Action of finite groups on (generalized) cluster categories Laurent Demonet Max Planck Institut f¨ ur Mathematik - Germany 1 / 19
� � triangulated or exact 2-Calabi-Yau category categorification cluster characters (MRZ, BMRRT, (CC, CK, GLS, P, GLS, . . . ) FK, . . . ) Skew-symmetric cluster algebra 2 / 19
� � � � 2-Calabi-Yau category triangulated or exact with an action 2-Calabi-Yau category of a finite group categorification cluster characters (MRZ, BMRRT, (CC, CK, GLS, P, categorification cluster character GLS, . . . ) FK, . . . ) Skew-symmetric Skew-symmetrizable cluster algebra cluster algebra 2 / 19
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Equivariant category Let k be an algebraically closed field and C a k -category. Let G be a finite group such that char k does not divide # G . 4 / 19
Equivariant category Let k be an algebraically closed field and C a k -category. Let G be a finite group such that char k does not divide # G . Definition (category G = mod Fun( G )) simple objects : { g | g ∈ G } ; morphisms : Hom G ( g , h ) = k δ gh ; tensor product : g ⊗ h = gh . 4 / 19
Equivariant category Let k be an algebraically closed field and C a k -category. Let G be a finite group such that char k does not divide # G . Definition (category G = mod Fun( G )) simple objects : { g | g ∈ G } ; morphisms : Hom G ( g , h ) = k δ gh ; tensor product : g ⊗ h = gh . Definition (action of G over C ) Structure of G -module category on C . 4 / 19
Equivariant category Definition (Equivariant object ( X , ψ )) X ∈ C ; ( ψ g ) g ∈ G with ψ g : g ⊗ X − → X . 5 / 19
� � Equivariant category Definition (Equivariant object ( X , ψ )) X ∈ C ; ( ψ g ) g ∈ G with ψ g : g ⊗ X − → X . Id g ⊗ ψ h � g ⊗ X g ⊗ ( h ⊗ X ) α ψ g ψ gh � X gh ⊗ X 5 / 19
� � � � Equivariant category Definition (Equivariant object ( X , ψ )) X ∈ C ; ( ψ g ) g ∈ G with ψ g : g ⊗ X − → X . Id g ⊗ ψ h � g ⊗ X g ⊗ ( h ⊗ X ) α ψ g ψ gh � X gh ⊗ X Definition (morphism f from ( X , ψ ) to ( Y , χ )) ψ g � X g ⊗ X Id g ⊗ f f χ g � Y g ⊗ Y 5 / 19
Equivariant category Proposition C G is a mod k [ G ] -module category. 6 / 19
Equivariant category Proposition C G is a mod k [ G ] -module category. Example If C is the category of k -vector spaces, then C G ≃ mod k [ G ]. 6 / 19
Equivariant category Proposition C G is a mod k [ G ] -module category. Example If C is the category of k -vector spaces, then C G ≃ mod k [ G ]. Q = 1 ← 2 → 1 ′ G = Z / 2 Z 6 / 19
� � � Equivariant category Proposition C G is a mod k [ G ] -module category. Example If C is the category of k -vector spaces, then C G ≃ mod k [ G ]. Q = 1 ← 2 → 1 ′ G = Z / 2 Z ( X , ψ ) ∈ (mod kQ ) G indecomposable : 1 ⊕ 1 ′ 2 , Id 2 , − Id ⊕ 2 2 � � � � � 1 1 ′ 2 2 � � 1 ′ , ψ ′ � � 1 ′ , ψ � � � � � � 1 1 6 / 19
Case of exact categories Suppose that: Q is a dynkin quiver; Λ is the preprojective algebra of Q ; C = mod Λ; G acts on Q . 7 / 19
