Covering Theory and Cluster Categories. International Workshop on Cluster Algebras and Related Topics

Transcription

1 Covering Theory and Cluster Categories Shiping Liu (Université de Sherbrooke) joint with Fang Li, Jinde Xu and Yichao Yang International Workshop on Cluster Algebras and Related Topics Chern Institute of Mathematics Nankai University (Tianjin, China) July 10-13, 2017

2 Objective 1 Construct a universal covering for each valued quiver.

3 Objective 1 Construct a universal covering for each valued quiver. 2 Introduce covering of species and the associated push-down functor.

4 Objective 1 Construct a universal covering for each valued quiver. 2 Introduce covering of species and the associated push-down functor. 3 As an application, we shall construct a cluster category of non simply laced type C.

5 Motivation 1 Unfolding the exchange matrix of a cluster algebra is important in the study of cluster algebras, for example,

6 Motivation 1 Unfolding the exchange matrix of a cluster algebra is important in the study of cluster algebras, for example, classification of mutation-finite skew-symmetrizable cluster algebras by Felikson-Shapiro-Tumarkin.

7 Motivation 1 Unfolding the exchange matrix of a cluster algebra is important in the study of cluster algebras, for example, classification of mutation-finite skew-symmetrizable cluster algebras by Felikson-Shapiro-Tumarkin. 2 The unfolding of an exchange matrix is a covering of the corresponding valued quiver.

8 Motivation 1 Unfolding the exchange matrix of a cluster algebra is important in the study of cluster algebras, for example, classification of mutation-finite skew-symmetrizable cluster algebras by Felikson-Shapiro-Tumarkin. 2 The unfolding of an exchange matrix is a covering of the corresponding valued quiver. 3 There is a growing interest in cluster categories with a cluster structure of infinite rank, while cluster categories of non simply laced type seems unseen.

9 Part I Covering theory for species and their representations joint with Fang Li, Jinde Xu and Yichao Yang

10 Valued quivers 1 A valued quiver is a pair (, v), where

11 Valued quivers 1 A valued quiver is a pair (, v), where = ( 0, 1 ) is a locally finite quiver without multiple arrows, loops or 2-cycles;

12 Valued quivers 1 A valued quiver is a pair (, v), where = ( 0, 1 ) is a locally finite quiver without multiple arrows, loops or 2-cycles; v is a valuation, that is, each arrow x y is endowed with (v xy, v yx ) N N.

13 Valued quivers 1 A valued quiver is a pair (, v), where = ( 0, 1 ) is a locally finite quiver without multiple arrows, loops or 2-cycles; v is a valuation, that is, each arrow x y is endowed with (v xy, v yx ) N N. 2 The valuation v is called trivial if v xy = v yx = 1 for every x y 1.

14 Valued quivers 1 A valued quiver is a pair (, v), where = ( 0, 1 ) is a locally finite quiver without multiple arrows, loops or 2-cycles; v is a valuation, that is, each arrow x y is endowed with (v xy, v yx ) N N. 2 The valuation v is called trivial if v xy = v yx = 1 for every x y 1. Remark A non-valued quiver without multiple arrows, loops or 2-cycles is regarded as a trivially valued quiver.

15 Example A valued quiver of type C with a zigzag orientation: 0 (2,1) where trivial valuations are omitted.

16 Valued quiver morphisms Definition 1 Let (Γ, u) and (, v) be valued quivers.

17 Valued quiver morphisms Definition 1 Let (Γ, u) and (, v) be valued quivers. 2 A quiver morphism ϕ : Γ is valued quiver morphism

18 Valued quiver morphisms Definition 1 Let (Γ, u) and (, v) be valued quivers. 2 A quiver morphism ϕ : Γ is valued quiver morphism if, for any x (uxy,uyx ) y Γ 1, we have (u xy, u yx ) (v ϕ(x),ϕ(y), v ϕ(y),ϕ(x) ).

