Skew group categories, algebras associated to Cartan matrices and folding of root lattices
aa r X i v : . [ m a t h . R T ] F e b SKEW GROUP CATEGORIES, ALGEBRAS ASSOCIATED TOCARTAN MATRICES AND FOLDING OF ROOT LATTICES
XIAO-WU CHEN, REN WANG ∗ Abstract.
For a finite group action on a finite EI quiver, we construct its‘orbifold’ quotient EI quiver. The free EI category associated to the quotientEI quiver is equivalent to the skew group category with respect to the givengroup action. Specializing the result to a finite group action on a finite acyclicquiver, we prove that, under reasonable conditions, the skew group categoryof the path category is equivalent to a finite EI category of Cartan type. Ifthe ground field is of characteristic p and the acting group is a cyclic p -group,we prove that the skew group algebra of the path algebra is Morita equivalentto the algebra associated to a Cartan matrix, defined in [C. Geiss, B. Leclerc,and J. Schr¨oer, Quivers with relations for symmetrizable Cartan matrices I:Foundations , Invent. Math. (2017), 61–158]. We apply the Morita equiv-alence to construct a categorification of the folding projection between theroot lattices with respect to a graph automorphism. In the Dynkin cases, therestriction of the categorification to indecomposable modules corresponds tothe folding of positive roots. Introduction
The background.
The folding of root lattices is classic [25] and plays a signif-icant role in Lie theory when getting from the simply-laced cases to the non-simply-laced cases. The starting point is the fact that a symmetrizable generalized Cartanmatrix C is determined by a finite graph Γ with an admissible automorphism σ [25, 19]. There is a surjective homomorphism, called the folding projection , f : Z Γ −→ Z (Γ / h σ i )from the root lattice of Γ to that of C , which preserves simple roots; see [24,Section 10.3]. Here, Γ denotes the set of vertices in Γ, and the orbit set Γ / h σ i indexes both the rows and columns of C , so that we identify Z (Γ / h σ i ) with theroot lattice of C . It is proved by [14, Proposition 15] that the folding projectionrestricts to a surjective map f : Φ(Γ) −→ Φ( C )between the root systems [16], known as the folding of roots.Let K be a field, and ∆ be a finite acyclic quiver such that its underlying graph isΓ. The path algebra K ∆ is finite dimensional and hereditary. It is well known thatthe category of finite dimensional K ∆-modules, denoted by K ∆-mod, categorifiesthe root lattice Z Γ in the following manner [9]: the dimension vector dim( M ) ofany K ∆-module M belongs to Z Γ , where simple K ∆-modules correspond to simple Date : February 17, 2021.2010
Mathematics Subject Classification.
Key words and phrases. folding, Cartan matrix, graph with automorphism, free EI category,skew group category. ∗ The corresponding author.E-mail: [email protected]; [email protected]. roots. Gabriel’s theorem [9, 1.2 Satz], one of the foundations in modern represen-tation theory of algebras, states that if ∆ is of Dynkin type, then indecomposable K ∆-modules correspond bijectively to positive roots in Φ(Γ).Associated to a symmetrizable generalized Cartan matrix C , a finite dimen-sional 1-Gorenstein algebra H is defined in [11]. The category of finite dimen-sional τ -locally free H -modules, denoted by H -mod τ -lf, categorifies the root lattice Z (Γ / h σ i ) in a similar manner: the rank vector rank( X ) of any τ -locally free H -module X belongs to Z (Γ / h σ i ), where generalized simple H -modules correspondto simple roots. [11, Theorem 1.3], a remarkable analogue of Gabriel’s theorem,states that if C is of Dynkin type, then indecomposable τ -locally free H -modulescorrespond bijectively to positive roots in Φ( C ).We mention that the categorification in [11] works over an arbitrary ground field.In particular, it works for algebraically closed fields, and then certain geometricconsideration for K ∆ carries over to H ; see [10]. The traditional categorification of Z (Γ / h σ i ) for a non-symmetric Cartan matrix uses species [8], where the groundfield has to be chosen suitably and can not be algebraically closed.In view of the above work, the following question is natural and fundamental:how to categorify the folding projection f between the root lattices? More pre-cisely, is there an additive functor Θ : K ∆-mod → H -mod τ -lf making the followingdiagram K ∆-mod dim (cid:15) (cid:15) Θ / / H -mod τ -lf rank (cid:15) (cid:15) Z Γ f / / Z (Γ / h σ i )commute? Such a functor Θ might be called a categorification of f .We will construct such a categorification under the assumptions that the charac-teristic char( K ) = p of the field is positive and that the automorphism σ is of order p a for some a ≥
1. Moreover, if ∆ is of Dynkin type, Θ preserves indecomposablemodules and categorifies the folding of positive roots.For our purpose, it is very natural to require that σ preserves the orientation,that is, it acts on ∆ by quiver automorphisms. We will work in a slightly moregeneral setting, namely, finite group actions on finite free EI categories.Recall that a finite category is EI provided that each endomorphism is invert-ible; in particular, the endomorphism monoid of each object is a finite group. Forexample, the path category of a finite acyclic quiver is EI. The study of finite EIcategories goes back to [20], and is used to reformulate and extend Alperin’s weightconjecture [29, 18]. We mention that EI categories are very similar to graphs ofgroups in the sense of Bass-Serre [2, 23].As an EI analogue of a path category, the notion of a finite free EI category isintroduced in [17]. We are mostly interested in EI categories of Cartan type [4],which are certain finite free EI categories associated to symmetrizable generalizedCartan matrices. The construction of the categorification Θ relies on the isomor-phism [4] between the category algebra of an EI category of Cartan type and thealgebra H in [11].1.2. The main results.
Let C be a finite category and G be a finite group. Assumethat G acts on C by categorical automorphisms. As a very special case of theGrothendieck construction, we have the skew group category C ⋊ G . The terminologyis justified by the following fact: the category algebra K ( C ⋊ G ) is isomorphic to K C G , the skew group algebra of the category algebra K C with respect to theinduced G -action. KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 3
Following [17, Definition 2.1], a finite EI quiver ( Q, U ) consists of a finite acyclicquiver Q and an assignment U on Q . The assignment U assigns to each vertex i of Q a finite group U ( i ), and to each arrow α , a finite ( U ( tα ) , U ( sα ))-biset U ( α ). Here, tα and sα denote the terminating vertex and starting vertex of α , respectively.In a natural manner, each finite EI quiver ( Q, U ) gives rise to a finite EI category C ( Q, U ) such that the objects of C ( Q, U ) are precisely the vertices of Q , the au-tomorphism group of i coincides with U ( i ), and that elements of U ( α ) correspondto unfactorizable morphisms. By [17, Definition 2.2 and Proposition 2.8], a finiteEI category C is said to be free , provided that it is equivalent to C ( Q, U ) for somefinite EI quiver (
Q, U ).Let G be a finite group acting on ( Q, U ) by EI quiver automorphisms. Then G acts naturally on the EI category C ( Q, U ). We form the skew group category C ( Q, U ) ⋊ G . Inspired by [2, Section 3], we construct the ‘orbifold’ quotient EIquiver (
Q, U ). Here, Q is the quotient quiver Q by G , and the construction of theassignment U is quite involved. We mention that for each vertex i of Q , the finitegroup U ( i ) is a semi-direct product of U ( i ) with the stabilizer G i for some vertex i of Q . For details, we refer to Subsection 5.1.The first main result identifies the category associated to the quotient EI quiverwith the skew group category, and thus justifies the ‘orbifold’ quotient construction. Theorem A.
Let ( Q, U ) be a finite EI quiver with a G -action, and ( Q, U ) be itsquotient EI quiver. Then there is an equivalence of categories C ( Q, U ) ≃ C ( Q, U ) ⋊ G. We mention that Theorem A (= Theorem 5.1) might be viewed as a combina-torial analogue to the well-known fact: the skew group algebra of a commutativealgebra with respect to a finite group action is closely related to the correspondingquotient singularity; for example, see [30].Let ∆ be a finite acyclic quiver. Denote by P ∆ its path category. We view ∆ asa finite EI quiver (∆ , U tr ) with trivial assignment U tr . Then we have C (∆ , U tr ) = P ∆ . Assume that G acts on ∆ by quiver automorphisms. It induces a G -action on(∆ , U tr ). Denote by (∆ , U tr ) the corresponding quotient EI quiver, where ∆ isthe quotient quiver ∆ by G . Theorem A implies that there is an equivalence ofcategories C (∆ , U tr ) ≃ P ∆ ⋊ G. (1.1)By a Cartan triple ( C, D,
Ω), we mean that C is a symmetrizable generalizedCartan matrix, D is its symmetrizer and that Ω is an acyclic orientation of C .Following [11, Section 1.4], we denote by H ( C, D,
Ω) the 1-Gorenstein K -algebraassociated to any Cartan triple ( C, D,
Ω). Similarly, we associate a finite free EIcategory C ( C, D,
Ω), called an
EI category of Cartan type , to any Cartan triple(
C, D,
Ω); see [4, Definition 4.1].As is well known, there is a Cartan triple (
C, D,
Ω) associated to the above G -action on ∆ such that both the rows and columns of C and D are indexed by theorbit set ∆ = ∆ /G . Here, ∆ denotes the set of vertices in ∆. Moreover, foreach G -orbit i of vertices, the corresponding diagonal entry of D is | G | / | i | ; thecorresponding off-diagonal entry of C is c i , j = − N i , j | j | , XIAO-WU CHEN, REN WANG where | i | denotes the cardinality of the G -orbit i and N i , j denotes the number ofarrows in ∆ between the G -orbit i and G -orbit j . The orientation of Ω is inducedfrom the one of ∆.The second main theorem establishes an equivalence between the skew groupcategory and the EI category of Cartan type. Based on [4], we obtain a Moritaequivalence between the skew group algebra K ∆ G and H ( C, D,
Ω).
Theorem B.
Let ∆ be a finite acyclic quiver with a G -action that satisfies ( † - ( † in Subsection 6.2. Assume that ( C, D, Ω) is the associated Cartan triple. Then wehave the following statements.(1) There is an equivalence of categories P ∆ ⋊ G ≃ C ( C, D, Ω) . (2) Assume that char( K ) = p > and that G is a p -group. Then the skewgroup algebras K ∆ G and H ( C, D, Ω) are Morita equivalent. The above technical conditions ( † †
3) are easily satisfied when G is cyclic.On the other hand, examples where they do hold seem to be ubiquitous; see Ex-ample 6.6. In view of (1.1), the core of the proof of Theorem B is to describe theassignment U tr in the quotient EI quiver. We refer to Theorem 6.5 for more details.The equivalence and the Morita equivalence in Theorem B indicate that bothEI categories of Cartan type [4] and the algebra H ( C, D,
Ω) [11] arise naturally inthe representation theory of quivers with automorphisms [19, 14].The Morita equivalence in Theorem B(2) yields an equivalence between modulecategories Ψ : K ∆ G -mod ∼ −→ H ( C, D,
Ω)-mod . We have the obvious induction functor − G : K ∆-mod −→ K ∆ G -mod , M M G. For τ -locally free modules over H = H ( C, D,
Ω), we refer to [11, Definition 1.1and Section 11]. Denote by H -mod τ -lf the full subcategory of H -mod consistingof τ -locally free modules. In contrast to [11], we do not require τ -locally free H -modules to be indecomposable.Recall that Z ∆ and Z (∆ /G ) denote the root lattices of ∆ and C , respectively.The sets of positive roots are denoted by Φ + (∆) and Φ + ( C ), respectively.The third main result shows that the composite functor Ψ ◦ ( − G ) is the pursuedcategorification of the folding projection f ; see Theorem 7.8 and Proposition 7.9. Theorem C.
