A non-simply laced version for cluster structures on 2-Calabi-Yau categories

arX iv:0910.5077v2 [math.RT] 6 Dec 2013 A non-simply laced version fo r cluster str uctures on 2-Cala bi-Y au catego ries Bertrand Nguefack 1 1 Universit y of Y aoun de I; b.nguefack@uy1.uninet.cm ngc.bertrand@gmail.com Beamer versi on of the a rticle published online a t Journal of Pure and Applied Algebra DOI: 10.10 16/j.jpaa.20 13.11.027 B. Nguefack (Univ Y aounde I) Non-simply laced cluster structures December 20 13 1 / 15 Extende d abstract Somma ry 1 Extended a bstract 2 Some pr eliminar ies on t riangulated and exact categor ies 3 Minimal ap pro ximatio n sequences Minimal map s, a p pro ximatio n sequences Exchange sequences and irreducible maps 4 Main statements This pap er investigates a non simply-laced version of cluster structures for 2-Calabi-Y au o r stably 2- Calab i-Y au cat ego ries over a rbitrary fields. It results t h at 2-Calabi-Y au or stably 2- Calab i-Y au cat ego ries having a cluster tilting sub category with neither loops n o r 2-cycles do have the generalized version of cluster structure. This is in p articula r the case of cluster categor ies over non-algebraical ly closed fields. Keywo rds Calabi-Y au cat ego ry , cluster structure, mo d ulated quivers, cluster algebra. MSC (2010 ) Primary: 16G70; Secondary : 18E30, 18E10, 13F60 , 12E15 B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 2 / 15 Extende d abstract Brief intro duction The theo ry of cluster algebr as, intro duced by Fomi n and Zelevinsky , is connected with many areas of math ematics. In the representation t h eo ry of algebras, the philosophy h as b een to prov ide a categori cal interpretation of the main combinato rics, called mutation of seeds, used to define cluster algebras. Most of the connections b etw een cluster algeb ras and representation theory are proved only in th e simply-laced case which corres p onds to cluster algeb ras asso ciated with skew -symmetric matrices an d t o rep resentations of quivers. The general case deals with skew -symmetrizable matrices and one should consider cat ego ries over arbitra ry fields, a n d hence wo rk with rep resentations of mo dulated quivers. In this pap er and some of our fo rthcoming wo rk ( P otentials and Jacobian algebras for tensor algebras of bimodules ) , we investigate cluster t ilting within the framew or k of 2-Calabi-Y au o r exact stab ly 2-Calab i- Y au catego ries C over an a rbitrary field k . Simply-laced prep rojective algeb ras are p roven useful to get examples of exact stably 2-Calabi-Y au cat ego ries having cluster tilting sub catego ries without loops o r 2-cycles. A step forw a rd, motivated by this wo rk, could b e a further study of the repres entation theor y of n on simply-laced prep rojective algebras [10]. A recent account of th e study of prep rojec tive algeb ras over a non algebrai cally closed field with resp ect to cluster tilting is given in [21, § 4.3]. B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 3 / 15 Some prelimina ries on triangulate d and exact categories F ramew o r k and conv entions Fix a ground field k and write D = Hom k ( - , k ) fo r the stan dard dualit y . An exact cat ego ry is an additive categor y endow ed with a set of short exact sequences d enoted b y L h M f N and satisfy ing the Q uillen’s axioms, see [23] and [5]. An exact cat ego ry is F roben ius if it has enough p rojectives and enough injective and th e projectives coincide with the injectives . Fo r a triangulated catego ry C , we shall often denote a triangle X U Y X [ 1 ] simply b y X U Y and when this happ ens we also omit t o specify the fourth component a [ 1 ] in a morphism of triangle ( a , u , b , a [ 1 ]) . We also write Ex t 1 C ( X , Y ) = C ( X , Y [ 1 ]) ∼ = C ( X [ − 1 ] , Y ) for