19 Valued quiver covering Definition A valued quiver morphism ϕ : (Γ, u) (, v) is called a valued quiver covering provided that

20 Valued quiver covering Definition A valued quiver morphism ϕ : (Γ, u) (, v) is called a valued quiver covering provided that 1 Given a 0, ϕ (a) := {x Γ 0 ϕ(x) = a} ;

21 Valued quiver covering Definition A valued quiver morphism ϕ : (Γ, u) (, v) is called a valued quiver covering provided that 1 Given a 0, ϕ (a) := {x Γ 0 ϕ(x) = a} ; 2 Given any arrow α : a (v ab,v ba ) b 1,

22 Valued quiver covering Definition A valued quiver morphism ϕ : (Γ, u) (, v) is called a valued quiver covering provided that 1 Given a 0, ϕ (a) := {x Γ 0 ϕ(x) = a} ; 2 Given any arrow α : a (v ab,v ba ) b 1, for any x ϕ (a), we have v ab = β:x y ϕ (α) u xy;

23 Valued quiver covering Definition A valued quiver morphism ϕ : (Γ, u) (, v) is called a valued quiver covering provided that 1 Given a 0, ϕ (a) := {x Γ 0 ϕ(x) = a} ; 2 Given any arrow α : a (v ab,v ba ) b 1, for any x ϕ (a), we have v ab = β:x y ϕ (α) u xy; for any y ϕ (b), we have v ba = γ:x y ϕ (α) u yx.

24 Example of valued quiver covering (1,1). b 1 a 0 (1,2) b (1,1) 0 a 1 (1,2) b 1. ϕ a (2,3) b

25 Universal cover of a non-valued quiver Theorem (Bongartz, Gabriel) Given non-valued quiver Q, one has unique quiver covering π : Q Q, where Q is a tree, called universal cover of Q.

26 Unfold a valued quiver Let (, v) be a connected valued quiver.

27 Unfold a valued quiver Let (, v) be a connected valued quiver. Definition Define unfolding quiver ˆ of (, v) to be non-valued quiver.

28 Unfold a valued quiver Let (, v) be a connected valued quiver. Definition Define unfolding quiver ˆ of (, v) to be non-valued quiver. ˆ 0 = 0 ;

29 Unfold a valued quiver Let (, v) be a connected valued quiver. Definition Define unfolding quiver ˆ of (, v) to be non-valued quiver. ˆ 0 = 0 ; For each arrow α : x (vxy,vyx ) y in, one draws v xy v yx arrows α ij : x y in ˆ, arranged as a (v yx v xy )-matrix α 11 α 12 α 1,vxy α 21 α 22 α 2,vxy α vyx,1 α vyx,2 α vyx,v xy.

30 Universal cover of a valued quiver Theorem 1 There exists a valued quiver covering π :,

31 Universal cover of a valued quiver Theorem 1 There exists a valued quiver covering π :, where is a full subquiver of the universal cover of ˆ.

32 Universal cover of a valued quiver Theorem 1 There exists a valued quiver covering π :, where is a full subquiver of the universal cover of ˆ. In particular, is a trivially valued tree.

33 Universal cover of a valued quiver Theorem 1 There exists a valued quiver covering π :, where is a full subquiver of the universal cover of ˆ. In particular, is a trivially valued tree. 2 π covers every valued quiver covering φ : Γ.

34 Example: Universal cover of C x 1 x 2 x 3 A : x 0 x 1 x 2 x 3 ϕ C : 0 (2,1) 1 2 3

35 Species (, v): symmetrizable valued quiver without infinite paths.

36 Species (, v): symmetrizable valued quiver without infinite paths. Definition Let k be a field. A k-species S of (, v) consists of

37 Species (, v): symmetrizable valued quiver without infinite paths. Definition Let k be a field. A k-species S of (, v) consists of 1 finite dimensional division k-algebra S(i), for i 0 ;

38 Species (, v): symmetrizable valued quiver without infinite paths. Definition Let k be a field. A k-species S of (, v) consists of 1 finite dimensional division k-algebra S(i), for i 0 ; 2 S(i)-S(j)-bimodules S(α), for α : i (v ij,v ji ) j 1, with

39 Species (, v): symmetrizable valued quiver without infinite paths. Definition Let k be a field. A k-species S of (, v) consists of 1 finite dimensional division k-algebra S(i), for i 0 ; 2 S(i)-S(j)-bimodules S(α), for α : i (v ij,v ji ) j 1, with dim S(α) S(j) = v ij ;

40 Species (, v): symmetrizable valued quiver without infinite paths. Definition Let k be a field. A k-species S of (, v) consists of 1 finite dimensional division k-algebra S(i), for i 0 ; 2 S(i)-S(j)-bimodules S(α), for α : i (v ij,v ji ) j 1, with dim S(α) S(j) = v ij ; dim S(i) S(α) = v ji.