Assume that char( K ) = p > and that G is a cyclic p -group. Assumethat G acts on a finite acyclic quiver ∆ such that G α = G s ( α ) ∩ G t ( α ) for each arrow α in ∆ . Assume that ( C, D, Ω) is its associated Cartan triple. Then we have thefollowing commutative diagram. K ∆ - mod Ψ ◦ ( − G ) / / dim (cid:15) (cid:15) H ( C, D, Ω) - mod τ - lfrank (cid:15) (cid:15) Z ∆ f / / Z (∆ /G ) KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 5
Assume further that ∆ is of Dynkin type. Then the above commutative diagramrestricts to the following one. (1.2) K ∆ - ind Ψ ◦ ( − G ) / / dim (cid:15) (cid:15) H ( C, D, Ω) - ind τ - lfrank (cid:15) (cid:15) Φ + (∆) f / / Φ + ( C )Here, G α , G s ( α ) and G t ( α ) denote the stabilizers of an arrow α , its startingvertex s ( α ) and terminating vertex t ( α ), respectively. The natural condition G α = G s ( α ) ∩ G t ( α ) implies that the technical conditions ( † †
3) in Theorem B hold.In (1.2), we denote by K ∆-ind a complete set of representatives of indecompos-able K ∆-modules. Similarly, H ( C, D,
Ω)-ind τ -lf is a complete set of representativesof indecomposable τ -locally free H ( C, D,
Ω)-modules.In the Dynkin cases, by [9, 1.2 Satz] and [11, Theorem 1.3], the vertical arrowsin (1.2) are both bijections. Since f : Φ + (∆) → Φ + ( C ) is surjective, we infer thatup to the equivalence Ψ, every τ -locally free H ( C, D,
Ω)-module is induced from K ∆-ind. This yields a new interpretation of those H ( C, D,
Ω)-modules [11] thatcategorify the root system Φ + ( C ).In view of [19, Section 14.1] and [14], it has been expected that skew groupalgebras play a role in categorifying the root lattice for symmetrizable generalizedCartan matrices. We observe that in [14, Section 4] the characteristic of the groundfield is assumed to be coprime to the order of the acting group. In contrast, thefeature of Theorem C is the assumptions that the ground field K is of characteristic p and that the order of the acting group G is a p -power.1.3. The structure.
The paper is structured as follows. In Section 2, we provethat for a finite group action on a finite category, the skew group category is EI ifand only if so is the given category; see Proposition 2.5. In Section 3, we study theunique factorization property of morphisms and free EI categories. We prove thatfor a finite group action on a finite category, the skew group category is free EI ifand only if so is the given category; see Proposition 3.4. In Section 4, we recallfinite EI quivers introduced in [17], and prove a universal property of the free EIcategory associated to a finite EI quiver; see Proposition 4.2.For a finite group action on a finite EI quiver, we construct its ‘orbifold’ quotientEI quiver explicitly in Section 5. Theorem 5.1 states that the category associatedto the quotient EI quiver is equivalent to the skew group category.In Section 6, we recall the algebras H [11] and EI categories [4] associated toCartan triples. For a finite group action on a finite acyclic quiver, we give sufficientconditions on when the quotient EI quiver is of Cartan type. Consequently, theskew group algebra of the path algebra is Morita equivalent to the algebra H ; seeTheorem 6.5.In the final section, we first study induced modules over skew group algebras. Weapply Theorem 6.5 to the case where a finite cyclic p -group acts on a finite acyclicquiver. Theorem 7.8 obtains a categorification of the folding projection f , namelyan additive functor from the module category over the path algebra to the categoryof τ -locally free H -modules. In the Dynkin cases, restricting the categorificationto indecomposable modules, we obtain a categorification of the folding of positiveroots; see Proposition 7.9. In the end, we give an explicit example to illustrate thecategorification.By default, a module means a finite dimensional left module. For a finite dimen-sional algebra A , we denote by A -mod the abelian category of finite dimensional left XIAO-WU CHEN, REN WANG A -modules. We use rad( A ) to denote the Jacobson radical of A . The unadornedtensor ⊗ means the tensor product over the ground field K .2. Skew group categories
In this section, we recall basic facts about finite group actions on finite categories.The EI property of a skew group category is studied in Proposition 2.5.2.1.
Finite G -categories. Let C be a finite category, that is, a category with onlyfinitely many morphisms. As any object is determined by its identity endomor-phism, the finite category C necessarily has only finitely many objects. Denote byObj( C ) ( resp. Mor( C )) the finite set of objects ( resp. morphisms) in C . We denoteby Aut( C ) the automorphism group of C .Let G be a finite group with its unit 1 G . A finite G -category C is a finite categoryequipped with a group homomorphism ρ : G −→ Aut( C ) . To simplify the notation, the following convention will be used: for g ∈ G and x ∈ Obj( C ), we write g ( x ) = ρ ( g )( x ); for α ∈ Mor( C ), we write g ( α ) = ρ ( g )( α ).For a finite G -category C , we will recall the skew group category C ⋊ G ; compare[22, Subsection 3.1] and [5, Definition 2.3]. It has the same objects as C ; for twoobjects x and y , the corresponding Hom set is defined to beHom C ⋊ G ( x, y ) = { ( α, g ) | g ∈ G, α ∈ Hom C ( g ( x ) , y ) } . For any morphisms ( α, g ) ∈ Hom C ⋊ G ( x, y ) and ( β, h ) ∈ Hom C ⋊ G ( y, z ), the compo-sition is defined by ( β, h ) ◦ ( α, g ) = ( β ◦ h ( α ) , hg ) . (2.1)We observe that the identity endomorphism of x in C ⋊ G is given by (Id x , G ),where Id x is the identity endomorphism of x in C . We mention that the formationof a skew group category might be viewed as a very special case of the Grothendieckconstruction; compare [13, VI.8] and [28, Section 7].Let K be a field and C be a finite category. The category algebra K C of C is a finitedimensional K -algebra defined as follows. As a K -vector space, K C = L α ∈ Mor( C ) K α ,and the product between the basis elements is given by the following rule: αβ = (cid:26) α ◦ β, if α and β can be composed in C ;0 , otherwise.The unit of K C is given by 1 K C = P x ∈ Obj( C ) Id x .Denote by ( K -mod) C the category of covariant functors from C to K -mod. Thereis a canonical equivalence can : K C -mod ∼ −→ ( K -mod) C , (2.2)sending a K C -module M to the functor can( M ) : C → K -mod described as follows:can( M )( x ) = Id x .M for each object x in C ; for any morphism α : x → y , we havecan( M )( α ) : can( M )( x ) −→ can( M )( y ) , m α.m. For details, we refer to [28, Proposition 2.1].Denote by Aut( K C ) the group of algebra automorphisms on K C . Each categoricalautomorphism on C induces uniquely an algebra automorphism on K C . Therefore,there is a canonical embedding of groupsAut( C ) ֒ → Aut( K C ) . KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 7
Assume that C is a finite G -category. The group homomorphism ρ : G → Aut( C )induces a group homomorphism ρ ′ : G → Aut( K C ). In other words, the group G acts on the algebra K C by algebra automorphisms. We denote by K C G thecorresponding skew group algebra . Here, we recall that K C G = K C ⊗ K G as a K -vector space, where the tensor product α ⊗ g is written as α g . The multiplicationis given by ( β h )( α g ) = βh ( α ) hg for any α, β ∈ Mor( C ) and g, h ∈ G . We emphasize that on the right hand side, βh ( α ) means the product of β and h ( α ) in K C , namely, the composition β ◦ h ( α )in C .The following easy observation, extending [31, Lemma 2.3.2], justifies the termi-nology ‘skew group category’. Proposition 2.1.
Let C be a finite G -category. Then there is an isomorphism ofalgebras K ( C ⋊ G ) ∼ −→ K C G, sending a morphism ( α, g ) in C ⋊ G to the element α g in K C G . (cid:3) In the following lemma, we collect elementary facts on skew group categories.
Lemma 2.2.
Let C be a finite G -category. Then the following two statements hold.(1) A morphism ( α, g ) in C ⋊ G is an isomorphism if and only if α is anisomorphism in C .(2) For two objects x and y in C , they are isomorphic in C ⋊ G if and only if x is isomorphic to g ( y ) in C for some g ∈ G .Proof. (1) For the “if” part, we assume that α − is the inverse of α in C . Then( g − ( α − ) , g − ) is a well-defined morphism in C ⋊ G ; moreover, it is the requiredinverse of ( α, g ).For the “only if” part, we observe that the inverse of ( α, g ) has to be of the form( β, g − ). Then it is direct to see that g ( β ) is the inverse of α , as required.(2) For the “if” part, we assume that α : g ( y ) → x is an isomorphism in C . Then( α, g ) is a morphism from y to x in C ⋊ G ; moreover, by (1) it is an isomorphismbetween y and x .For the “only if” part, we assume that ( α, g ) ∈ Hom C ⋊ G ( y, x ) is an isomorphism.By (1), we deduce that α is an isomorphism from g ( y ) to x in C . (cid:3) The EI property.
Let C be a finite G -category as above. For each object x in C , we denote by G x = { g ∈ G | g ( x ) = x } its stabilizer. We observe that G x actson the monoid Hom C ( x, x ) by monoid automorphisms. Denote by Hom C ( x, x ) ⋊ G x the corresponding semi-direct product. There is an inclusion between monoidsinc x : Hom C ( x, x ) ⋊ G x ֒ → Hom C ⋊ G ( x, x ) , ( α, g ) ( α, g ) . The following terminology is inspired by [19, Subsection 12.1.1].
Definition 2.3.
A finite G -category C is admissible , provided that for any x ∈ Obj( C ) and g ∈ G , Hom C ( g ( x ) , x ) = ∅ whenever g ( x ) = x . (cid:3) Lemma 2.4.
A finite G -category C is admissible if and only if inc x is surjectivefor each object x in C .Proof. The inclusion inc x is not surjective if and only if there exists g ∈ G satisfying g ( x ) = x and Hom C ( g ( x ) , x ) = ∅ . Then the result follows immediately. (cid:3) Recall from [28] that a finite category C is EI if every endomorphism is anisomorphism. Therefore, for each object x , Hom C ( x, x ) = Aut C ( x ) is a finite group.Finite EI categories are of interest from many different perspectives; for example,see [29, 31]. XIAO-WU CHEN, REN WANG
Proposition 2.5.
Let C be a finite G -category. Then C is an EI category if andonly if so is C ⋊ G .Proof. For the “if” part, we assume that C ⋊ G is an EI category. For any α ∈ Hom C ( x, x ), ( α, G ) is an endomorphism of x in C ⋊ G . Since C ⋊ G is EI, ( α, G )is an isomorphism. By Lemma 2.2(1), the endomorphism α is an isomorphism in C , as required.For the “only if” part, we assume that C is an EI category. Any endomorphismof x in C ⋊ G is of the form ( α, g ), where α : g ( x ) → x is a morphism in C . Assumethat g d = 1 G for some d ≥
1. Then we have a chain x = g d ( x ) g d − ( α ) −→ g d − ( x ) −→ · · · −→ g ( x ) g ( α ) −→ g ( x ) α −→ x of morphisms in C . Since C is EI, it follows that all the morphisms in the chainare isomorphisms. In particular, the morphism α is an isomorphism. ApplyingLemma 2.2(1), we infer that the endomorphism ( α, g ) is an isomorphism, provingthat C ⋊ G is an EI category. (cid:3) The following corollary follows immediately from Lemma 2.4.