all X , Y ∈ C . Calabi-Y au cond ition. Recall that C is 2 -Calabi-Y au (weakly 2-Calabi-Y au in [24]) if fo r all objects X , Y ∈ C we have a functor ial isomo rphism D Ext 1 C ( X , Y ) ∼ Ext 1 C ( Y , X ) , or equivalently , w e have a functorial isomo rphism D Hom C ( X , Y ) ∼ Hom C ( Y , X [ 2 ]) . Recall from [3] that a stably 2 -Calabi-Y au catego ry is a Hom-finite Frobenius categor y C whose stable catego ry C , which is a triangulated catego ry acco rding to [17], is 2-Calabi-Y au. B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 4 / 15 Some prelimina ries on triangulate d and exact categories F ramew o r k and conventions Assumption : C stands fo r a Hom-finite Krull-Schmidt k -category which moreover is assumed to b e either triangulated 2-Calabi-Y au or exact stably 2-Calabi-Y au. Fo r all X ∈ C indecomp osable, k X is the residue division algebra of C ( X , X ) , in other terms k X = C ( X , X ) / J C ( X , X ) where J C is the ( Jacobson) radical bifunctor of C . By a sub cat egory of C we alwa ys mean an ad ditive full sub category of C which is stable by direct summand s, direct sums and isomo rph isms. B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 4 / 15 Some prelimina ries on triangulate d and exact categories Octaedral axiom in action in exact catego ries Lemma Fo r any diagram ( ∆) , (∆ ′ ) , (∆ ◦ ) o r (∆ ′ ◦ ) b elow (∆) : (∆ ′ ) : (∆ ◦ ) : (∆ ′ ◦ ) : M Z X M B N Y Y h ′ f ′ h f u ′ u v pb , Y Y M Z X M B N h ′ f ′ h f s ′ s t ′ t pb , Y Y N B M X Z M f ◦ h ◦ v ◦ u ◦ p o , N B M X Z M Y Y f ◦ h ◦ t ◦ s ◦ p o , in which a short exact plain ro w ∗ ∗ ∗ and a short exact plain column ∗ ∗ ∗ a re given, the dashed mo rphisms exist an d complete the diagram to a commutative diagram in which the d ashed row ∗ ∗ ∗ and the dashed column ∗ ∗ ∗ a re also short exact. The square marked by pb (resp., po ) is a p u ll-back (resp., pu sh-out) squar e. Notice that (∆ ◦ ) exp resses "Noether isomo rphism B / N ∼ = ( B / Y ) / ( N / Y ) " ([5, Lem 3.5]). B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 5 / 15 Some prelimina ries on triangulate d and exact categories Octaedral axiom in action in exact catego ries Lemma Fo r any diagram ( ∆) , (∆ ′ ) , (∆ ◦ ) o r (∆ ′ ◦ ) b elow (∆) : (∆ ′ ) : (∆ ◦ ) : (∆ ′ ◦ ) : M Z X M B N Y Y h ′ f ′ h f u ′ u v pb , Y Y M Z X M B N h ′ f ′ h f s ′ s t ′ t pb , Y Y N B M X Z M f ◦ h ◦ v ◦ u ◦ p o , N B M X Z M Y Y f ◦ h ◦ t ◦ s ◦ p o , in which a short exact plain ro w ∗ ∗ ∗ and a short exact plain column ∗ ∗ ∗ a re given, the dashed mo rphisms exist an d complete the diagram to a commutative diagram in which the d ashed row ∗ ∗ ∗ and the dashed column ∗ ∗ ∗ a re also short exact. The square marked by pb (resp., po ) is a p u ll-back (resp., pu sh-out) squar e. Notice that (∆ ◦ ) exp resses "Noether isomo rphism B / N ∼ = ( B / Y ) / ( N / Y ) " ([5, Lem 3.5]). B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 5 / 15 Some prelimina ries on triangulate d and exact categories T riangulated structure of stable catego ries In the exact case ( ou category C is t hen assumed Frobenius), the triangulated structure of C is given as follo ws. Fo r all u ∈ C ( X , Y ) we denote by u ∈ C ( X , Y ) the residue class of u with respect to the ideal of morphisms in C factoring th rough projective -injective. Each object X ∈ C is associated with a chosen short exact sequ ence denoted by ( i X , p X ) : X i X i X p X X [ 1 ] with i X a proje ctive-injective object. Each sho rt exact sequence ( ξ ) : X h B f Y defines a standard triangle ( ξ ) : X h B f Y δ X [ 1 ] and fo r all a ∈ C ( X , Z ) , the shift a [ 1 ] ∈ C ( X [ 1 ] , Z [ 1 ]) is given by a ′ , where δ ∈ C ( Y , X [ 1 ]) and a ′ ∈ C ( X [ 1 ] , Z [ 1 ]) are any maps o ccurring as in commutative diagrams of th e forms: ( ξ ) : X B Y ( i X , p X ) : X i X X [ 1 ] h f i X p X h ′ δ and ( i X , p X ) : X i X X [ 1 ] . ( i Z , p Z ) : Z i Z Z [ 1 ] . i X p X i Z p Z a α a ′ (2.1) B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 6 / 15 Some prelimina ries on triangulate d and exact categories T riangulated structure of stable catego ries In the exact case ( ou category C is t hen assumed Frobenius), the triangulated structure of C is given as follo ws. Fo r all u ∈ C ( X , Y ) we denote by u ∈ C ( X , Y ) the residue class of u with respect to the ideal of morphisms in C factoring th rough projective -injective. Each object X ∈ C is associated with a chosen short exact sequ ence denoted by ( i X , p X ) : X i X i X p X X [ 1 ] with i X a proje ctive-injective object. Each sho rt exact sequence ( ξ ) : X h B f Y defines a standard triangle ( ξ ) : X h B f Y δ X [ 1 ] and fo r all a ∈ C ( X , Z ) , the shift a [ 1 ] ∈ C ( X [ 1 ] , Z [ 1 ]) is given by a ′ , where δ ∈ C ( Y , X [ 1 ]) and a ′ ∈ C ( X [ 1 ] , Z [ 1 ]) are any maps o ccurring as in commutative diagrams of th e forms: ( ξ ) : X B Y ( i X , p X ) : X i X X [ 1 ] h f i X p X h ′ δ and ( i X , p X ) : X i X X [ 1 ] . ( i Z , p Z ) : Z i Z Z [ 1 ] . i X p X i Z p Z a α a ′ (2.1) B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 6 / 15 Some prelimina ries on triangulate d and exact categories T riangulated structure of stable catego ries We then p oint out the follo wing observation. Remark Keeping ab ove notations, fo r all X , Y ∈ C we have a canonical isomo rp h ism Ext 1 C ( Y , X ) ∼ Ext 1 C ( Y , X ) : ( ξ ) 7→ δ . Mo reover, if X an d Y have no proj ective direct summand , then J C ( X , Y ) = J C ( X , Y ) = { u : u ∈ J C ( X , Y ) } , and in par ticular, w e have a n atural identification of residue k -algebras: k X ∼ C ( X , X ) / J C ( X , X ) : x + J C ( X , X ) 7→ x + J C ( X , X ) . B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 6 / 15 Some prelimina ries on triangulate d and exact categories T riangulated structure of stable catego ries As a consequence of prev ious remark, the 2-Calabi-Y au condition on C amounts to the follow ing: we h ave functorial isomo rphisms D Ext 1 C ( X , Y ) ∼ Ext 1 C ( Y , X ) , with X , Y ∈ C . The follo wing easy fact is needed . Lemma Keeping previous notations, for any endomorphism ( a , b , c ) of the sho rt exact sequ ence ( ξ ) , the residue class ( a , b , c , a [ 1 ]) is an endomorphism of the induced standard triangle ( ξ ) . B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 6 / 15 Minimal approx imation sequences Minimal maps, appro ximation sequen ces Minimal maps Definition A mo rphism f : X Y in C is called right minimal if fo r any direct summand U 0 of f w e have U = 0. D ually , f is called left minimal if fo r any direct summand 0 U of f w e ha ve U = 0. The follo wing and its d ual version are valid over arbitra ry Krull-Schmidt catego ries. Lemma Fo r any mo rphism f : X Y in C , w e have: f ri ght minimal if and only if every section s such th at fs = 0 is zero, if and only if every morphism u such that fu = 0 is a radical ma p, if and only if every u ∈ C ( X , X ) such that fu = f is an automo rphism. The follo wing also ap plies t o all Krull-Schmidt categor ies o r Hom -finite cat egories. Lemma Let f ∈ C ( X , Y ) . Then th ere are isomo rphisms γ : X 1 ⊕ X 2 ∼ X and λ : Y ∼ Y 1 ⊕ Y 2 such that f γ = [ f ′ 0 ] : X 1 ⊕ X 2 Y and λ f =  f ′′ 0  : Y Y 1 ⊕ Y 2 with f ′ right minimal and f ′′ left minimal. B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 7 / 15 Minimal approx imation sequences Minimal maps, appro ximation sequen ces App ro ximation sequences Let M b e a sub categor y of C and B ∈ M . A map f ∈ C ( B , X ) is a right M -appro ximation if each object Z ∈ M yields an exact sequence C ( Z , B ) f - C ( Z , X ) 0. In addition, it