41 Example: R-species of C C : 0 (2,1) C : C C R R R R R R R R

42 Representations of a species 1 A representation X of a speceis S consists of

43 Representations of a species 1 A representation X of a speceis S consists of right S(i)-vector space X (i), for each i 0 ;

44 Representations of a species 1 A representation X of a speceis S consists of right S(i)-vector space X (i), for each i 0 ; S(j)-linear map X (α) : X (i) S(i) S(α) X (j), for each α : i j 1.

45 Representations of a species 1 A representation X of a speceis S consists of right S(i)-vector space X (i), for each i 0 ; S(j)-linear map X (α) : X (i) S(i) S(α) X (j), for each α : i j 1. 2 Set dim X = i 0 dim X (i) S(i).

46 Representations of a species 1 A representation X of a speceis S consists of right S(i)-vector space X (i), for each i 0 ; S(j)-linear map X (α) : X (i) S(i) S(α) X (j), for each α : i j 1. 2 Set dim X = i 0 dim X (i) S(i). Proposition (Dlab-Ringel) The category rep(s) of finite dimensional representations of S is a hereditary abelian category.

47 Morphisms of species T : species of another valued quiver no infinite path (Γ, u).

48 Morphisms of species T : species of another valued quiver no infinite path (Γ, u). Definition A species morphism Φ : T S consists of

49 Morphisms of species T : species of another valued quiver no infinite path (Γ, u). Definition A species morphism Φ : T S consists of 1 a valued quiver morphism ϕ : (Γ, u) (, v);

50 Morphisms of species T : species of another valued quiver no infinite path (Γ, u). Definition A species morphism Φ : T S consists of 1 a valued quiver morphism ϕ : (Γ, u) (, v); 2 an algebra morphism ϕ i : T (i) S(ϕ(i)), for each i Γ 0 ;

51 Morphisms of species T : species of another valued quiver no infinite path (Γ, u). Definition A species morphism Φ : T S consists of 1 a valued quiver morphism ϕ : (Γ, u) (, v); 2 an algebra morphism ϕ i : T (i) S(ϕ(i)), for each i Γ 0 ; 3 For each arrow β : i j Γ with α = ϕ(β) : a b,

52 Morphisms of species T : species of another valued quiver no infinite path (Γ, u). Definition A species morphism Φ : T S consists of 1 a valued quiver morphism ϕ : (Γ, u) (, v); 2 an algebra morphism ϕ i : T (i) S(ϕ(i)), for each i Γ 0 ; 3 For each arrow β : i j Γ with α = ϕ(β) : a b, T (i)-s(b)-bilinear map βϕ : T (β) T (j) S(b) S(α);

53 Morphisms of species T : species of another valued quiver no infinite path (Γ, u). Definition A species morphism Φ : T S consists of 1 a valued quiver morphism ϕ : (Γ, u) (, v); 2 an algebra morphism ϕ i : T (i) S(ϕ(i)), for each i Γ 0 ; 3 For each arrow β : i j Γ with α = ϕ(β) : a b, T (i)-s(b)-bilinear map S(a)-T (j)-bilinear map βϕ : T (β) T (j) S(b) S(α); ϕ β : S(a) T (i) T (β) S(α).

54 Species covering Definition A species morphism Φ : T S is called species covering if

55 Species covering Definition A species morphism Φ : T S is called species covering if 1 ϕ : (Γ, u) (, v) is valued quiver covering;

56 Species covering Definition A species morphism Φ : T S is called species covering if 1 ϕ : (Γ, u) (, v) is valued quiver covering; 2 Given any α : a b 1, the following are satisfied.

57 Species covering Definition A species morphism Φ : T S is called species covering if 1 ϕ : (Γ, u) (, v) is valued quiver covering; 2 Given any α : a b 1, the following are satisfied. For each i ϕ (a), there exists an isomorphism ( β ϕ) : T (β) S(b) S(α). β:i j ϕ (α) T (j)

58 Species covering Definition A species morphism Φ : T S is called species covering if 1 ϕ : (Γ, u) (, v) is valued quiver covering; 2 Given any α : a b 1, the following are satisfied. For each i ϕ (a), there exists an isomorphism ( β ϕ) : T (β) S(b) S(α). β:i j ϕ (α) T (j) For each j ϕ (b), there exists isomorphism (ϕ γ ) : S(a) T (γ) S(α). γ:i j ϕ (α) T (i)

59 Example of species covering R R R R R R R R R R R R R Φ C C R R R R R

60 Push-down functor and pull-up functor Fix a species covering Φ : T S with vqc ϕ : Γ.

61 Push-down functor and pull-up functor Fix a species covering Φ : T S with vqc ϕ : Γ. 1 The push-down functor Φ λ : rep(t ) rep(s) as follows.