Corollary 2.6.
Let C be a finite admissible G -category. Assume that C is EI. Thenfor each object x , we have an identification of groups Aut C ( x ) ⋊ G x = Aut C ⋊ G ( x ) . (cid:3) Free EI categories
In this section, we study the unique factorization property of morphisms andfree EI categories [17]. We prove that a skew group category is free EI if and onlyif so is the given category; see Proposition 3.4.Let C be a finite category. Recall from [17, Definition 2.3] that a morphism α : x → y in C is unfactorizable , if it is a non-isomorphism and whenever it has afactorization x β → z γ → y , then either β or γ is an isomorphism. We observe that if α : x → y is unfactorizable, then so is h ◦ α ◦ g for any isomorphism h and g .As the notion of an unfactorizable morphism is categorical, it is preserved byany categorical automorphism. Then the following observation is clear. Lemma 3.1.
Let G be a finite group and C be a finite G -category. Then for anymorphism α in C and g ∈ G , α is unfactorizable if and only if so is g ( α ) . (cid:3) We say that a morphism α in a finite category C satisfies the Unique FactorizationProperty (UFP), if it is either an isomorphism or, whenever it has two factorizationsinto unfactorizable morphisms: x = x α → x α → · · · α m → x m = y and x = y β → y β → · · · β n → y n = y, then m = n , and there are isomorphisms γ i : x i → y i in C for 1 ≤ i ≤ m −
1, suchthat the following diagram commutes. x = x α / / x α / / γ (cid:15) (cid:15) x α / / γ (cid:15) (cid:15) · · · α m − / / x m − α m / / γ m − (cid:15) (cid:15) x m = yx = y β / / y β / / y β / / · · · β m − / / y m − β m / / y m = y We mention that, in general, a non-isomorphism in a finite category C mightnot have a factorization into unfactorizable morphisms. However, if C is EI, any KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 9 non-isomorphism in C has a factorization into unfactorizable morphisms; see [17,Proposition 2.6]. Lemma 3.2.
Let G be a finite group and C be a finite G -category. Then a morphism ( α, g ) in C ⋊ G is unfactorizable if and only if α is unfactorizable in C .Proof. By Lemma 2.2(1), we observe that ( α, g ) is a non-isomorphism if and onlyif so is α .For the “if” part, we assume that α is unfactorizable. Suppose we have a factor-ization ( α, g ) = ( β, h ) ◦ ( γ, k ) = ( β ◦ h ( γ ) , hk )in C ⋊ G . The factorization α = β ◦ h ( γ ) in C implies that either β or h ( γ ) isan isomorphism. As h ∈ G induces a categorical automorphism on C , we inferthat h ( γ ) is an isomorphism if and only if so is γ . In view of Lemma 2.2(1), weinfer that either ( β, h ) or ( γ, k ) is an isomorphism in C ⋊ G , proving that ( α, g ) isunfactorizable.For the “only if” part, we assume that ( α, g ) is unfactorizable. Assume on thecontrary that α = β ◦ γ with both β and γ non-isomorphisms in C . Then we have( α, g ) = ( β, G ) ◦ ( γ, g ) . By Lemma 2.2(1), we have that both ( β, G ) and ( γ, g ) are non-isomorphisms in C ⋊ G . This contradicts to the unfactorizability of ( α, g ). (cid:3) The following result characterizes the UFP of morphisms in a skew group cate-gory.
Proposition 3.3.
Let G be a finite group and C be a finite G -category. Then amorphism ( α, g ) in C ⋊ G satisfies the UFP if and only if α satisfies the UFP in C .Proof. By Lemma 2.2(1), the morphism ( α, g ) is an isomorphism if and only if sois α . In the following proof, we will assume that both ( α, g ) and α : g ( x ) → y arenon-isomorphisms.For the “if” part, we assume that α : g ( x ) → y satisfies the UFP in C . Supposethat ( α, g ) : x → y has two factroizations into unfactorizable morphisms in C ⋊ G : x = x α ,g ) −→ x α ,g ) −→ · · · ( α n ,g n ) −→ x n = y and x = y β ,h ) −→ y β ,h ) −→ · · · ( β m ,h m ) −→ y m = y. The factorizations imply g n · · · g = g = h m · · · h in G . Moreover, the morphism α : g ( x ) → y has two factorizations in C : g ( x ) g n ··· g ( α ) −→ g n · · · g ( x ) g n ··· g ( α ) −→ · · · g n ( α n − ) −→ g n ( x n − ) α n −→ x n = y and g ( x ) h m ··· h ( β ) −→ h m · · · h ( y ) h m ··· h ( β ) −→ · · · h m ( β m − ) −→ h m ( y m − ) β m −→ y m = y. Here, in the first factorization we use g ( x ) = g n · · · g g ( x ), and in the secondone we use g ( x ) = h m · · · h h ( x ). By Lemmas 3.1 and 3.2, all the morphismsappearing in the above two factorizations are unfactorizable in C .Since α satisfies the UFP, we infer that n = m , and that there are isomorphisms θ i : g n · · · g i +1 ( x i ) → h n · · · h i +1 ( y i ), 1 ≤ i ≤ n −
1, such that the following diagram in C commutes. g ( x ) g n ··· g ( α ) / / g n · · · g ( x ) g n ··· g ( α ) / / θ (cid:15) (cid:15) · · · g n ( α n − ) / / g n ( x n − ) α n / / θ n − (cid:15) (cid:15) x n = yg ( x ) h n ··· h ( β ) / / h n · · · h ( y ) h n ··· h ( β ) / / · · · h n ( β n − ) / / h n ( y n − ) β n / / y n = y Set θ = Id g ( x ) and θ n = Id y . Then the above commutativity implies that θ i ◦ g n · · · g i +1 ( α i ) = h n · · · h i +1 ( β i ) ◦ θ i − (3.1)for each 1 ≤ i ≤ n . The identity for the case i = n means α n = β n ◦ θ n − .For each 1 ≤ i ≤ n −
1, we set a i = h − i +1 · · · h − n g n · · · g i +1 . In addition, we set a = 1 G = a n . Then we have a i g i = h i a i − (3.2)for each 1 ≤ i ≤ n . Here, to see a g = h , we use the fact that g n · · · g = h n · · · h .For each 1 ≤ i ≤ n , we set η i = h − i +1 · · · h − n ( θ i ). We observe η = Id x and η n = Id y . Then each η i : a i ( x i ) → y i is an isomorphism in C . Consequently, byLemma 2.2(1) we have an isomorphism ( η i , a i ) : x i → y i in C ⋊ G .Applying h − i +1 · · · h − n to (3.1), we have η i ◦ a i ( α i ) = β i ◦ h i ( η i − )(3.3)for each 1 ≤ i ≤ n . By (3.2) and (3.3), we have the following commutative diagramin C ⋊ G . x = x α ,g ) / / x α ,g ) / / ( η ,a ) (cid:15) (cid:15) x / / ( η ,a ) (cid:15) (cid:15) · · · ( α n − ,g n − ) / / x n − α n ,g n ) / / ( η n − ,a n − ) (cid:15) (cid:15) x n = yx = y β ,h ) / / y β ,h ) / / y / / · · · ( β n − ,h n − ) / / y n − β n ,h n ) / / y n = y This implies that the morphism ( α, g ) satisfies the UFP.The proof of the “only if” is similar and actually easier. We assume that( α, g ) : x → y satisfies the UFP in C ⋊ G . Suppose that α : g ( x ) → y has twofactorizations into unfactorizable morphisms in C : g ( x ) = x α −→ x α −→ · · · α n −→ x n = y and g ( x ) = y β −→ y β −→ · · · β m −→ y m = y. Then the morphism ( α, g ) : x → y in C ⋊ G has two factorizations: x = g − ( x ) ( α ,g ) −→ x α , G ) −→ · · · ( α n , G ) −→ x n = y and x = g − ( y ) ( β ,g ) −→ y β , G ) −→ · · · ( β m , G ) −→ y m = y. By Lemma 3.2, all the morphisms ( α , g ), ( β , g ), ( α i , G ) and ( β j , G ) are unfac-torizable, for 2 ≤ i ≤ n and 2 ≤ j ≤ m .Since the morphism ( α, g ) satisfies the UFP, then m = n and there are isomor-phisms ( γ i , g i ) : x i → y i , 1 ≤ i ≤ n −
1, such that the following diagram in C ⋊ G commutes. x = g − ( x ) ( α ,g ) / / x α , G ) / / ( γ ,g ) (cid:15) (cid:15) x / / ( γ ,g ) (cid:15) (cid:15) · · · ( α n − , G ) / / x n − α n , G ) / / ( γ n − ,g n − ) (cid:15) (cid:15) x n = yx = g − ( y ) ( β ,g ) / / y β , G ) / / y / / · · · ( β n − , G ) / / y n − β n , G ) / / y n = y KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 11
The commutativity implies g = g = · · · = g n − = 1 G . By Lemma 2.2(1), each γ i : x i → y i is an isomorphism in C . Consequently, the isomorphisms γ i make thefollowing diagram in C commute. g ( x ) = x α / / x α / / γ (cid:15) (cid:15) x / / γ (cid:15) (cid:15) · · · α n − / / x n − α n / / γ n − (cid:15) (cid:15) x n = yg ( x ) = y β / / y β / / y / / · · · β n − / / y n − β n / / y n = y This proves that α satisfies the UFP, as required. (cid:3) Recall that a finite EI category C is free provided that each morphism satisfiesthe UFP; compare [17, Definition 2.7 and Proposition 2.8]. For an alternativecharacterization of a free EI category, we refer to [27, Proposition 4.5].The following result follows immediately from Propositions 2.5 and 3.3. Proposition 3.4.
Let C be a finite G -category. Then C is a free EI category if andonly if so is C ⋊ G . (cid:3) Remark 3.5.
Let us sketch a shorter proof of Proposition 3.4 using categoryalgebras. By Proposition 2.5, we may assume that both C and C ⋊ G are EIcategories.Take an arbitrary field K of characteristic zero. By Proposition 2.1, we identifythe category algebra K ( C ⋊ G ) with the skew group algebra K C G . It is wellknown that K C is hereditary if and only if so is K C G ; see [22, Theorems 1.3(c)and 1.4]. Then Proposition 3.4 follows immediately from the following result dueto [17, Theorem 5.3]: the EI category C ( resp. C ⋊ G ) is free if and only if thecorresponding category algebra K C ( resp. K ( C ⋊ G )) is hereditary.4. Finite EI quivers and G -actions In this section, we recall basic facts on finite EI quivers. We prove a universalproperty of the free EI category associated to a finite EI quiver; see Proposition 4.2.We study finite group actions on finite EI quivers.4.1.
Categories associated to finite EI quivers.