is called minimal if f is right minimal. We also have t he d u al notion of (minimal) left M -ap pro ximation . By an extension from Y to X in the exact case ( resp., in the triangulated case) we mean any short exact sequence (resp., any triangle) ( ξ ) : X h B f Y . An extension X f Z g Y is called an M -ap pro ximation sequence if f is a left M -appro ximation and g a right M -appro ximation. In addition, it is called minimal if f i s left minimal and g right minimal. Notice that minimal ap pro ximation sequences never split. B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 8 / 15 Minimal approx imation sequences Minimal maps, appro ximation sequen ces App ro ximation sequences Remark Assuming the exact case, let M b e the sub catego ry of C induced by M . Then a sho rt exact sequence ( ξ ) : X f Z g Y is an M -appro ximation sequence (resp., a minimal M -appro ximation sequence) if and only if the indu ced stan d ar d triangle ( ξ ) is an M -appro ximation sequence (resp., a minimal M -appro ximation sequence). A direct application of p revious Lemma yields the follo wing result. Lemma Any extension ( ξ ) : X h B f Y in C is a direct sum of a minimal extension ( ξ min ) : X 1 h 1 B 1 f 1 Y 1 and a split extension ( ξ 0 ) : X 2 [ 1 0 ] X 2 ⊕ Y 2 [ 0 1 ] Y 2 . Corolla ry Fo r any non-split extension ( ξ ) : X h B f Y in C , if Y is indecomposable then h is a left minimal radical mor phism, if X is indecomp osable then f is a right minimal radical mo rphism. B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 8 / 15 Minimal approx imation sequences Minimal maps, appro ximation sequen ces Minimalit y and Rigid sub catego ries We recall (fo r example from [4, 9, 19]) that a subcat ego ry T ⊂ C is called rigid if Ext 1 C ( X , Y ) = 0 fo r all X , Y ∈ T . It is called cluster tilting if it is functorially finite (i.e. all X ∈ C ad mits left and right T - appro ximations) and T =  X ∈ C : ∀ T ∈ T , Ext 1 C ( X , T ) = 0  (equivalently , T =  X ∈ C : ∀ T ∈ T , Ext 1 C ( T , X ) = 0  ). It is immediate that a cluster tilting sub category is also rigid. B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 9 / 15 Minimal approx imation sequences Minimal maps, appro ximation sequen ces Minimalit y and Rigid sub catego ries The follo wing consequence is also derived. Lemma Supp ose t h at M ⊂ C is rigid and ( ξ ) : X h U f Y and ( ξ ′ ) : Y h ′ U ′ f ′ X a re M -appro ximation sequences. Then there is a common direct summand X 2 ⊕ Y 2 of U and U ′ such that ( ξ ) is t he d irect sum of a minimal M - a ppro ximation sequence ( ξ min ) : X 1 h 1 U 1 f 1 Y 1 with the split extension X 2 X 2 ⊕ Y 2 Y 2 , and ( ξ ′ ) is the d irect sum of a minimal M - a ppro ximation sequence ( ξ ′ min ) : Y 1 h ′ 1 U ′ 1 f ′ 1 X 1 with the split extension Y 2 Y 2 ⊕ X 2 X 2 . B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures Decemb er 20 13 9 / 15 Minimal approx imation sequences Minimal maps, appro ximation sequen ces F rom minimal app ro ximation sequences to isomo rphisms of Divisio n algeb ras We also p rove t he follow ing lemma, crucial for the rest of th is section. Lemma Let M b e a subcategory of C . If ( ξ ) : X h B f Y and ( ξ ′ ) : X ′ h ′ B ′ f ′ Y ′ ar e minimal M -appro ximation sequences then the follo wing assertions hold. • Any morphism u : X X ′ (resp., v : Y Y ′ ) extends to a morphism ϕ : ( ξ ) ( ξ ′ ) of appro ximation sequences. Moreo ver, u (resp., v ) is an isomo rphism if and only if ϕ is an isomor phism. • X is indecomp osable if and only if Y is. In this case, ( ξ ) induces an isomorphis m φ ξ : k X ∼ k Y taking u + J C ( X , X ) to v + J C ( Y , Y ) whenever the pair ( u , v ) extends to a n endomorphism of ( ξ ) . Sp ecializing to the exact setting, w e also prove t h e follo wing fact. Lemma Assuming the exact case, let M b