62 Push-down functor and pull-up functor Fix a species covering Φ : T S with vqc ϕ : Γ. 1 The push-down functor Φ λ : rep(t ) rep(s) as follows. If X rep(t ), then Φ λ (X ) rep(s) such, for a 0,

63 Push-down functor and pull-up functor Fix a species covering Φ : T S with vqc ϕ : Γ. 1 The push-down functor Φ λ : rep(t ) rep(s) as follows. If X rep(t ), then Φ λ (X ) rep(s) such, for a 0, Φ λ (X )(a) = i ϕ (a) X (i) T (i) S(a);

64 Push-down functor and pull-up functor Fix a species covering Φ : T S with vqc ϕ : Γ. 1 The push-down functor Φ λ : rep(t ) rep(s) as follows. If X rep(t ), then Φ λ (X ) rep(s) such, for a 0, Φ λ (X )(a) = i ϕ (a) X (i) T (i) S(a); 2 The pull-up functor Φ µ : rep(s) rep(t ) as follows.

65 Push-down functor and pull-up functor Fix a species covering Φ : T S with vqc ϕ : Γ. 1 The push-down functor Φ λ : rep(t ) rep(s) as follows. If X rep(t ), then Φ λ (X ) rep(s) such, for a 0, Φ λ (X )(a) = i ϕ (a) X (i) T (i) S(a); 2 The pull-up functor Φ µ : rep(s) rep(t ) as follows. If M rep(s), then Φ µ (M) rep(t ) such, for i Γ 0,

66 Push-down functor and pull-up functor Fix a species covering Φ : T S with vqc ϕ : Γ. 1 The push-down functor Φ λ : rep(t ) rep(s) as follows. If X rep(t ), then Φ λ (X ) rep(s) such, for a 0, Φ λ (X )(a) = i ϕ (a) X (i) T (i) S(a); 2 The pull-up functor Φ µ : rep(s) rep(t ) as follows. If M rep(s), then Φ µ (M) rep(t ) such, for i Γ 0, Φ µ (M)(i) = M(ϕ(i)),

67 Push-down functor and pull-up functor Fix a species covering Φ : T S with vqc ϕ : Γ. 1 The push-down functor Φ λ : rep(t ) rep(s) as follows. If X rep(t ), then Φ λ (X ) rep(s) such, for a 0, Φ λ (X )(a) = i ϕ (a) X (i) T (i) S(a); 2 The pull-up functor Φ µ : rep(s) rep(t ) as follows. If M rep(s), then Φ µ (M) rep(t ) such, for i Γ 0, Φ µ (M)(i) = M(ϕ(i)), which is considered as T (i)-vector space.

68 Adjoint pairs Theorem Let Φ : T S be a species covering. 1 The push-down functor Φ λ : rep(t ) rep(s) induces an exact functor Φ D λ : Db (rep(t )) D b (rep(s)).

69 Adjoint pairs Theorem Let Φ : T S be a species covering. 1 The push-down functor Φ λ : rep(t ) rep(s) induces an exact functor Φ D λ : Db (rep(t )) D b (rep(s)). 2 The pull-up functor Φ µ : rep(s) rep(t ) induces an exact functor Φ D µ : D b (rep(s)) D b (rep(t )).

70 Adjoint pairs Theorem Let Φ : T S be a species covering. 1 The push-down functor Φ λ : rep(t ) rep(s) induces an exact functor Φ D λ : Db (rep(t )) D b (rep(s)). 2 The pull-up functor Φ µ : rep(s) rep(t ) induces an exact functor Φ D µ : D b (rep(s)) D b (rep(t )). 3 (Φ λ, Φ µ ) and (Φ D λ, ΦD µ ) are adjoint pairs.

71 Example of push-down R R R R R 1I R 0 R R R R R R R 1I R 0 Φ Φ λ C C R R R R C R R 0

72 Part II Cluster category of type C joint with Jinde Xu and Yichao Yang

73 Orbit category 1 A : Hom-finite Krull-Schmidt additive k-category.

74 Orbit category 1 A : Hom-finite Krull-Schmidt additive k-category. 2 G: group acting on A such, for objects X, Y, that A(X, g Y ) = 0 for almost all g G.