Let Q = ( Q , Q ; s, t ) be afinite quiver, where Q and Q are the finite sets of vertices and arrows, respec-tively. The maps s, t : Q → Q assign to each arrow α its starting vertex s ( α ) andterminating vertex t ( α ), respectively.A path p = α n · · · α α of length n in Q consists of arrows α i satisfying t ( α i ) = s ( α i +1 ) for each 1 ≤ i ≤ n −
1. Here, we write concatenation from right to left.We set s ( p ) = s ( α ) and t ( p ) = t ( α n ). An arrow is identified with a path of lengthone. To each vertex i ∈ Q , we associate a trivial path e i of length zero, satisfying s ( e i ) = i = t ( e i ).A finite quiver Q is said to be acyclic , provided that there is no oriented cyclein Q , that is, there is no nontrivial path with the same starting and terminatingvertex. This is equivalent to the condition that there are only finitely many pathsin Q .Let H be a finite group, and let X be a right H -set, that is, H acts on X on theright. Let Y be a left H -set. The biset product X × H Y is defined to be the set X × Y / ∼ of equivalence classes with respect to the equivalence relation ∼ given by ( x.h, y ) ∼ ( x, h.y ) for x ∈ X, h ∈ H and y ∈ Y . By abuse of notation, the elements in X × Y / ∼ are still denoted by ( x, y ) for x ∈ X and y ∈ Y . Let G and K be finite groups. By a ( G, H )-biset X , we mean a set X which is aleft G -set and a right H -set satisfying ( g.x ) .h = g. ( x.h ) for any g ∈ G , x ∈ X and h ∈ H . Here, we use the dot to denote the group actions. Let Y be a ( H, K )-biset.Then the biset product X × H Y is naturally a ( G, K )-biset.
Example 4.1.
Let C be a finite EI category. For any two objects x and y , the Hom -set
Hom C ( x, y ) is naturally an (Aut C ( y ) , Aut C ( x )) -biset, where the actions are givenby the composition of morphisms in C .Denote by Hom C ( x, y ) the subset of Hom C ( x, y ) consisting of unfactorizable mor-phisms. As unfactorizable morphisms are closed under composition with isomor-phisms, Hom C ( x, y ) is an (Aut C ( y ) , Aut C ( x )) -sub-biset of Hom C ( x, y ) . Recall from [17, Definition 2.1] that a finite EI quiver ( Q, U ) consists of a finiteacyclic quiver Q and an assignment U = ( U ( i ) , U ( α )) i ∈ Q ,α ∈ Q . In more details,for each vertex i ∈ Q , U ( i ) is a finite group, and for each arrow α ∈ Q , U ( α ) is afinite ( U ( tα ) , U ( sα ))-biset. Here, we emphasize that each U ( α ) is nonempty.For any path p = α n · · · α α in Q , we define U ( p ) = U ( α n ) × U ( tα n − ) U ( α n − ) × U ( tα n − ) · · · × U ( tα ) U ( α ) × U ( tα ) U ( α ) . Then U ( p ) is naturally a ( U ( tp ) , U ( sp ))-biset. A typical element in U ( p ) will bedenoted by ( u n , · · · , u , u ) with each u i ∈ U ( α i ). For each vertex i ∈ Q , weidentify U ( e i ) with U ( i ).For two paths p, q satisfying s ( p ) = t ( q ), we have a natural isomorphism of( U ( tp ) , U ( sq ))-bisets U ( p ) × U ( tq ) U ( q ) ∼ −→ U ( pq ) , (4.1)sending (( u ′ m , · · · , u ′ ) , ( u n , · · · , u )) to ( u ′ m , · · · , u ′ , u n , · · · , u ), where pq denotesthe concatenation of paths.Each finite EI quiver ( Q, U ) gives rise to a finite EI category C ( Q, U ); see [17,Section 2]. The objects of C ( Q, U ) coincide with the vertices of Q . For two objects i and j , we have a disjoint unionHom C ( Q,U ) ( i, j ) = G { p paths in Q with s ( p )= i and t ( p )= j } U ( p ) . The composition of morphisms is induced by the concatenation of paths and theisomorphism (4.1). Since Q has only finitely many paths, we infer that C ( Q, U ) isa finite category. As e i is the only path starting and terminating at i , we infer thatHom C ( Q,U ) ( i, i ) = U ( e i ) = U ( i ) , (4.2)which is a finite group. We conclude that the category C ( Q, U ) is indeed finite EI.We mention the following immediate factHom C ( Q,U ) ( i, j ) = G { α ∈ Q | s ( α )= i, t ( α )= j } U ( α ) . (4.3)By [17, Proposition 2.8], the EI category C ( Q, U ) is free. Moreover, a finite EIcategory is free if and only if it is equivalent to C ( Q, U ) for some finite EI quiver(
Q, U ).4.2.
A universal property.
The free EI category C ( Q, U ) enjoys a certain uni-versal property; compare [17, Proposition 2.9].
Proposition 4.2.
Let D be a finite EI category. Assume that φ : Q → Obj( D ) isa map, ψ i : U ( i ) → Aut D ( φ ( i )) is a group homomorphism for each vertex i ∈ Q ,and that ψ α : U ( α ) → Hom D ( φ ( sα ) , φ ( tα )) is a map of ( U ( tα ) , U ( sα )) -bisets foreach arrow α ∈ Q . Then there is a unique functor Φ : C ( Q, U ) → D subject to thefollowing constraints: KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 13 (1) Φ( i ) = φ ( i ) for each i ∈ Q = Obj( C ( Q, U )) ;(2) Φ( x ) = ψ i ( x ) for each x ∈ U ( i ) = Aut C ( Q,U ) ( i ) ;(3) Φ( u ) = ψ α ( u ) for each u ∈ U ( α ) ⊆ Hom C ( Q,U ) ( s ( α ) , t ( α )) .Moreover, Φ is an equivalence of categories if and only if all the following con-ditions are satisfied:(E1) The finite EI category D is free;(E2) Whenever φ ( i ) and φ ( j ) are isomorphic in D , we have i = j ;(E3) Every object in D is isomorphic to φ ( i ) for some i ∈ Q ;(E4) Each ψ i is an isomorphism, and the maps ψ α induce a bijection, for any i, j ∈ Q , G { α ∈ Q | s ( α )= i,t ( α )= j } U ( α ) −→ Hom D ( φ ( i ) , φ ( j )) . Before giving the proof, we leave two comments to clarify the statements. The( U ( tα ) , U ( sα ))-biset structure on Hom D ( φ ( sα ) , φ ( tα )) is given as follows: for amorphism f : φ ( sα ) → φ ( tα ) in D , x ∈ U ( tα ) and x ′ ∈ U ( sα ), we have x.f.x ′ = ψ t ( α ) ( x ) ◦ f ◦ ψ s ( α ) ( x ′ ) . In the condition (E4), the domain of the bijection is a disjoint union; moreover, itimplies that ψ α ( u ) is unfactorizable in D for any u ∈ U ( α ). Proof.
Set C = C ( Q, U ). For any path p = α n · · · α α in Q and an element( u n , · · · , u , u ) ∈ U ( p ), we defineΦ( u n , · · · , u , u ) = ψ α n ( u n ) ◦ · · · ◦ ψ α ( u ) ◦ ψ α ( u ) . We claim that this is independent of the choice of the representatives; compare [17,the proof of Proposition 2.9].Assume that ( u n , · · · , u , u ) = ( v n , · · · , v , v ) in U ( p ). This means that thereare elements x i ∈ U ( tα i ) for each 1 ≤ i ≤ n −
1, such that the following identitieshold: v n = u n .x n − , v i = x − i .u i .x i − , and v = x − .u . Since each ψ α i is a map of bisets, we have ψ α n ( v n ) = ψ α n ( u n ) ◦ ψ t ( α n − ) ( x n − ) , ψ α i ( v i ) = ψ t ( α i ) ( x i ) − ◦ ψ α i ( u i ) ◦ ψ t ( α i − ) ( x i − ) , and ψ α ( v ) = ψ t ( α ) ( x ) − ◦ ψ α ( u ) . Then the following identity follows immediately.Φ( v n , · · · , v , v ) = ψ α n ( v n ) ◦ · · · ◦ ψ α ( v ) ◦ ψ α ( v )= ψ α n ( u n ) ψ t ( α n − ) ( x n − ) ◦ · · · ◦ ψ t ( α ) ( x ) − ψ α ( u ) ψ t ( α ) ( x ) ◦ ψ t ( α ) ( x ) − ψ α ( u )= ψ α n ( u n ) ◦ · · · ◦ ψ α ( u ) ◦ ψ α ( u ) = Φ( u n , · · · , u , u ) . The above claim yields a well-defined functor Φ. The uniqueness of Φ is clear,as ( u n , · · · , u , u ) might be viewed as the composition u n ◦ · · · ◦ u ◦ u in C .For the “only if” part of the second statement, we assume that Φ is an equiva-lence. Then (E1) is clear, since C is free. Since C is skeletal and Φ respects isomor-phism classes, (E2) follows immediately. The condition (E3) is just the densenessof Φ. For (E4), we observe that the equivalence Φ necessarily induces isomorphismsAut C ( i ) ≃ Aut D ( φ ( i ))of groups and bijections Hom C ( i, j ) ≃ Hom D ( φ ( i ) , φ ( j ))between the sets of unfactorizable morphisms. Then we apply (4.2) and (4.3). For the “if” part, we assume the conditions (E1)-(E4). By (E3), the functor Φis dense. It suffices to prove that for any i, j ∈ Q , the following mapΦ i,j : Hom C ( i, j ) −→ Hom D ( φ ( i ) , φ ( j )) , f Φ( f )is bijective.By (E4), each ψ i is an isomorphism, and then the case i = j follows. We nowassume that i = j . Then by (E2), φ ( i ) and φ ( j ) are not isomorphic.Recall from [17, Proposition 2.6] that each morphism in D has a factorizationinto unfactorizable morphisms. Since Φ is dense, any morphism g : φ ( i ) → φ ( j )admits a factorization φ ( i ) g −→ φ ( i ) g −→ φ ( i ) −→ · · · −→ φ ( i n − ) g n − −→ φ ( j )with each g k unfactorizable. By (E4), each g k belongs to the image of Φ. It followsthat there is a morphism f : i → j in C satisfying Φ( f ) = g . This proves that Φ i,j is surjective.It remains to show that Φ i,j is injective. Assume that p = α n · · · α α and q = β m · · · β β are two paths from i to j , and that ( u n , · · · , u , u ) ∈ U ( p ) and( v m , · · · , v , v ) ∈ U ( q ) satisfyΦ( u n , · · · , u , u ) = Φ( v m , · · · , v , v ) = g ′ . We claim that p = q and ( u n , · · · , u , u ) = ( v m , · · · , v , v ). Then we are done.For the claim, we observe that the morphism g ′ admits two factorizations: φ ( i ) ψ α ( u ) −→ φ ( i ) ψ α ( u ) −→ φ ( i ) −→ · · · −→ φ ( i n − ) ψ αn ( u n ) −→ φ ( j )and φ ( i ) ψ β ( v ) −→ φ ( j ) ψ β ( v ) −→ φ ( j ) −→ · · · −→ φ ( j m − ) ψ βm ( v m ) −→ φ ( j ) . Here, i k = t ( α k ) and j k = t ( β k ). By (E4), all the morphisms appearing in thetwo factorizations are unfactorizable. By (E1), the EI category D is free, that is,any morphism satisfies the UFP. Consequently, m = n and there are isomorphisms g ′ k : φ ( i k ) → φ ( j k ) making the following diagram commute. φ ( i ) ψ α ( u ) / / φ ( i ) g ′ (cid:15) (cid:15) ψ α ( u ) / / φ ( i ) g ′ (cid:15) (cid:15) / / · · · / / φ ( i n − ) g ′ n − (cid:15) (cid:15) ψ αn ( u n ) / / φ ( j ) φ ( i ) ψ β ( v ) / / φ ( j ) ψ β ( v ) / / φ ( j ) / / · · · / / φ ( j n − ) ψ βn ( v n ) / / φ ( j )By (E2), we infer that i k = j k ; moreover, by (E4) we obtain automorphisms a k ∈ Aut C ( i k ) = U ( i k ) satisfying ψ i k ( a k ) = g ′ k . The commutativity yields ψ β k +1 ( v k +1 ) ◦ ψ i k ( a k ) = ψ i k +1 ( a k +1 ) ◦ ψ α k +1 ( u k +1 )for each 0 ≤ k ≤ n −