e a subcat egory of C and ( ξ ) : X h B f Y a minimal M -appro ximation sequence with X and Y indecomp osable non-pro jective. Then fo r the indu ced minimal M -appro ximation standard triangle ( ξ ) we have φ ξ = φ ξ . B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures December 20 13 10 / 15 Minimal approx imation sequences Exchange sequen ces and irreducible maps Irreducible maps We let T b e a sub category of C and remind the reader that, b y assumption, T is full, stable by direct summands, direct sums and isomo rphisms. Given any sub catego ry X ⊂ T , the facto r category T / X is the sub category of T whose non-zero objects consist of all objects of T not b elonging to X . In case X = add ( X ) fo r some X ∈ T , w e simply write T / X for T / X . Let X , Y ∈ T . We have J T ( X , Y ) = J C ( X , Y ) , and the square J 2 T of the rad ical bifunctor of T is such that J 2 T ( X , Y ) = P Z ∈ T J C ( Z , Y ) J C ( X , Z ) = { vu : ∃ Z ∈ T , u ∈ J C ( X , Z ) , v ∈ J C ( Z , Y ) } . Next, define th e k Y - k X -bimod ule of T -irreducible morphisms from X to Y t o be Irr T ( X , Y ) = J C ( X , Y ) / J 2 T ( X , Y ) . Fo r an indecomp osable X ∈ T , t he category T is said to have n o loop at X if Irr T ( X , X ) = 0. B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures December 20 13 11 / 15 Minimal approx imation sequences Exchange sequen ces and irreducible maps Irreducible maps We therefor e get the follo wing. Remark The catego ry T h as no lo op at the ind ecomposable object X if and only if any radical mo rphism u ∈ J C ( X , X ) factor s through some object in T / X . Consequently , in this case, any left (resp., right) ( T / X ) -a ppro ximation h : X B (resp., f : B X ) is left almost split (resp., right almost split). B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures December 20 13 11 / 15 Minimal approx imation sequences Exchange sequen ces and irreducible maps Exchange sequences Definition Fo r a sub cat ego ry M of C , a minimal M -appro ximation sequence ( ξ ) : X B Y is called a n exchange sequence if X is indecomposable, X 6∈ M and add ( X , M ) is rigid, or, equivalently , Y is ind ecomp osable, Y 6∈ M and add ( Y , M ) is rigid. More precisely , ( ξ ) is called a n exchange sequence from add ( X , M ) to add ( Y , M ) . Lemma Let ( ξ ) : X h B f Y b e an exchange sequence from a rigid sub category T to a rigid sub catego ry T ′ of C . Then t he follow ing are equivalent: ( a ) The cat ego ry T has no lo op at X . ( b ) The category T ′ has no lo op at Y . ( c ) Ext 1 C ( Y , X ) ∼ = k X ∼ = k Y . Mo reover, in t h is case, any non-split extension from Y to X is isomorphic to ( ξ ) . B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures December 20 13 12 / 15 Minimal approx imation sequences Exchange sequen ces and irreducible maps T race maps pla y a determinant role A k -linear trace on K is a central element of the stan d ard dual b imodu le D ( K ) . Remark Any finite-dimensional division k -algebra has a n on-zero t race. In p artic ular, fo r finite-dimensional divisio n k - algeb ras K and K ′ and fo r an y K-K ′ -bimod ule B (finite-dimensional over k ), the choice of tw o non- zero traces t ∈ D ( K ) and t ′ ∈ D ( K ′ ) induces canonical isomo rphisms Hom K ( B , K ) ∼ D ( B ) : u 7→ t ◦ u , Hom K ′ ( B , K ′ ) ∼ D ( B ) : u 7→ t ′ ◦ u from the left dual (resp., right dual) of B to its standard dual. We no w pro ve the follo wing connection b etw een exchange sequences and b imodu les of irreducible map s. Prop osition Let ( ξ ) : M h B f M ′ b e an exchange sequence b etw een rigid sub catego ries T and T ′ and assume th a t T has no loop at M . Und er the canonical isomo rphism φ ξ : k M k M ′ , fo r all X ∈ T ∩ T ′ w e have an isomorphis m of k M - k X -bimod ules φ ξ, X : Irr T ′ ( X , M ′ ) ∼ D Irr T ( M , X ) . B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures December 20 13 13 / 15 Minimal approx imation sequences Exchange sequen ces and irreducible maps T race maps pla y a determinant role Pro o f. Let X ∈ T ∩ T ′ b e indecomp osable, put M B X = Irr T ( M , X ) and X B M ′ = φ Irr T ′ ( X , M ′ ) . Fo r all u ∈ J C ( M , X ) and v ∈ J C ( X , M ′ ) , put e u = u + J 2 T ( M , X ) and e v = v + J 2 T ′ ( X , M ′ ) . B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures December 20 13 13 / 15 Minimal approx imation sequences Exchange sequen ces and irreducible maps T race maps pla y a determinant role Pro o f. Thus, in view of Remark ?? , h is left minimal almost split in T while f is right minimal a lmost split in T ′ . Hence, fo r any tw o radical maps u ∈ J C ( M , X ) and v ∈ J C ( X , M ′ ) as b efore and all a ∈ C ( M , M ) , invoking the factorizatio n property of h and f together with Lemma 3.6 w e get a commutative diagram with the follo wing shap e: X ( ξ ) : M B M ′ ( ξ ) : M B M ′ X h f h f u v a a ′ v ∗ b u ∗ (D1) B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures December 20 13 13 / 15 Minimal approx imation sequences Exchange sequen ces and irreducible maps T race maps pla y a determinant role Pro o f. We claim that there is an isomo rp h ism of k M -k X -bimod ules given b y ϕ : X B M ′ Hom k X ( M B X , k X ) : e v 7→ ( e u 7→ ϕ ( e v )( e u ) = u ∗ v ∗ ) . ....... ......... ................ B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures December 20 13 13 / 15 Main state ments Comaptible exchange sequences and generalized cluster structures T o each sub category T of C can b e associated t he follo wing combinatorial data. Definition The exchange matrix of T is the (not necessa rily finite) matrix B ( T ) which has coefficients b X , Y = dim k X Irr T ( X , Y ) − dim k X Irr T ( Y , X ) fo r all X , Y ∈ ind T , Y b eing non-projective ( in the exact case). The p rincipal part of th e exchange matrix B ( T ) is skew symmetrizable via the diagonal matrix with co efficients dim k ( k X ) fo r X ind ecomposable n on-proj ective. Notice th a t t h e exchange matrix B ( T ) uniquely determines t h e valued quiver (or the quiver in the simply-laced case) of T . B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures December 20 13 14 / 15 Main state ments Comaptible exchange sequences and generalized cluster structures No w let M ∈ ind T an d supp ose that w e have tw o exchange sequences ( ξ ) : M ∗ B M and ( ξ ′ ) : M B ′ M ∗ relating T to a sub category T ∗ = add ( T / M , M ∗ ) for some indecomp osable object M ∗ 6∈ T . When T and T ∗ ar e cluster t ilting, it is pro ved in the simply-laced context of [3, Theorem 1.6] th at B ( T ) is related to B ( T ∗ ) via the Fo min-Zelevinsky mutation [13]. But when th e base field k is not algebraically closed, the matrix B ( T ) alone do es not give enough infor mation ab out irreducible map s in T , the corresponding info rmation is encoded by the mod ulated q uiver of T , (see fo r example [11] fo r the notion of mo dulated quiver). Then, w e introduce the follo wing notion of compatibilit y in order t o get a more finer relation b et ween bimodu les of irreducible maps in T and in T ∗ . Definition The exchange sequences ( ξ ) and ( ξ ′ ) are called compatible if the indu ced isomorphisms φ ξ : k M ∗ k M and φ ξ ′ : k M k M ∗ and a re inverse from each other. B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures December 20 13 14 / 15 Main state ments Comaptible exchange sequences and generalized cluster structures The follo wing proposition sheds more light on the compat ibilit y b etw een exchange sequences ( ξ ) and ( ξ ′ ) ; they may fail to b e compatible especially when th e residue division a lgebra k M is not commutative. Prop