75 Orbit category 1 A : Hom-finite Krull-Schmidt additive k-category. 2 G: group acting on A such, for objects X, Y, that A(X, g Y ) = 0 for almost all g G. 3 Define G-orbit category A/G as follows.

76 Orbit category 1 A : Hom-finite Krull-Schmidt additive k-category. 2 G: group acting on A such, for objects X, Y, that A(X, g Y ) = 0 for almost all g G. 3 Define G-orbit category A/G as follows. The objects of A/G are those of A;

77 Orbit category 1 A : Hom-finite Krull-Schmidt additive k-category. 2 G: group acting on A such, for objects X, Y, that A(X, g Y ) = 0 for almost all g G. 3 Define G-orbit category A/G as follows. The objects of A/G are those of A; (A/G)(X, Y ) = g G A(X, g Y ).

78 Orbit category 1 A : Hom-finite Krull-Schmidt additive k-category. 2 G: group acting on A such, for objects X, Y, that A(X, g Y ) = 0 for almost all g G. 3 Define G-orbit category A/G as follows. The objects of A/G are those of A; (A/G)(X, Y ) = g G A(X, g Y ). Lemma A/G is Hom-finite Krull-Schmidt additive k-category with a canonical embedding σ : A A/G : X X ; f f.

79 G-covering Definition A k-linear functor π : A B is called G-covering provided

80 G-covering Definition A k-linear functor π : A B is called G-covering provided commutative diagram A σ A/G π B.

81 Cluster structure Assume now that A is a triangulated k-category.

82 Cluster structure Assume now that A is a triangulated k-category. A non-empty collection C of subcategories of A is called cluster structure provided, for any C C, that

83 Cluster structure Assume now that A is a triangulated k-category. A non-empty collection C of subcategories of A is called cluster structure provided, for any C C, that 1 The quiver Q C of C has no loop or 2-cycle;

84 Cluster structure Assume now that A is a triangulated k-category. A non-empty collection C of subcategories of A is called cluster structure provided, for any C C, that 1 The quiver Q C of C has no loop or 2-cycle; 2 for any M indc,! M inda ( = M) such that µ M (C) := add(c M, M ) lies in C;

85 Cluster structure Assume now that A is a triangulated k-category. A non-empty collection C of subcategories of A is called cluster structure provided, for any C C, that 1 The quiver Q C of C has no loop or 2-cycle; 2 for any M indc,! M inda ( = M) such that µ M (C) := add(c M, M ) lies in C; 3 Q µm (C) is obtained from Q C by FZ-mutation at M;

86 Cluster structure Assume now that A is a triangulated k-category. A non-empty collection C of subcategories of A is called cluster structure provided, for any C C, that 1 The quiver Q C of C has no loop or 2-cycle; 2 for any M indc,! M inda ( = M) such that µ M (C) := add(c M, M ) lies in C; 3 Q µm (C) is obtained from Q C by FZ-mutation at M; 4 There exist in A two exact triangles : M f N g M M[1]; M u L v M M [1], where f, u are minimal left C M -approximations; g, v are minimal right C M -approximations.

87 Cluster tilting subcategories Definition (Koenig, Zhu) A subcategory C of A is called cluster-tilting provided that

88 Cluster tilting subcategories Definition (Koenig, Zhu) A subcategory C of A is called cluster-tilting provided that 1 C is functorially finite in A;

89 Cluster tilting subcategories Definition (Koenig, Zhu) A subcategory C of A is called cluster-tilting provided that 1 C is functorially finite in A; 2 For any object X A, we have Hom A (C, X [1]) = 0 X C Hom A (X, C[1]) = 0.

90 Cluster categories Definition A triangulated category A is called cluster category if it admits a cluster structure.

91 Cluster categories Definition A triangulated category A is called cluster category if it admits a cluster structure. Theorem (Buan, Iyama, Reiten, Scott) If A is 2-CY category with cluster tilting subcategories, then

92 Cluster categories Definition A triangulated category A is called cluster category if it admits a cluster structure. Theorem (Buan, Iyama, Reiten, Scott) If A is 2-CY category with cluster tilting subcategories, then cluster tilting subcategories in A form a cluster structure

93 Cluster categories Definition A triangulated category A is called cluster category if it admits a cluster structure. Theorem (Buan, Iyama, Reiten, Scott) If A is 2-CY category with cluster tilting subcategories, then cluster tilting subcategories in A form a cluster structure no loop or 2-cycle in quiver of any cluster tilting subcategory.