1. Here, a and a n are the identity elements in U ( i ) and U ( j ),respectively. The above identity is equivalent to ψ β k +1 ( v k +1 .a k ) = ψ α k +1 ( a k +1 .u k +1 ) . By the bijection in (E4), we infer that β k +1 = α k +1 and that v k +1 .a k = a k +1 .u k +1 for each 0 ≤ k ≤ n −
1. It follows that p = q ; moreover, in view of the definition of U ( p ) via biset products, we infer that ( u n , · · · , u , u ) = ( v n , · · · , v , v ), provingthe claim. (cid:3) KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 15 G -actions on finite EI quivers. Let (
Q, U ) be a finite EI quiver. An auto-morphism σ = ( σ , σ ) of ( Q, U ) consists of an automorphism σ : Q → Q of theacyclic quiver Q and an assignment σ = ( σ i , σ α ) i ∈ Q ,α ∈ Q of isomorphisms. Moreprecisely, for each i ∈ Q , σ i : U ( i ) ∼ −→ U ( σ ( i ))is an isomorphism of groups; for each arrow α ∈ Q , σ α : U ( α ) ∼ −→ U ( σ ( α ))is an isomorphism of ( U ( tα ) , U ( sα ))-bisets. Here, the ( U ( tα ) , U ( sα ))-biset struc-ture on U ( σ ( α )) is induced by the group isomorphisms σ t ( α ) and σ s ( α ) .The composition of two automorphisms σ = ( σ , σ ) and θ = ( θ , θ ) on ( Q, U )is given by θ ◦ σ = ( θ ◦ σ , θ ⋆ σ ) , where the assignment θ ⋆ σ is given by( θ ⋆ σ ) i = θ σ ( i ) ◦ σ i and ( θ ⋆ σ ) α = θ σ ( α ) ◦ σ α . We denote by Aut(
Q, U ) the group of automorphisms of (
Q, U ), whose multiplica-tion is given by the composition of automorphisms.We observe that each automorphism σ = ( σ , σ ) on ( Q, U ) induces an au-tomorphism ˜ σ on C ( Q, U ) in the following natural manner: the action of ˜ σ onobjects is given by σ ; for u ∈ U ( i ), we have ˜ σ ( u ) = σ i ( u ) ∈ U ( σ ( i )); for a path p = α n · · · α α and a morphism ( u n , · · · , u , u ) ∈ U ( p ), we have˜ σ ( u n , · · · , u , u ) = ( σ α n ( u n ) , · · · , σ α ( u ) , σ α ( u )) ∈ U ( σ ( p )) . This actually gives rise to an injective group homomorphismAut(
Q, U ) ֒ → Aut( C ( Q, U )) , σ ˜ σ. (4.4)Let G be a finite group. By a G -action on a finite EI quiver ( Q, U ), we mean agroup homomorphism ρ : G −→ Aut(
Q, U ) , g ρ ( g ) = ( ρ ( g ) , ρ ( g ) ) . Composing ρ with (4.4), the G -action makes C ( Q, U ) into a G -category.The following convention for the G -action ρ will simplify the notation. For g ∈ G and i ∈ Q = Obj( C ( Q, U )), we write g ( i ) = ρ ( g ) ( i ) ∈ Q . (4.5)Similarly, for α ∈ Q , we write g ( α ) = ρ ( g ) ( α ) ∈ Q . For a ∈ U ( i ) = Aut C ( Q,U ) ( i )with i ∈ Q , we write g ( a ) = ρ ( g ) i ( a ) ∈ U ( g ( i )) . (4.6)For u ∈ U ( α ) ⊆ Hom C ( Q,U ) ( i, j ) with an arrow α ∈ Q from i to j , we write g ( u ) = ρ ( g ) α ( u ) ∈ U ( g ( α )) . (4.7)For admissible G -categories, we refer to Definition 2.3. Lemma 4.3.
Let G be a finite group with a G -action ρ on ( Q, U ) as above. Thenthe corresponding G -category C ( Q, U ) is admissible.Proof. Let g ∈ G and i ∈ Obj( C ( Q, U )) = Q such that g ( i ) = i . Recall thatHom C ( Q,U ) ( g ( i ) , i ) = G { p paths in Q with s ( p )= g ( i ) and t ( p )= i } U ( p ) . Since Q is acyclic, there is no path p satisfying s ( p ) = g ( i ) and t ( p ) = i . Therefore,the set Hom C ( Q,U ) ( g ( i ) , i ) is actually empty, proving that the G -category C ( Q, U )is admissible. (cid:3) The quotient EI quiver
In this section, we fix a finite group G and a finite EI quiver ( Q, U ) with a G -action ρ . We will construct its ‘orbifold’ quotient EI quiver ( Q, U ) explicitly. Weprove that the free EI category C ( Q, U ) is equivalent to the skew group category C ( Q, U ) ⋊ G ; see Theorem 5.1.For each vertex i and each arrow α in Q , their stabilizers are denoted by G i and G α , respectively. We observe that G α ⊆ G s ( α ) ∩ G t ( α ) .5.1. The construction of ( Q, U ) . The finite quiver Q = ( Q , Q ; s, t ) is just thequotient quiver of Q by G . In more details, Q = Q /G and Q = Q /G are thecorresponding sets of G -orbits, and the maps s and t are induced by the ones of Q . Since Q is acyclic, we infer that the finite quiver Q is acyclic. By definition, wehave the canonical projections π : Q −→ Q and π : Q −→ Q . The vertices and arrows in Q are written in the bold form. For example, the verticesare usually denoted by i and j .To define the assignment U , we have to fix three maps ι : Q −→ Q , ι : Q −→ Q , and g ( − ) : Q −→ G (5.1)satisfying the following conditions: π ◦ ι = Id Q , π ◦ ι = Id Q , t ( ι ( α )) = ι ( t ( α )) and s ( ι ( α )) = g α ( ι ( s α ))(5.2)for each arrow α ∈ Q . Here, we use the convention (4.5) for g α ( ι ( s α )).The inclusion G ι ( α ) ⊆ G ι ( t α ) makes G ι ( t α ) a right G ι ( α ) -set. The injectivegroup homomorphism G ι ( α ) ⊆ G s ( ι ( α )) ∼ −→ G ι ( s α ) , k g − α kg α makes G ι ( s α ) a left G ι ( α ) -set. Therefore, we have the biset product G ι ( t α ) × G ι α ) G ι ( s α ) , which is naturally a ( G ι ( t α ) , G ι ( s α ) )-biset. A typical element in the above bisetproduct is written as ( h, g ) with h ∈ G ι ( t α ) and g ∈ G ι ( s α ) . By definition, wehave ( hk, g ) = ( h, g − α kg α g )(5.3)for each k ∈ G ι ( α ) .Finally, we choose a right coset decomposition G ι ( t α ) = m α G r =1 h α ,r G ι ( α ) . (5.4)Consequently, any element in G ι ( t α ) × G ι α ) G ι ( s α ) is uniquely written as ( h α ,r , k )for 1 ≤ r ≤ m α and k ∈ G ι ( s α ) .The construction of the assignment U is as follows. For each vertex i of Q , weset U ( i ) = U ( ι ( i )) ⋊ G ι ( i ) . Here, we note that G ι ( i ) acts on U ( ι ( i )) by group automorphisms. Therefore, thesemi-direct product is well defined. For each arrow α : i → j in Q , we set U ( α ) = U ( ι ( α )) × ( G ι ( t α ) × G ι α ) G ι ( s α ) )= U ( ι ( α )) × ( G ι ( j ) × G ι α ) G ι ( i ) ) . KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 17
A typical element in U ( α ) is denoted by ( u, ( h α ,r , k )). The right U ( i )-action isgiven by ( u, ( h α ,r , k )) . ( a, g ) = ( u. ( g α k ( a )) , ( h α ,r , kg ))for any ( a, g ) ∈ U ( i ) = U ( ι ( i )) ⋊ G ι ( i ) . Here, using the convention (4.6), weobserve that g α k ( a ) lies in U ( g α ι ( i )) = U ( sι ( α )), and that u. ( g α k ( a )) means theright U ( sι ( α ))-action on U ( ι ( α )).To describe the left U ( j )-action, we take an arbitrary element ( b, h ) ∈ U ( j ) = U ( ι ( j )) ⋊ G ι ( j ) . Assume that hh α ,r = h α ,p k ′ for some 1 ≤ p ≤ m α and k ′ ∈ G ι ( α ) . The left U ( j )-action is given by( b, h ) . ( u, ( h α ,r , k )) = ( h − α ,p ( b ) .k ′ ( u ) , ( h α ,p k ′ , k ))= ( h − α ,p ( b ) .k ′ ( u ) , ( h α ,p , g − α k ′ g α k )) . Here, we observe that h − α ,p ( b ) lies in U ( ι ( j )) = U ( tι ( α )), and that k ′ ( u ) lies in U ( ι ( α )). The convention (4.7) is used for k ′ ( u ). Finally, h − α ,p ( b ) .k ′ ( u ) denotes theleft U ( tι ( α ))-action on U ( ι ( α )).The above actions make U ( α ) a ( U ( j ) , U ( i ))-biset. In summary, we have definedthe finite EI quiver ( Q, U ).5.2.
An equivalence of categories.
The EI quiver (
Q, U ) might be viewed asa certain ‘orbifold’ quotient of (
Q, U ). We mention a similar construction in theclassic work [2, Section 3] on graphs of groups. The EI quiver (
Q, U ) depends onthe choices in (5.1) and (5.4).The following result justifies the quotient construction.
Theorem 5.1.
Let ( Q, U ) be a finite EI quiver with a G -action ρ , and let ( Q, U ) be its quotient as above. Then there is an equivalence of categories C ( Q, U ) ≃ C ( Q, U ) ⋊ G. Proof.
