M reg A := E(T A) reg /W ( T A ) { C χ A > 0 H χ A < 0 E(T A ) reg := E(T A ) \ {reflection hyperplanes}. Want to understand the universal covering of M reg A. 22 / 29
reg M A The universal covering of should be the space of Bridgeland s stability conditions on some triangulated category associated to the derived preprojective algebra (2-CY completion) of C T A,Λ. M reg The fundamental group of A should determine the autoequivalence group of the derived category. 23 / 29
Artin groups Definition 18 (cf. Yamada 00 when χ A = 0) The Artin group G( T A ) is a group defined by the generators { g v v T A } and relations: g v g v = g v g v if I ( α v, α v ) = 0, (A1.0) g v g v g v = g v g v g v if I ( α v, α v ) = 1, (A1.1) g (i,1) s 1 g (i,1) s 1 = s 1 g (i,1) s 1 g (i,1), 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. where s 1 := g 1 g 1 and s (i,1) := g (i,1) s 1 g (i,1) s 1 1. (A2) (A3) 24 / 29
Proposition 19 The correspondence g v w v for v T A induces an isomorphism / G( T A ) g v 2 v T A = W ( T A ). Remark 20 The 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). ( t v ( λ) := λ I ( λ, α v )( α 1 α 1 )) 25 / 29
Artin groups as fundamental groups Generalizing Yamada s result for χ A = 0, we have Theorem 21 (STW) If χ A 0, then there is an isomorphism G( T A ) ( ) = π Mreg 1 A, = π1 (E(T A ) reg /W ( T ) A ),. Key: Van der Lek s description of π 1 (E(T A ) reg /W ( T ) A ),. 26 / 29
Lemma 22 (Van der Lek 83) The group π 1 (E(T A ) reg /W ( T ) A ), can be described by the generators {g v, s v v T A } and the following relations: g v g v = g v g v if I (α v, α v ) = 0, (A 1.0) g v g v g v = g v g v g v if I (α v, α v ) = 1, (A 1.1) g v s v = s v g v if I (α v, α v ) = 0, (A 2) g v s v g v = s v s v if I (α v, α v ) = 1. (A 3) Roughly speaking, the correspondence ( g v, s v ) (g v, s v ) gives the isomorphism G( T A ) = π 1 (E(T A ) reg /W ( T ) A ),. 27 / 29
Thank you very much! 28 / 29
Happy 65th birthday Prof. Yamagata! 29 / 29