94 2-CY category associated with hereditary category 1 Let H be Hom-finite, hereditary, abelian k-category having AR-sequences.

95 2-CY category associated with hereditary category 1 Let H be Hom-finite, hereditary, abelian k-category having AR-sequences. 2 Then D b (H) has AR-triangles, and its AR-translation is auto-equivalence.

96 2-CY category associated with hereditary category 1 Let H be Hom-finite, hereditary, abelian k-category having AR-sequences. 2 Then D b (H) has AR-triangles, and its AR-translation is auto-equivalence. 3 Let D b (H) be full subcategory of D b (H), containing exactly one object of each isoclass of objects of D b (H).

97 2-CY category associated with hereditary category 1 Let H be Hom-finite, hereditary, abelian k-category having AR-sequences. 2 Then D b (H) has AR-triangles, and its AR-translation is auto-equivalence. 3 Let D b (H) be full subcategory of D b (H), containing exactly one object of each isoclass of objects of D b (H). 4 Then D b (H) has AR-triangles, and its AR-translation τ D is automorphism.

98 2-CY category associated with hereditary category 1 Let H be Hom-finite, hereditary, abelian k-category having AR-sequences. 2 Then D b (H) has AR-triangles, and its AR-translation is auto-equivalence. 3 Let D b (H) be full subcategory of D b (H), containing exactly one object of each isoclass of objects of D b (H). 4 Then D b (H) has AR-triangles, and its AR-translation τ D is automorphism. Theorem(Keller) Setting F = τ 1 D is 2-CY triangulated category. [1], the canonical orbit category C (H) = D b (H)/< F >

99 Known cluster categories from hereditary category Theorem Let Q be a locally finite quiver without infinite paths.

100 Known cluster categories from hereditary category Theorem Let Q be a locally finite quiver without infinite paths. 1 Then rep(q) has AR-sequences.

101 Known cluster categories from hereditary category Theorem Let Q be a locally finite quiver without infinite paths. 1 Then rep(q) has AR-sequences. 2 The cluster tilting subcategories in C (rep(q)) form a cluster structure in case Q is

102 Known cluster categories from hereditary category Theorem Let Q be a locally finite quiver without infinite paths. 1 Then rep(q) has AR-sequences. 2 The cluster tilting subcategories in C (rep(q)) form a cluster structure in case Q is finite (BMRRT);

103 Known cluster categories from hereditary category Theorem Let Q be a locally finite quiver without infinite paths. 1 Then rep(q) has AR-sequences. 2 The cluster tilting subcategories in C (rep(q)) form a cluster structure in case Q is finite (BMRRT); of type A or A (Liu, Paquette);

104 Known cluster categories from hereditary category Theorem Let Q be a locally finite quiver without infinite paths. 1 Then rep(q) has AR-sequences. 2 The cluster tilting subcategories in C (rep(q)) form a cluster structure in case Q is finite (BMRRT); of type A or A (Liu, Paquette); of type D (Yang).

105 1 Consider the valued quiver C : 0 (2,1) and its following R-species : C : C C R R R R R R R R

106 1 Consider the valued quiver C : 0 (2,1) and its following R-species : C : C C R R R R R R R R 2 We shall show that C (rep(c )) is a cluster category.

107 Recall the following covering: A : x 0 x 1 x 2 x 1 x 2 A : R R R R R R R R R R R ϕ Φ C : 0 (2,1) 1 2 C : C C R R R R

108 Group action on C (A ) Consider the following automorphism σ : A A : x n x n.

109 Group action on C (A ) Consider the following automorphism Then < σ >= {e, σ} := G. σ : A A : x n x n.

110 Group action on C (A ) Consider the following automorphism σ : A A : x n x n. Then < σ >= {e, σ} := G. Remark The G-action on A G-action on rep(a );

111 Group action on C (A ) Consider the following automorphism σ : A A : x n x n. Then < σ >= {e, σ} := G. Remark The G-action on A G-action on rep(a ); G-action on C (A );

112 Group action on C (A ) Consider the following automorphism σ : A A : x n x n. Then < σ >= {e, σ} := G. Remark The G-action on A G-action on rep(a ); G-action on C (A ); G-action on Γ C (A ).