We write C = C ( Q, U ) in this proof. We will apply Proposition 4.2 to deducethe equivalence. We use the map ι : Q → Q = Obj( C ⋊ G ), and the identification U ( i ) = U ( ι ( i )) ⋊ G ι ( i ) = Aut C ⋊ G ( ι ( i )) , where the right equality follows by combining Lemma 4.3 and Corollary 2.6.To apply Proposition 4.2, it remains to construct for each arrow α in Q , a mapbetween bisets U ( α ) −→ Hom C ⋊ G ( ι ( s α ) , ι ( t α )) . We will see in the following construction that these maps between bisets yield therequired bijection in (E4).By Proposition 3.4, the category C ⋊ G is EI free, therefore the condition (E1) issatisfied. For two different vertices i and j , the vertices ι ( i ) and ι ( j ) are not inthe same G -orbit. By Lemma 2.2(2), ι ( i ) and ι ( j ) are not isomorphic in C ⋊ G ,proving the condition (E2). For each vertex i ∈ Q , the corresponding object i is isomorphic to ι ( π ( i )) in C ⋊ G , proving (E3). Once we construct the abovemaps between bisets, we will infer by Proposition 4.2 the required equivalence ofcategories. To construct the required maps, we take arbitrary vertices i and j in Q . ByLemma 3.2, we haveHom C ⋊ G ( ι ( i ) , ι ( j )) = { ( θ, g ) | g ∈ G, θ ∈ Hom C ( g ( ι ( i )) , ι ( j )) } = G g ∈ G Hom C ( g ( ι ( i )) , ι ( j )) × { g } . = G g ∈ G G { α ∈ Q | s ( α )= g ( ι ( i )) , t ( α )= ι ( j ) } U ( α ) × { g } . Here, for the last equality we use (4.3).For each arrow α : i → j , we define the following subset of Hom C ⋊ G ( ι ( i ) , ι ( j )) S ( α ) = G { α ∈ Q | π ( α )= α , t ( α )= ι ( j ) } G { g ∈ G | s ( α )= g ( ι ( i ) } U ( α ) × { g } . Recall from Example 4.1 that the ( U ( j ) , U ( i ))-biset structure on Hom C ⋊ G ( ι ( i ) , ι ( j ))is induced by composition (2.1) of morphisms in C ⋊ G . Then we infer that S ( α )is a ( U ( j ) , U ( i ))-sub-biset. Now, we have the following disjoint unionHom C ⋊ G ( ι ( i ) , ι ( j )) = G { α ∈ Q | s ( α )= i , t ( α )= j } S ( α ) . (5.5)We will complete the proof by establishing an isomorphism of ( U ( j ) , U ( i ))-bisets U ( α ) ≃ S ( α )for any arrow α : i → j . Indeed, in view of (5.5), the isomorphism yields therequired bijection in (E4). Then we are done.To analyze S ( α ), we observe that in the index set of the outer disjoint union, thearrows α are of the form h ( ι ( α )) for some h ∈ G ι ( j ) . By the coset decomposition(5.4), there is a unique 1 ≤ r ≤ m α satisfying α = h α ,r ( ι ( α )) . For simplicity, we write h r for h α ,r . In the inner disjoint union, we have g ( ι ( i )) = s ( α ) = h r ( sι ( α )) = h r g α ( ι ( i )) . Consequently, there is a unique k ∈ G ι ( i ) satisfying g = h r g α k. The above analysis implies that S ( α ) = m α G r =1 G k ∈ G ι i ) U ( h r ( ι ( α ))) × { h r g α k } . We observe that U ( ι ( α )) is bijective to each U ( h r ( ι ( α ))) via ρ ( h r ) ι ( α ) , whichsends u ∈ U ( ι ( α )) to h r ( u ) ∈ U ( h r ( ι ( α ))). Moreover, the biset product G ι ( j ) × G ι α ) G ι ( i ) is bijective to { , , · · · , m α } × G ι ( i ) , which is further bijective to the following disjoint union m α G r =1 G k ∈ G ι i ) { h r g α k } . Using these bijections, we infer that the following map U ( α ) = U ( ι ( α )) × ( G ι ( j ) × G ι α ) G ι ( i ) ) −→ S ( α ) , ( u, ( h r , k )) ( h r ( u ) , h r g α k ) KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 19 is a bijection. We omit the routine verification that this explicit bijection is indeeda map of ( U ( j ) , U ( i ))-bisets. This is the required isomorphism of bisets. (cid:3) The above construction of the quotient EI quiver (
Q, U ) is rather general. Inwhat follows, we impose conditions which will simplify the construction.
Remark 5.2.
Assume that the G -action ρ on ( Q, U ) satisfies the following trivialityconditions:(1) for each i ∈ Q , g ∈ G i and a ∈ U ( i ), we have g ( a ) = a ;(2) for each α ∈ Q , g ∈ G α and u ∈ U ( α ), we have g ( u ) = u .Then the quotient EI quiver ( Q, U ) is described as follows: U ( i ) = U ( ι ( i )) × G ι ( i ) is the direct product; for each arrow α : i → j in Q , we have U ( α ) = U ( ι ( α )) × ( G ι ( j ) × G ι α ) G ι ( i ) ) . Its typical element is denoted by ( u, ( h, k )) for u ∈ U ( ι ( α )), h ∈ G ι ( j ) and k ∈ G ι ( i ) . The right U ( i )-action is given by( u, ( h, k )) . ( a, g ) = ( u.g α ( a ) , ( h, kg )) . The left U ( j )-action is given by( b, g ′ ) . ( u, ( h, k )) = ( b.u, ( g ′ h, k )) . Here, u.g α ( a ) and b.u mean the right U ( sι ( α ))-action and the left U ( tι ( α ))-actionon U ( ι ( α )), respectively.Let ∆ = (∆ , ∆ ; s, t ) be a finite acyclic quiver. Recall that the path category P ∆ is defined as follows: Obj( P ∆ ) = ∆ and Hom P ∆ ( i, j ) consists of all paths from i to j ; the composition is given by concatenation of paths.Denote by (∆ , U tr ) the EI quiver with trivial assignment U tr , that is, each group U tr ( i ) is trivial and each biset U tr ( α ) has only one element. We observe C (∆ , U tr ) = P ∆ . (5.6)Let G be a finite group which acts on ∆ by quiver automorphisms. Then G actson the associated EI quiver (∆ , U tr ). Denote by (∆ , U tr ) the quotient EI quiver.Therefore, ∆ is the quotient quiver of ∆ by G . Fix the choices (5.1). In view ofRemark 5.2, the assignment U tr is described as follows: for each vertex i of ∆, wehave U tr ( i ) = G ι ( i ) ;for each arrow α : i → j in ∆, we have U tr ( α ) = G ι ( i ) × G ι α ) G ι ( j ) , whose ( G ι ( i ) , G ι ( j ) )-biset structure is given by the multiplication of G ι ( i ) fromthe left, and of G ι ( j ) ) from the right.In view of (5.6), we have the following special case of Theorem 5.1. Corollary 5.3.
Let ∆ be a finite acyclic quiver with a G -action. Keep the notationas above. Then there is an equivalence of categories C (∆ , U tr ) ≃ P ∆ ⋊ G. (cid:3) Categories and algebras associated to Cartan triples
In this section, we will first recall the algebras [11] and EI categories [4] asso-ciated to Cartan triples. For a finite group action on a finite acyclic quiver, wegive sufficient conditions on when the quotient EI quiver is of Cartan type. Con-sequently, the skew group algebra of the path algebra is Morita equivalent to thealgebra studied in [11]; see Theorem 6.5.For two nonzero integers a and b , we denote by gcd( a, b ) their greatest commondivisor, which is always assumed to be positive.6.1. Cartan triples.
Let n ≥ n × n matrix C =( c ij ) with integer coefficients is called a symmetrizable generalized Cartan matrix ,provided that the following conditions are satisfied:(C1) c ii = 2 for all i ;(C2) c ij ≤ i = j , and c ij < c ji < D = diag( c , · · · , c n ) with c i ∈ Z ≥ for all i such that the product matrix DC is symmetric.The matrix D appearing in (C3) is called a symmetrizer of C . For brevity, asymmetrizable generalized Cartan matrix is called a Cartan matrix.Let C = ( c ij ) be a Cartan matrix. An (acyclic) orientation of C is a subsetΩ ⊂ { , , · · · , n } × { , , · · · , n } such that the following conditions are satisfied:(O1) { ( i, j ) , ( j, i ) } ∩ Ω = ∅ if and only if c ij < i , i ) , ( i , i ) , · · · , ( i t , i t +1 )) with t ≥ i s , i s +1 ) ∈ Ω for all 1 ≤ s ≤ t , we have i = i t +1 .Following [4], we will call ( C, D,
Ω) a
Cartan triple , where C is a Cartan matrix, D its symmetrizer and Ω an orientation of C .In what follows, we recall that, associated to each Cartan triple, there are a finitefree EI category C ( C, D,
Ω) and a finite dimensional algebra H ( C, D,
Ω).Let Q = Q ( C, Ω) be the finite quiver with the set of vertices Q = { , , · · · , n } and with the set of arrows Q = { α ( g ) ij : j → i | ( i, j ) ∈ Ω , ≤ g ≤ gcd( c ij , c ji ) } ⊔ { ε i : i → i | ≤ i ≤ n } . Let Q ◦ = Q ◦ ( C, Ω) be the quiver obtained from Q by deleting all the loops ε i . Bythe condition (O2), we infer that the finite quiver Q ◦ is acyclic.We recall the finite EI quiver ( Q ◦ , X ). The assignment X is given as follows: X ( i ) = h η i | η c i i = 1 i is a cyclic group of order c i ; for each ( i, j ) ∈ Ω, we set G ij = h η ij | η gcd( c i ,c j ) ij = 1 i to be a cyclic group of order gcd( c i , c j ). There areinjective group homomorphisms G ij ֒ → X ( i ) , η ij η ci gcd( ci,cj ) i and G ij ֒ → X ( j ) , η ij η cj gcd( ci,cj ) j . Then we have the ( X ( i ) , X ( j ))-biset X ( i ) × G ij X ( j ). We set X ( α ( g ) ij ) = X ( i ) × G ij X ( j )for each 1 ≤ g ≤ gcd( c ij , c ji ). Definition 6.1. ([4, Definition 4.1]) Associated to a Cartan triple (
C, D,
Ω), thefinite EI category C ( C, D,
Ω) is defined to be the free EI category C ( Q ◦ , X ) associ-ated to the above EI quiver ( Q ◦ , X ). We say that such EI quivers ( Q ◦ , X ) and EIcategories C ( C, D,
Ω) are of
Cartan type . (cid:3) KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 21
Let K be a field. The following algebras [11] play a fundamental role in categori-fying the root lattices for non-symmetric Caran matrices. For more background,we refer to [10]. Definition 6.2. ([11, Section 1.4]) Let (
C, D,
Ω) be a Cartan triple with Q = Q ( C, Ω). Consider the following K -algebra H ( C, D,
Ω) = K Q/I, where K Q is the path algebra of Q , and I is the two-sided ideal of K Q generatedby the following set { ε c k k , ε ci gcd( ci,cj ) i α ( g ) ij − α ( g ) ij ε cj gcd( ci,cj ) j | k ∈ Q , ( i, j ) ∈ Ω , ≤ g ≤ gcd( c ij , c ji ) } . (cid:3) We will recall from [4, Subsection 4.2] the construction of a new Cartan triple( C ′ , D ′ , Ω ′ ) from a given one ( C, D,
Ω), which depends on the characteristic of K .Recall that D = diag( c , · · · , c n ). Construction ( ‡ ) for the case char( K ) = p >
0. Assume that c i = p r i d i satisfying r i ≥ p, d i ) = 1. For each 1 ≤ i, j ≤ n , we setΣ pij = { ( l i , l j ) | ≤ l i < d i , ≤ l j < d j , l i p r i ≡ l j p r j (mod gcd( d i , d j )) } . The rows and columns of the Cartan matrix C ′ and its symmetrizer D ′ areindexed by the following set M = G ≤ i ≤ n { ( i, l i ) | ≤ l i < d i } . The diagonal entries of C ′ are 2, and the off-diagonal entries are given as follows: c ′ ( i,l i ) , ( j,l j ) = ( − gcd( c ij , c ji ) p r j − min( r i ,r j ) , if ( l i , l j ) ∈ Σ pij ;0 , otherwise.Let D ′ be a diagonal matrix, whose ( i, l i )-th component is given by p r i . SetΩ ′ = { (( i, l i ) , ( j, l j )) | ( i, j ) ∈ Ω , ( l i , l j ) ∈ Σ pij } , which is an orientation of C ′ . Construction ( ‡ ) for the case char( K ) = 0. This is very similar to the aboveconstruction. We put d i = c i and replace Σ pij byΣ ij = { ( l i , l j ) | ≤ l i < c i , ≤ l j < c j , l i ≡ l j (mod gcd( c i , c j )) } . (6.1)The off-diagonal entries of C ′ is given by c ′ ( i,l i ) , ( j,l j ) = ( − gcd( c ij , c ji ) , if ( l i , l j ) ∈ Σ ij ;0 , otherwise.We observe that C ′ is symmetric and that D ′ is the identity matrix.We say that K has enough roots of unity for D , if for each 1 ≤ i ≤ n , thepolynomial t c i − K [ t ]. Theorem 6.3.
