Action of finite groups on (generalized) cluster categories Laurent - - PowerPoint PPT Presentation

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Action of finite groups on (generalized) cluster categories Laurent - - PowerPoint PPT Presentation

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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,

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Action of finite groups on (generalized) cluster categories

Laurent Demonet Max Planck Institut f¨ ur Mathematik - Germany

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triangulated or exact 2-Calabi-Yau category

cluster characters (CC, CK, GLS, P, FK, . . . )

  • Skew-symmetric

cluster algebra

categorification (MRZ, BMRRT, GLS, . . . )

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triangulated or exact 2-Calabi-Yau category

cluster characters (CC, CK, GLS, P, FK, . . . )

  • 2-Calabi-Yau category

with an action

  • f a finite group

cluster character

  • Skew-symmetric

cluster algebra

categorification (MRZ, BMRRT, GLS, . . . )

  • Skew-symmetrizable

cluster algebra

categorification

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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.

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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.

Definition (category G = mod Fun(G))

simple objects : {g | g ∈ G} ; morphisms : HomG(g, h) = kδgh ; tensor product : g ⊗ h = gh.

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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.

Definition (category G = mod Fun(G))

simple objects : {g | g ∈ G} ; morphisms : HomG(g, h) = kδgh ; tensor product : g ⊗ h = gh.

Definition (action of G over C)

Structure of G-module category on C.

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Equivariant category

Definition (Equivariant object (X, ψ))

X ∈ C ; (ψg)g∈G with ψg : g ⊗ X − → X.

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Equivariant category

Definition (Equivariant object (X, ψ))

X ∈ C ; (ψg)g∈G with ψg : g ⊗ X − → X. g ⊗ (h ⊗ X)

Idg ⊗ψh

g ⊗ X

ψg

  • gh ⊗ X

α

  • ψgh

X

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Equivariant category

Definition (Equivariant object (X, ψ))

X ∈ C ; (ψg)g∈G with ψg : g ⊗ X − → X. g ⊗ (h ⊗ X)

Idg ⊗ψh

g ⊗ X

ψg

  • gh ⊗ X

α

  • ψgh

X

Definition (morphism f from (X, ψ) to (Y , χ))

g ⊗ X

Idg ⊗f

  • ψg

X

f

  • g ⊗ Y

χg

Y

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Equivariant category

Proposition

CG is a mod k[G]-module category.

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Equivariant category

Proposition

CG is a mod k[G]-module category.

Example

If C is the category of k-vector spaces, then CG ≃ mod k[G].

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Equivariant category

Proposition

CG is a mod k[G]-module category.

Example

If C is the category of k-vector spaces, then CG ≃ mod k[G]. Q = 1 ← 2 → 1′ G = Z/2Z

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Equivariant category

Proposition

CG is a mod k[G]-module category.

Example

If C is the category of k-vector spaces, then CG ≃ mod k[G]. Q = 1 ← 2 → 1′ G = Z/2Z (X, ψ) ∈ (mod kQ)G indecomposable : 1 ⊕ 1′ 2, Id 2, − Id 2

  • 1

⊕ 2

  • 1′

2

  • 1

1′ , ψ′ 2

  • 1

1′ , ψ

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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.

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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 (Ext1

C(X, Y ) ≃ Ext1 C(Y , X)∗);

CG has the same properties.

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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 (Ext1

C(X, Y ) ≃ Ext1 C(Y , X)∗);

CG has the same properties. (can be generalized to other exact categories)

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Equivariant category

Notation

Add(C) = {T ⊂ C additive, full, stable under isomorphisms and direct summand}

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Equivariant category

Notation

Add(C) = {T ⊂ C additive, full, stable under isomorphisms and direct summand}

Proposition

{T ∈ Add(C) G-stable}

F

← →{T ∈ Add(CG) mod k[G]-stable}

Notation (forgetful functor)

F : CG → C (X, ψ) → X

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Stable cluster-tilting subcategories

Definition (stable cluster-tilting subcategory)

T ∈ Add(C) (resp. ∈ Add(CG)) is G-stable (resp. mod k[G]-stable) cluster-tilting if

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Stable cluster-tilting subcategories

Definition (stable cluster-tilting subcategory)

T ∈ Add(C) (resp. ∈ Add(CG)) is G-stable (resp. mod k[G]-stable) cluster-tilting if ∀X ∈ C, X ∈ T ⇔ ∀Y ∈ T , Ext1

C (resp. CG)(X, Y ) = 0;

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Stable cluster-tilting subcategories

Definition (stable cluster-tilting subcategory)

T ∈ Add(C) (resp. ∈ Add(CG)) is G-stable (resp. mod k[G]-stable) cluster-tilting if ∀X ∈ C, X ∈ T ⇔ ∀Y ∈ T , Ext1

C (resp. CG)(X, Y ) = 0;

G ⊗ T = T (resp. mod k[G] ⊗ T = T ).