113 Properties of the cluster category C (A ) Theorem (Liu, Paquette) The AR-quiver Γ C (A ) of C (A ) consists of

114 Properties of the cluster category C (A ) Theorem (Liu, Paquette) The AR-quiver Γ C (A ) of C (A ) consists of a connecting component C A ( = ZA ) ;

115 Properties of the cluster category C (A ) Theorem (Liu, Paquette) The AR-quiver Γ C (A ) of C (A ) consists of a connecting component C A ( = ZA ) ; two orthogonal regular components R R, R L ( = ZA ).

116 Properties of the cluster category C (A ) Theorem (Liu, Paquette) The AR-quiver Γ C (A ) of C (A ) consists of a connecting component C A ( = ZA ) ; two orthogonal regular components R R, R L ( = ZA ). Observation Consider the σ-action on Γ C (A ).

117 Properties of the cluster category C (A ) Theorem (Liu, Paquette) The AR-quiver Γ C (A ) of C (A ) consists of a connecting component C A ( = ZA ) ; two orthogonal regular components R R, R L ( = ZA ). Observation Consider the σ-action on Γ C (A ). 1 σ R R = R L.

118 Properties of the cluster category C (A ) Theorem (Liu, Paquette) The AR-quiver Γ C (A ) of C (A ) consists of a connecting component C A ( = ZA ) ; two orthogonal regular components R R, R L ( = ZA ). Observation Consider the σ-action on Γ C (A ). 1 σ R R = R L. 2 The σ-action on C A is reflection across τ C -orbit of P[x 0 ].

119 Induced G-coverings Theorem Consider the species covering Φ : A C.

120 Induced G-coverings Theorem Consider the species covering Φ : A C. 1 The push-down Φ λ : rep(a ) rep(c ) is G-covering.

121 Induced G-coverings Theorem Consider the species covering Φ : A C. 1 The push-down Φ λ : rep(a ) rep(c ) is G-covering. 2 The push-down Φ λ induces G-covering Φ D λ : D b (rep(a )) D b (rep(c )),

122 Induced G-coverings Theorem Consider the species covering Φ : A C. 1 The push-down Φ λ : rep(a ) rep(c ) is G-covering. 2 The push-down Φ λ induces G-covering Φ D λ : D b (rep(a )) D b (rep(c )), which induces G-covering Φ C λ : C (rep(a )) C (rep(c )).

123 Induced G-coverings Theorem Consider the species covering Φ : A C. 1 The push-down Φ λ : rep(a ) rep(c ) is G-covering. 2 The push-down Φ λ induces G-covering Φ D λ : D b (rep(a )) D b (rep(c )), which induces G-covering Φ C λ : C (rep(a )) C (rep(c )). 3 The functor Φ C λ induces valued translation quiver covering Φ C λ : Γ C (rep(a )) Γ C (rep(c )).

124 The cluster category C (rep(c )) Theorem 1 The cluster tilting subcategories in C (rep(c )) form a cluster structure.

125 The cluster category C (rep(c )) Theorem 1 The cluster tilting subcategories in C (rep(c )) form a cluster structure. 2 The AR-quiver Γ C (rep(c )) of C (rep(c )) consists of

126 The cluster category C (rep(c )) Theorem 1 The cluster tilting subcategories in C (rep(c )) form a cluster structure. 2 The AR-quiver Γ C (rep(c )) of C (rep(c )) consists of a connecting component C C ( = ZC ), obtained by folding C A along the τ C -orbit of P[x 0 ] ;

127 The cluster category C (rep(c )) Theorem 1 The cluster tilting subcategories in C (rep(c )) form a cluster structure. 2 The AR-quiver Γ C (rep(c )) of C (rep(c )) consists of a connecting component C C ( = ZC ), obtained by folding C A along the τ C -orbit of P[x 0 ] ; a regular component R ( = ZA ), obtained by identifying R R with R L.

128 The cluster category C (rep(c )) Theorem 1 The cluster tilting subcategories in C (rep(c )) form a cluster structure. 2 The AR-quiver Γ C (rep(c )) of C (rep(c )) consists of a connecting component C C ( = ZC ), obtained by folding C A along the τ C -orbit of P[x 0 ] ; a regular component R ( = ZA ), obtained by identifying R R with R L. Remark In a similar fashion, we can construct a cluster category of type B.