Assume that ( C, D, Ω) is a Cartan triple and that K has enoughroots of unity for D . Keep the notation in Construction ( ‡ ) . Then there is anisomorphism of algebras K C ( C, D, Ω) ≃ H ( C ′ , D ′ , Ω ′ ) . Proof.
This result is due to [4, Theorem 4.3]. We mention that the assumption hereon K is slightly weaker than the one therein. Since each polynomial t c i − ‡ ) for each case, the ground field K has a ( Q ni =1 d i )-th primitive root of unity. Then the proof of [4, Theorem 4.3], in particular, theargument in [4, Section 5], carries through under the weaker assumption here. (cid:3) We are interested in the following special case.
Proposition 6.4.
Assume that char( K ) = p > and that ( C, D, Ω) is a Cartantriple such that each c i is a p -power. Then there is an isomorphism of algebras K C ( C, D, Ω) ≃ H ( C, D, Ω) , which identifies Span K { Id i , η i , · · · , η c i − i } with Span K { e i , ε i , · · · , ε c i − i } . Here, Span K means the subspace spanned by the mentioned elements. Both Id i and η i are viewed as automorphisms of i in C ( C, D,
Ω). Similarly, the trivial path e i and the loop ε i are viewed as elements in H ( C, D,
Ω).
Proof.
The assumption on entries of D implies that ( C ′ , D ′ , Ω ′ ) = ( C, D,
Ω), wherewe identify ( i, ∈ M with i ; see [4, Example 6.7]. For the same reason, thepolynomials t c i − K has enough roots of unity for D . Then theisomorphism follows from Theorem 6.3. By the proof of [4, Theorem 4.3], theisomorphism clearly identifies the above two subspaces. (cid:3) From quotient to Cartan type.
We study the situation of Corollary 5.3.Let G be a finite group and let ∆ be a finite acyclic quiver with a G -action. Wegive conditions on when the quotient EI quiver (∆ , U tr ) is of Cartan type.The following natural conditions will be imposed on the quiver ∆.( †
1) For each i , the stabilizer G i = h ξ i | ξ a i i = 1 i is cyclic with order a i .( †
2) For each arrow α : i → j , we have that ξ ai gcd( ai,aj ) i = ξ aj gcd( ai,aj ) j and bothbelong to G α .( †
3) For each g ∈ G , we have ξ g ( i ) = gξ i g − .The condition ( †
3) means that the choice of the specific generators ξ i for G i iscompatible with the G -action. It follows from ( †
2) that the inclusion G α ⊆ G i ∩ G j is an equality.Associated to the G -action on ∆, we will define a Cartan triple ( C, D,
Ω); com-pare [19, Section 14.1]. The rows and columns of C and D are indexed by the orbitset ∆ = ∆ /G . For each G -orbit i of vertices, the corresponding diagonal entryof D is c i = | G || i | , where | i | denotes the cardinality of the G -orbit. For any vertices i and j , wedenote by N i , j the number of arrows in ∆ between the G -orbit i and G -orbit j .The corresponding off-diagonal entry of C is given by c i , j = − N i , j | j | . The orientation Ω is consistent with that of ∆, that is, ( j , i ) ∈ Ω if and only ifthere is an arrow from i to j in ∆.By the equality c i c i , j = c j c j , i , we infer the following useful identity − c i , j gcd( c i , j , c j , i ) = c j gcd( c i , c j ) . (6.2) Theorem 6.5.
Assume that the G -action on ∆ satisfies ( † - ( † and that the as-sociated Cartan triple is ( C, D, Ω) . Denote by ( Q ◦ , X ) the corresponding EI quiverof Cartan type. Then there is an isomorphism of EI quivers (∆ , U tr ) ≃ ( Q ◦ , X ) . Moreover, we have the following immediate consequences.
KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 23 (1) There is an equivalence of categories P ∆ ⋊ G ≃ C ( C, D, Ω) . (2) Assume that K has enough roots of unity for D and that the Cartan triple ( C ′ , D ′ , Ω ′ ) is given in Construction ( ‡ ) . Then the algebras K ∆ G and H ( C ′ , D ′ , Ω ′ ) are Morita equivalent.(3) Assume that char( K ) = p > and that G is a p -group. Then the algebras K ∆ G and H ( C, D, Ω) are Morita equivalent.Proof. We first prove the isomorphism of EI quivers. Take two vertices i = Gi and j = Gj in ∆. We observe c i = a i and c j = a j . For each arrow α between i and j in ∆, we have observed that G α = G i ∩ G j , which is of order gcd( | G i | , | G j | ). Thenwe have | Gα | = | G | gcd( c i , c j ) . It follows that the number of arrows between i and j in ∆ equals N i , j | Gα | = − c i , j | j | gcd( c i , c j ) | G | = − c i , j gcd( c i , c j ) c j = gcd( c i , j , c j , i ) . Here, the last equality uses (6.2). Recall that the vertex set of Q ◦ is bijective tothe index set of rows of C , namely the vertex set ∆ . By comparing the number ofarrows, we identify ∆ with Q ◦ .We now compare the assignments U tr and X . By ( † U tr ( i ) = G ι ( i ) is cyclic of order c i = a i . For each arrow α : i → j in ∆, we write ι ( i ) = i and ι ( j ) = j . By (5.2), we have the arrow ι ( α ) : g α ( i ) → j in ∆. By ( †
2) and( † g α ξ ai gcd( ai,aj ) i g − α = ξ aj gcd( ai,aj ) j . Recall that U tr ( α ) = G i × G ι α ) G j . In view of (5.3), the above identity impliesthat, in U tr ( α ), we have( ξ ai gcd( ai,aj ) i , G ) = (1 G , ξ aj gcd( ai,aj ) j ) . This actually implies that the following map is well defined U tr ( α ) −→ X ( i ) × G ij X ( j ) = X ( α ) , ( ξ ai , ξ bj ) ( η ai , η bj ) . The above map is bijective and respects the ( G i , G j )-biset structures. We readilydeduce that the assignment U tr is isomorphic to X , as required.For (1), we recall that C ( C, D,
Ω) = C ( Q ◦ , X ). Then the equivalence of categoriesfollows from the obtained isomorphism of EI quivers and Corollary 5.3.For (2) and (3), we recall that the path algebra K ∆ is identified with the categoryalgebra K P ∆ of the path category. By Proposition 2.1, we identify K ( P ∆ ⋊ G ) with K P ∆ G . Recall from [28, Proposition 2.2] that the category algebras of two equiv-alent categories are Morita equivalent. Applying (1), we infer that K C ( C, D,
Ω) and K ( P ∆ ⋊ G ) are Morita equivalent. In summary, we have obtained that K C ( C, D,
Ω)and K ∆ G are Morit equivalent.Now, the required statement in (2) follows from the isomorphism in Theorem 6.3.For (3), we observe that each c i is a p -power, as G is a p -group. We apply theisomorphism in Proposition 6.4. (cid:3) Although the conditions ( † †
3) are technical, as we will see, natural examplesare ubiquitous. The following construction is inspired by [23, Section 5.3].
Example 6.6.
Let n ≥ and G be a finite group. For each ≤ i ≤ n , we fix ξ i ∈ G and assume that ξ i is of order a i . The cyclic subgroup generated by ξ i isdenoted by H i . The elements ξ i may not be distinct. Denote by G/H i the set ofleft H i -cosets, whose elements are denoted by gH i .We construct an acyclic quiver ∆ as follows: the set ∆ of vertices is a disjointunion F ni =1 G/H i × { i } ; only if i < j and ξ ai gcd( ai,aj ) i = ξ aj gcd( ai,aj ) j , each coset g ( H i ∩ H j ) is viewed as an arrow starting at ( gH i , i ) and terminating at ( gH j , j ) . Thenatural action of G on left cosets induces a G -action on ∆ . It is trivial to verifythat ( † - ( † do hold for the G -action. The following example is our main concern.
Example 6.7.
Let G = h ξ | ξ a = 1 i be a cyclic group of order a . Assume that G acts on ∆ such that G α = G s ( α ) ∩ G t ( α ) for each arrow α in ∆ . Then the conditions ( † - ( † are satisfied.For each vertex i with | G i | = a i , we take the generator ξ i to be ξ aai . Then it isdirect to check that ( † - ( † hold. Remark 6.8.
We observe that any Cartan triple does arise in the situation ofExample 6.7. More precisely, given any Cartan triple (
C, D,
Ω), we will constructan acyclic quiver with a cyclic group action such that its associated Cartan tripleis the given one; compare [19, Section 14.1].Assume that c = lcm( c , c , · · · , c n ) is the least common multiple of the entriesof D . Set d i = cc i . We construct an acyclic quiver ∆ as follows. The vertex set andarrow set are given by ∆ = { ( i, l i ) | ≤ i ≤ n, ≤ l i < d i } , and∆ = { α ( g )( i,l i ) , ( j,l j ) : ( j, l j ) → ( i, l i ) | ( i, j ) ∈ Ω , ( l i , l j ) ∈ Σ ij , ≤ g ≤ gcd( c ij , c ji ) } , respectively, where Σ ij is defined in (6.1). We observe the following identity | Σ ij | = d i d j gcd( d i , d j ) = c gcd( c i , c j ) = − c ij d j gcd( c ij , c ji ) . Let G = h σ | σ c = 1 i be a cyclic group of order c . Then G acts on ∆ such that σ ( i, l i ) = ( i, l i + 1) and σ ( α ( g )( i,l i ) , ( j,l j ) ) = α ( g )( i,l i +1) , ( j,l j +1) . Here, we identify ( i, d i ) with ( i, G -action on ∆. It is clear that G α = G s ( α ) ∩ G t ( α ) for each arrow α in ∆. Then by Example 6.7, Theorem 6.5applies to this G -action. Moreover, the associated Cartan triple coincides with thegiven one.The following counter-example shows that the condition G α = G s ( α ) ∩ G t ( α ) inExample 6.7 is necessary. Example 6.9.
Let ∆ be the following Kronecker quiver. α & & β Let G = { G , ξ } be a cyclic group of order which acts on ∆ by interchanging α and β . We observe that G α = G β = { G } ( G ∩ G = G . It follows that ( † isnot satisfied. KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 25
The quotient quiver ∆ is of type A . α / / The assignment U tr is described as follows: U tr ( ) = G = U tr ( ) , and U tr ( α ) = G × G. The quotient EI quiver (∆ , U tr ) is not of Cartan type. Induced modules and folding
In this final section, we first study induced modules on a skew group algebra.For a finite cyclic group action on a finite acyclic quiver, the main goal of the pa-per, Theorem 7.8, constructs a categorification of the folding projection betweenthe relevant root lattices. In the Dynkin cases, the restriction of the categorifica-tion to indecomposable modules corresponds to the folding of positive roots; seeProposition 7.9.7.1.
Generalities on induced modules.