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Stable cluster-tilting subcategories

Definition (stable cluster-tilting subcategory)

T ∈ Add(C) (resp. ∈ Add(CG)) is G-stable (resp. mod k[G]-stable) cluster-tilting if ∀X ∈ C, X ∈ T ⇔ ∀Y ∈ T , Ext1

C (resp. CG)(X, Y ) = 0;

G ⊗ T = T (resp. mod k[G] ⊗ T = T ).

Proposition

{T ∈ Add(C) G-stable cluster-tilting}

F

← →{T ∈ Add(CG) mod k[G]-stable cluster-tilting}

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Stable cluster-tilting subcategories

Definition (G-loop and G-2-cycle)

If T ∈ Add(C)G, G-loop : X → g ⊗ X irreducible (in T ) ;

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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.

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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 CG)

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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 CG)

Lemma

T ∈ Add(CG)mod k[G] has no mod k[G]-loop ⇔ FT has no G-loop. T ∈ Add(CG)mod k[G] has no mod k[G]-2-cycle ⇔ FT has no G-2-cycle.

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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 CG)

Lemma

T ∈ Add(CG)mod k[G] has no mod k[G]-loop ⇔ FT has no G-loop. T ∈ Add(CG)mod k[G] has no mod k[G]-2-cycle ⇔ FT has no G-2-cycle.

Proposition

There exist cluster-tilting T ∈ Add(CG)mod k[G]; Such T have neither mod k[G]-loop nor mod k[G]-2-cycle.

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Mutations

Theorem

Let T ′ ∈ Add(CG)mod k[G] and X / ∈ T ′ non projective and indecomposable such that add(T ′, k[G] ⊗ X) is mod k[G]-stable cluster-tilting. ∃!0 → X

f

− → T

g

− → Y → 0 such that f is a minimal left T ′-approximation ;

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Mutations

Theorem

Let T ′ ∈ Add(CG)mod k[G] and X / ∈ T ′ non projective and indecomposable such that add(T ′, k[G] ⊗ X) is mod k[G]-stable cluster-tilting. ∃!0 → X

f

− → T

g

− → Y → 0 such that f is a minimal left T ′-approximation ; Y / ∈ T ′ is non projective and indecomposable such that add(T ′, k[G] ⊗ Y ) is mod k[G]-stable cluster-tilting.

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Mutations

Definition

µX(add(T0, k[G] ⊗ X)) = add(T0, k[G] ⊗ Y )

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Mutations

Definition

µX(add(T0, k[G] ⊗ X)) = add(T0, k[G] ⊗ Y )

Proposition

µY (µX(add(T0, k[G] ⊗ X))) = add(T0, k[G] ⊗ X)

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Mutations

Definition

µX(add(T0, k[G] ⊗ X)) = add(T0, k[G] ⊗ Y )

Proposition

µY (µX(add(T0, k[G] ⊗ X))) = add(T0, k[G] ⊗ X)

Proposition

0 → X → T → Y → 0 0 → Y → T ′ → X → 0 add(T) ∩ add(T ′) = {0}

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Exchange matrix

Rows of B(T ) are indexed by G-orbit of indecomposable objects of T . Columns of B(T ) are indexed by G-orbit of non-projective indecomposable

  • bjects of T .

B(T )X,Y =

  • X ′∈X

(#{X ′ → Y irred. in T } − #{Y → X ′ irred. in T }).

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Exchange matrix

Rows of B(T ) are indexed by G-orbit of indecomposable objects of T . Columns of B(T ) are indexed by G-orbit of non-projective indecomposable

  • bjects of T .

B(T )X,Y =

  • X ′∈X

(#{X ′ → Y irred. in T } − #{Y → X ′ irred. in T }).

Proposition

B(µX(T )) = µX(B(T ))

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Cluster character

Iso(CG)/ mod k[G] Iso(C)

[FK]

  • [GLS]
  • Iso(C)/G
  • X
  • X
  • PX

PX

C

  • x±1

i

  • i∈I

C

  • x±1

i

  • i∈I/G

C [N0]

C [N]

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Cluster character

n = #{mod k[G]-stable indecomposable objects in T }

Theorem

∃CG/ mod k[G] → C

  • x±1

1 , x±1 2 , . . . , x±1 n

  • X → PX

such that PX⊕Y = PXPY and if 0 → X → Z → Y → 0 and 0 → Y → Z ′ → X → 0 are mutation sequences, then PXPY = PZ + PZ

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Cluster character

n = #{mod k[G]-stable indecomposable objects in T }

Theorem

∃CG/ mod k[G] → C

  • x±1

1 , x±1 2 , . . . , x±1 n

  • X → PX

such that PX⊕Y = PXPY and if 0 → X → Z → Y → 0 and 0 → Y → Z ′ → X → 0 are mutation sequences, then PXPY = PZ + PZ

′.