129 The infinite strip with marked points Consider the strip in the plane S[ 1, 1] = {(x, y) R 2 1 y 1} with marked points: l n = (n, 1), r i = ( n, 1); n Z.

130 The infinite strip with marked points Consider the strip in the plane S[ 1, 1] = {(x, y) R 2 1 y 1} with marked points: l n = (n, 1), r i = ( n, 1); n Z. l 6 l 5 l 4 l 3 l 2 l 1 l 0 l 1 l 2 l 3 l 4 l 5 r 6 r 5 r 4 r 3 r 2 r 1 r 0 r 1 r 2 r 3 r 4 r 5

131 Arcs in S[ 1, 1] There exist in S[ 1, 1] three types of arcs: upper arcs : [l i, l j ] with i j > 1; lower arcs : [r i, r j ] with i j > 1; connecting arcs : [l i, r j ] with i, j Z. l 6 l 5 l 4 l 3 l 2 l 1 l 0 l 1 l 2 l 3 l 4 l 5 r 6 r 5 r 4 r 3 r 2 r 1 r 0 r 1 r 2 r 3 r 4 r 5

132 Geometric realization of C (rep(a )) Theorem (Liu, Paquette) C (rep(a )) S[ 1, 1] {objects in R L } {objects in R R } {objects in C A } {upper arcs} {lower arcs} {connecting arcs}

133 Geometric realization of C (rep(a )) Theorem (Liu, Paquette) C (rep(a )) S[ 1, 1] {objects in R L } {objects in R R } {objects in C A } {maximal rigid subcategories} {upper arcs} {lower arcs} {connecting arcs} {triangulations}

134 Geometric realization of C (rep(a )) Theorem (Liu, Paquette) C (rep(a )) S[ 1, 1] {objects in R L } {objects in R R } {objects in C A } {maximal rigid subcategories} {cluster tilting subcategories} {upper arcs} {lower arcs} {connecting arcs} {triangulations} {compact triangulations}

135 The twisted strip with marked points 1 The group G =< σ > acts on S[ 1, 1], with σ acting as rotation around the origin of angle π.

136 The twisted strip with marked points 1 The group G =< σ > acts on S[ 1, 1], with σ acting as rotation around the origin of angle π. 2 The twisted strip is the quotient S[0, 1] = S[ 1, 1]/G.

137 Arcs in the twisted strip 1 S[0, 1] has a fundamental domain {(x, y) R 2 0 y 1}\{(x, 0) x < 0} with marked points l i = (i, 1); i Z.

138 Arcs in the twisted strip 1 S[0, 1] has a fundamental domain {(x, y) R 2 0 y 1}\{(x, 0) x < 0} with marked points l i = (i, 1); i Z. 2 There exist two types of arcs:

139 Arcs in the twisted strip 1 S[0, 1] has a fundamental domain {(x, y) R 2 0 y 1}\{(x, 0) x < 0} with marked points l i = (i, 1); i Z. 2 There exist two types of arcs: upper arcs : [l i, l j ], with i j 2, not passing through the origin O;

140 Arcs in the twisted strip 1 S[0, 1] has a fundamental domain {(x, y) R 2 0 y 1}\{(x, 0) x < 0} with marked points l i = (i, 1); i Z. 2 There exist two types of arcs: upper arcs : [l i, l j ], with i j 2, not passing through the origin O; connecting arcs : [l i, O, l j ], i, j Z, passing through the origin O.

141 Geometric realization of C (rep(c )) Theorem C (rep(c )) S[0, 1]

142 Geometric realization of C (rep(c )) Theorem C (rep(c )) S[0, 1] {objects in R} {upper arcs}

143 Geometric realization of C (rep(c )) Theorem C (rep(c )) S[0, 1] {objects in R} {objects in C C } {upper arcs} {connecting arcs}

144 Geometric realization of C (rep(c )) Theorem C (rep(c )) S[0, 1] {objects in R} {objects in C C } {maximal rigid subcategories} {upper arcs} {connecting arcs} {triangulations}

145 Geometric realization of C (rep(c )) Theorem C (rep(c )) S[0, 1] {objects in R} {objects in C C } {maximal rigid subcategories} {cluster tilting subcategories} {upper arcs} {connecting arcs} {triangulations} {compact triangulations}

146 Match between algebraic covering and topological covering C (A ) geometric realization S[ 1, 1] Φ C λ π C (C ) geometric realization S[0, 1]