Case of exact categories Suppose that: Q is a dynkin quiver; Λ is the preprojective algebra of Q ; C = mod Λ; G acts on Q . Then: C is exact, Hom-finite, Krull-Schmidt; C is stably 2-Calabi-Yau (Ext 1 C ( X , Y ) ≃ Ext 1 C ( Y , X ) ∗ ); C G has the same properties. 7 / 19
Case of exact categories Suppose that: Q is a dynkin quiver; Λ is the preprojective algebra of Q ; C = mod Λ; G acts on Q . Then: C is exact, Hom-finite, Krull-Schmidt; C is stably 2-Calabi-Yau (Ext 1 C ( X , Y ) ≃ Ext 1 C ( Y , X ) ∗ ); C G has the same properties. (can be generalized to other exact categories) 7 / 19
Equivariant category Notation Add ( C ) = {T ⊂ C additive, full, stable under isomorphisms and direct summand } 8 / 19
Equivariant category Notation Add ( C ) = {T ⊂ C additive, full, stable under isomorphisms and direct summand } Proposition {T ∈ Add ( C ) G-stable } F ← →{T ∈ Add ( C G ) mod k [ G ] -stable } Notation (forgetful functor) F : C G → C ( X , ψ ) �→ X 8 / 19
Stable cluster-tilting subcategories Definition (stable cluster-tilting subcategory) T ∈ Add ( C ) (resp. ∈ Add ( C G )) is G -stable (resp. mod k [ G ]-stable) cluster-tilting if 9 / 19
Stable cluster-tilting subcategories Definition (stable cluster-tilting subcategory) T ∈ Add ( C ) (resp. ∈ Add ( C G )) is G -stable (resp. mod k [ G ]-stable) cluster-tilting if ∀ X ∈ C , X ∈ T ⇔ ∀ Y ∈ T , Ext 1 C (resp. C G ) ( X , Y ) = 0; 9 / 19
Stable cluster-tilting subcategories Definition (stable cluster-tilting subcategory) T ∈ Add ( C ) (resp. ∈ Add ( C G )) is G -stable (resp. mod k [ G ]-stable) cluster-tilting if ∀ X ∈ C , X ∈ T ⇔ ∀ Y ∈ T , Ext 1 C (resp. C G ) ( X , Y ) = 0; G ⊗ T = T (resp. mod k [ G ] ⊗ T = T ). 9 / 19
Stable cluster-tilting subcategories Definition (stable cluster-tilting subcategory) T ∈ Add ( C ) (resp. ∈ Add ( C G )) is G -stable (resp. mod k [ G ]-stable) cluster-tilting if ∀ X ∈ C , X ∈ T ⇔ ∀ Y ∈ T , Ext 1 C (resp. C G ) ( X , Y ) = 0; G ⊗ T = T (resp. mod k [ G ] ⊗ T = T ). Proposition {T ∈ Add ( C ) G-stable cluster-tilting } F ← →{T ∈ Add ( C G ) mod k [ G ] -stable cluster-tilting } 9 / 19
Stable cluster-tilting subcategories Definition ( G -loop and G -2-cycle) If T ∈ Add ( C ) G , G -loop : X → g ⊗ X irreducible (in T ) ; 10 / 19
Stable cluster-tilting subcategories Definition ( G -loop and G -2-cycle) If T ∈ Add ( C ) G , G -loop : X → g ⊗ X irreducible (in T ) ; G -2-cycle : X → Y → g ⊗ X irreducible. 10 / 19
Stable cluster-tilting subcategories Definition ( G -loop and G -2-cycle) If T ∈ Add ( C ) G , G -loop : X → g ⊗ X irreducible (in T ) ; G -2-cycle : X → Y → g ⊗ X irreducible. (same definitions in C G ) 10 / 19
Stable cluster-tilting subcategories Definition ( G -loop and G -2-cycle) If T ∈ Add ( C ) G , G -loop : X → g ⊗ X irreducible (in T ) ; G -2-cycle : X → Y → g ⊗ X irreducible. (same definitions in C G ) Lemma T ∈ Add ( C G ) mod k [ G ] has no mod k [ G ] -loop ⇔ F T has no G-loop. T ∈ Add ( C G ) mod k [ G ] has no mod k [ G ] - 2 -cycle ⇔ F T has no G- 2 -cycle. 10 / 19
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