Corollary

T →

  • {PX | X ∈ ind(T )}, B(T )
  • realizes a cluster algebra. In our

exemple, it is of the form C[N].

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Cluster character

n = #{mod k[G]-stable indecomposable objects in T }

Theorem

∃CG/ mod k[G] → C

  • x±1

1 , x±1 2 , . . . , x±1 n

  • X → PX

such that PX⊕Y = PXPY and if 0 → X → Z → Y → 0 and 0 → Y → Z ′ → X → 0 are mutation sequences, then PXPY = PZ + PZ

′.

Corollary

T →

  • {PX | X ∈ ind(T )}, B(T )
  • realizes a cluster algebra. In our

exemple, it is of the form C[N].

Theorem

Cluster monomials are linearly independent (if B(T ) is full rank).

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Case of generalized cluster algebras

Aim: construct an analogue of what precedes for some triangulated categories.

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Case of generalized cluster algebras

Aim: construct an analogue of what precedes for some triangulated categories.

idea

C a 2-Calabi-Yau Hom-finite triangulated category and G a finite group acting on C. A triangulated structure on CG such that F : CG → C is a triangulated functor and CG is also 2-Calabi-Yau.

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Case of generalized cluster algebras

Let A be a dg k-algebra and G a group acting on A such that char A does not divide #G.

Definition (Skew group algebra)

AG = A, G | gag−1 = g · ak−alg.

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Case of generalized cluster algebras

Let A be a dg k-algebra and G a group acting on A such that char A does not divide #G.

Definition (Skew group algebra)

AG = A, G | gag−1 = g · ak−alg. Denote Ae = A ⊗ Aop. Suppose that A is homologically smooth (A ∈ per Ae); for p > 0, HpA = 0; dim H0A < ∞; A is bimodule G-3-Calabi-Yau that is RHommod Ae(A, Ae) ≃D(AeG) A[−3].

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Case of generalized cluster algebras

Let A be a dg k-algebra and G a group acting on A such that char A does not divide #G.

Definition (Skew group algebra)

AG = A, G | gag−1 = g · ak−alg. Denote Ae = A ⊗ Aop. Suppose that A is homologically smooth (A ∈ per Ae); for p > 0, HpA = 0; dim H0A < ∞; A is bimodule G-3-Calabi-Yau that is RHommod Ae(A, Ae) ≃D(AeG) A[−3].

Proposition

AG is also homologically smooth. Both A and AG are bimodule 3-Calabi-Yau.

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Case of generalized cluster algebras

Definition (Amiot)

Generalized cluster category: CA = per A/DbA; If (Q, W ) is a Jacobi-finite quiver with potential, C(Q,W ) = CA where A is the Ginzburg dg algebra of (Q, W ).

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Case of generalized cluster algebras

Definition (Amiot)

Generalized cluster category: CA = per A/DbA; If (Q, W ) is a Jacobi-finite quiver with potential, C(Q,W ) = CA where A is the Ginzburg dg algebra of (Q, W ).

Proposition (after Keller)

If G acts on the quiver Q and stabilizes the potential W then the Ginzburg dg algebra satisfies the previous conditions.

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Case of generalized cluster algebras

Definition (Amiot)

Generalized cluster category: CA = per A/DbA; If (Q, W ) is a Jacobi-finite quiver with potential, C(Q,W ) = CA where A is the Ginzburg dg algebra of (Q, W ).

Proposition (after Keller)

If G acts on the quiver Q and stabilizes the potential W then the Ginzburg dg algebra satisfies the previous conditions.

Proposition (Amiot)

CA is Hom-finite and 2-Calabi-Yau. The image of A in CA is a cluster tilting object with endomorphism algebra H0A.

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Case of generalized cluster algebras

Proposition

There is a commutative diagram: CBΓ

∼

  • ι∗
  • CBΓ

F

  • CB

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Case of generalized cluster algebras

Proposition

There is a commutative diagram: CBΓ

∼

  • ι∗
  • CBΓ

F

  • CB

Questions

For which dg algebra A and which action does the mutation work (without 2-cycles)? In particular, when does a “G-generic potential” exist? Are there obstacles to construct equivariant d-cluster categories?

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