osition Supp ose that ( η ) : M B ′′ M ∗ is an other exchange sequence from T t o T ∗ . Then any a ∈ C ( M , M ) extends t o endomorphisms ( c , b , a ) : ( ξ ) ( ξ ) , ( a , b ′ , c ′ ) : ( ξ ′ ) ( ξ ′ ) and ( a , u − 1 b ′ u , λ − 1 c ′ λ ) : ( η ) ( η ) such that c − c ′ ∈ J C ( M ∗ , M ∗ ) , λ is an automor phism of M ∗ and u ∈ C ( B ′ , B ′′ ) is an isomo rphism. Thus, when k M is commutative, an y t wo exchange sequences M ∗ U M and M U ′ M ∗ relating T and T ∗ ar e compatible. This is the case if k is algebr aically closed. Cluster structu re . The notion of cluster structure in the present framewo rk is a natural generalization of the one from [3], along the ab ove lines. Mutation of quivers from [3] should b e replaced by mutation of sk ew symmetrizable matrices and exchange sequen ces ar e required t o b e compatible. Recall that in t he simply laced context of [3], exchange sequences a re aut omatically compatible according to prev ious Proposition. B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures December 20 13 1 4 / 15 Main state ments Main statements Theor em ([20, Theorem 5.3] Iyama-Y oshino ) Let M b e a subcategory of C and M ∈ C r M b e indecomp osable (n on-proj ective in the exact case), such that T = add ( X , M ) is cluster tilting. Then there is a u nique (up to isomo rphism) M ∗ ∈ C r T indecomp osable such that th e sub category µ M ( T ) = add ( M ∗ , M ) is also cluster tilting. Moreov er, there exist exchange sequences ( ξ ) : M ∗ B M an d ( ξ ′ ) : M B ′ M ∗ relating T and µ M ( T ) . The subcategory T ∗ = µ M ( T ) is called the mutation of T at M and we also h ave T = µ M ∗ ( T ∗ ) . Our first contribution then specializes Iyama-Y oshino Theor em in this context as follo ws. Prop osition If T h as no lo op at M , then exchange sequences ( ξ ) and ( ξ ′ ) can b e chosen to be compatible. B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures December 20 13 1 5 / 15 Main state ments Main statements Befo re stating the next result of this pap er, recall that T h as no 2-cycle if fo r any X , Y ∈ T ind ecomposable, either Irr T ( X , Y ) = 0, either Irr T ( Y , X ) = 0. Theor em If T is a cluster tilting sub catego ry of C without loop o r 2 -cycle, a n d T ∗ the mu tation of T at some M ∈ ind T then B ( T ∗ ) = µ M ( B ( T )) . We should emphasize that the main contribu tion of this wo rk is tw o fold: Our main Theorem compares exchan ge ma trices (or equ ivalently , valued quivers) of T and T ∗ using Fomin-Zelevinsky muta t ion. The existence of compatible exchange sequences given by the Proposition more ab ove is crucial fro describing b imo dules of irreducible maps in T ∗ , indeed this result yields a first step to describe the mo dulated quiver of T ∗ using t hat of T . B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures December 20 13 1 5 / 15 Main state ments Main statements We get t he follo wing consequence where we use [4, Proposition 6.14] fo r the second part. Corolla ry Supp ose t h at none of cluster tilting subcategories of C h ave lo op or 2 -cycle. Then C h as a cluster structure induced by its cluster tilting sub categor ies. In particula r, it is the case fo r cluster catego ries associated with finite dimensional hereditary algebras. B. Nguefack (Univ Y ao unde I) Non- simply laced cluste r structures December 20 13 1 5 / 15 Main state ments I. Assem, D . Simson and A. Sko wronński. Elements of rep resentation theor y of associative algebras. I: T echniques of rep resentation theo ry . n o 65 London Mathematical Society stud ent text, Cambridge Universit y Press (2006). A. Berenstein, S. Fomin, A. Zelevinsky , Cluster algebras I I I: Upp er b ounds and double Bruhat cells, Duke Mathematical Journal 126 (1) (2005) 1–52. A. 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