Let A be a finite dimensional K -algebra,and let G be a finite group acting on A by algebra automorphisms. Denote by A G the skew group algebra. We view A as a subalgebra of A G by identifying a ∈ A with a G ∈ A G .For a left A -module M , we define a left A G -module M G , the induced module ,as follows: M G = M ⊗ K G as a vector space, and its left A G -action is given by( a g ) . ( m h ) = ( gh ) − ( a ) .m gh for any a g ∈ A G and m h ∈ M G . There is an isomorphism of left A G -modules ( A G ) ⊗ A M ∼ −→ M G, ( a g ) ⊗ m g − ( a ) .m g. (7.1)Similarly, for a right A -module N , we have a right A G -module N G = N ⊗ K G such that its right A G -action is given by( n h ) . ( a g ) = n.h ( a ) hg. There is an isomorphism of right A G -modules N ⊗ A ( A G ) ∼ −→ N G, n ⊗ ( a g ) n.a g. For each left A -module M and g ∈ G , the twisted A -module g M is defined asfollows: g M = M as a vector space, where an element m ∈ M corresponds to g m ∈ g M ; its left A -action is given by a. g m = g ( g ( a ) .m )for any a ∈ A . This yields the twisting endofunctor g ( − ) on A -mod.The following facts are contained in [22, Proposition 1.8]. Lemma 7.1.
Keep the notation as above. Then the following two statements hold.(1) For each h ∈ G , the A G -modules M G and ( h M ) G are isomorphic.(2) Assume that both M and M ′ are indecomposable A -modules. Then the A G -modules M G and M ′ G are isomorphic if and only if M and h ( M ′ ) are isomorphic for some h ∈ G .Proof. For (1), we mention that the isomorphism M G −→ ( h M ) G sends m g to ( h m ) gh .In view of (1), it remains to prove the “only if” part of (2). We have a decomposi-tion M G = L g ∈ G M g − of A -modules; moreover, the direct summand M g −
16 XIAO-WU CHEN, REN WANG is isomorphic to g M by identifying m g − with g m . We have an isomorphism of A -modules M G ≃ M g ∈ G g
M .
Similarly, we have M ′ G ≃ L g ∈ G g M ′ . Then the “only if” part of (2) follows fromthe Krull-Schmidt theorem. (cid:3) For an A -module M , we have a G -graded algebra M g ∈ G Hom A ( M, g M )(7.2)whose product is given by f f ′ = h ( f ) ◦ f ′ for f : M → g M and f ′ : M → h M . Here, we use the fact that h ( g M ) = gh M , andobserve that f f ′ : M → gh M is well defined.The following isomorphism is well known; see [22, Section 3] or [3, Proposi-tion 2.4]. Lemma 7.2.
Keep the notation as above. Then there is an isomorphism of algebras M g ∈ G Hom A ( M, g M ) ∼ −→ End A G ( M G ) , ( f : M → g M ) ( m h g − f ( m ) hg − ) . Here, g − f ( m ) means the element in M that corresponds to f ( m ) ∈ g M .Proof. We observe that M g − is naturally identified with g M as a left A -module.Then the above isomorphism follows from (7.1) and the Hom-tensor adjunction. (cid:3) Denote by D = Hom K ( − , K ) the duality of vector spaces, and by Tr A ( − ) thetranspose of left or right A -modules. Recall that the Auslander-Reiten translationsare given by τ A = D Tr A and τ − A = Tr A D ; see [1, IV].The following general facts seem to be well known; compare [22, Lemma 4.2]. Lemma 7.3.
Let M and N be a left A -module and a right A -module, respectively.(1) There are isomorphisms of right A G -modules: DM G ≃ D ( M G ) and Tr A ( M ) G ≃ Tr( M G ) .(2) There are isomorphisms of left A G -modules: DN G ≃ D ( N G ) and Tr A ( N ) G ≃ Tr( N G ) .(3) There are isomorphisms of left A G -modules τ A ( M ) G ≃ τ ( M G ) and τ − A ( M ) G ≃ τ − ( M G ) .Here, Tr and τ denote the transpose and Auslander-Reiten translation of A G -modules, respectively.Proof. We only prove (1), because the proof of (2) is similar, and that (3) followsimmediately from (1) and (2).The first isomorphism is given as follows DM G ∼ −→ D ( M G ) θ g ( m h θ ( m ) δ h,g − ) . Here, δ is the Kronecker symbol. For the second one, we first observe a naturalisomorphism c P : Hom A ( P, A ) G ∼ −→ Hom A G ( P G, A G ) θ g ( p h h ( θ ( p )) hg ) KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 27 of right A G -modules. Take a minimal projective presentation P → P → M → A ( M ) is defined by the following exact sequenceHom A ( P , A ) −→ Hom A ( P , A ) −→ Tr A ( M ) −→ . (7.3)As rad( A ) G ⊆ rad( A G ), we infer that P G −→ P G −→ M G −→ M G . Then the lower exact row of thefollowing commutative diagram follows from the definition of Tr( M G ). The upperexact row is obtained by applying − G to (7.3).Hom A ( P , A ) G / / c P (cid:15) (cid:15) Hom A ( P , A ) G c P (cid:15) (cid:15) / / Tr A ( M ) G / / A G ( P G, A G ) / / Hom A G ( P G, A G ) / / Tr( M G ) / / (cid:3) Recall that a finite dimensional algebra B is local provided that B/ rad( B ) is adivision algebra. Following [1, p.65], we say that B is elementary if B/ rad( B ) isisomorphic to a product of K . We observe that a finite dimensional algebra B islocal and elementary if and only if B/ rad( B ) is isomorphic to K .Recall from [21, Section 1.4] that a G -graded algebra Γ = L g ∈ G Γ g is a crossedproduct if each homogeneous component Γ g contains an invertible element. Such acrossed product Γ is often denoted by B ∗ G with B = Γ (1 G ) . Lemma 7.4.
Assume that K is perfect with char( K ) = p > and that G is a finite p -group. Let B be a finite dimensional algebra which is local and elementary. Thenany crossed product B ∗ G , as an ungraded algebra, is local and elementary.Proof. Take a normal subgroup N of G such that G/N is cyclic of order p . Then B ∗ G = L g ∈ G B g is naturally G/N -graded B ∗ G = M x ∈ G/N ( M g ∈ x B g ) . Under this new grading, it is also a cross product. In other words, we have B ∗ G = ( B ∗ N ) ∗ G/N.
By induction, it suffices to prove the statement for the case where G is cyclic oforder p .Assume now that G is cyclic of order p . We will prove that B ∗ G is local andelementary. We will first deal with a special case.We claim that any crossed product K ∗ G is always local and elementary. Take agenerator g of G and an invertible element u g in ( K ∗ G ) g . We have ( u g ) p = µ ∈ K for some nonzero µ ∈ K . Since K is perfect, there is some nonzero λ ∈ K satisfying λ p = µ . Now the algebra homomorphism K [ t ] / ( t p ) −→ K ∗ G, sending t to λ − u g −
1, is an isomorphism, proving the claim.For the general case, we observe that each homogeneous component B h of B ∗ G is a free B -module on each side. We observe that L h ∈ G rad( B h ) is a two-sidednilpotent ideal of B ∗ G . Therefore, we have M h ∈ G rad( B h ) ⊆ rad( B ∗ G ) . Recall that K ≃ B/ rad( B ). Combining the following obvious isomorphism B ∗ G/ M h ∈ G rad( B h ) ≃ K ∗ G and the above claim, we infer that B ∗ G is local and elementary. (cid:3) Lemma 7.5.
Let char( K ) = p > and G be a p -group. Then the following state-ments hold.(1) Assume that K is perfect. Then any crossed product K ∗ G is local andelementary.(2) Assume that G is cyclic. Then any crossed product K ∗ G is local.Proof. Since (1) is a special case of Lemma 7.4, we only prove (2). Assume that | G | = q for some p -power q . Take a generator g of G and an invertible element u g in ( K ∗ G ) g . We have ( u g ) q = µ ∈ K for some nonzero µ ∈ K . We observe a K -algebra isomorphism K [ t ] / ( t q − µ ) ∼ −→ K ∗ G, t u g . Then the required statement follows from a standard fact: the algebra K [ t ] / ( t q − µ )is always local. (cid:3) Let us come back to the situation where a finite group G acts on a finite dimen-sional algebra A . Proposition 7.6.
Let char( K ) = p > and G be a finite p -group. Assume that M is a left A -module such that End A ( M ) is local and elementary. Then the followingtwo statements hold.(1) Assume that K is perfect. Then End A G ( M G ) is local and elementary.(2) Assume that G is cyclic. Then End A G ( M G ) is local.In both cases, the A G -module M G is indecomposable.Proof. Denote the G -graded algebra in (7.2) by Γ. By Lemma 7.2, it suffices toprove that Γ is local and elementary under the assumption in (1), and local underthe assumption in (2), respectively.Consider the following G -graded subspace of Γ I = M g ∈ G { f ∈ Hom A ( M, g M ) | f is a non-isomorphism } . Since all the A -modules g M are indecomposable, it follows that I is a G -gradedtwo-sided ideal of Γ. Moreover, it is well known to be nilpotent; for example, see[1, VI, Corollary 1.3]. Consequently, we have I ⊆ rad(Γ).Consider the stabilizer G M = { g ∈ G | M ≃ g M } of M . Recall that K ≃ End A ( M ) / rad(End A ( M )), since End A ( M ) is local and elementary. We infer thatΓ /I is isomorphic to a crossed product K ∗ G M .By the inclusion I ⊆ rad(Γ), we haveΓ / rad(Γ) ≃ K ∗ G M / rad( K ∗ G M ) . As G M is a p -group, we can apply Lemma 7.5 to K ∗ G M . Then the requiredstatements follow immediately. (cid:3) Remark 7.7.
Assume that K is algebraically closed in Proposition 7.6. Then foreach indecomposable A -module M , End A ( M ) is local and automatically elementary.It follows that the A G -module M G is indecomposable. In other words, theinduction functor − G : A -mod −→ A G -modpreserves indecomposable modules. KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 29
The folding projection and categorification.
In this final subsection, wealways work in the following setup.
Setup ( ♣ ) . Let K be a field with char( K ) = p >
0, and let G = h σ | σ p a = 1 G i be a cyclic group of order p a for some a ≥
1. Let ∆ be a finite acyclic quiver with∆ = { , , · · · , n } . Assume that G acts on ∆ by quiver automorphisms such thatfor each arrow α ∈ ∆ , we have G α = G s ( α ) ∩ G t ( α ) .Denote by Z ∆ = L ni =1 Z ǫ i the root lattice of ∆. It is endowed with a symmetricbilinear form given by ( ǫ i , ǫ i ) = 2 and( ǫ i , ǫ j ) = −|{ arrows between i and j in ∆ }| for i = j . Denote by Φ + (∆) the set of positive roots [16].Denote by ∆ /G the orbit set of vertices. The elements in ∆ /G are denoted inthe bold form. The canonical projection π : ∆ → ∆ /G sends i to π ( i ) = Gi = i .Associated to the G -action on ∆, we have defined a Cartan triple ( C, D,
Ω) inSubsection 6.2. The rows and columns of C and D are indexed by ∆ /G . Theentries c i of D are determined by c i = | G || i | = p a i for some 0 ≤ a i ≤ a . The corresponding root lattice Z (∆ /G ) = L i ∈ ∆ /G Z E i isendowed with a symmetric bilinear form given by ( E i , E i ) = 2 c i and( E i , E j ) = c i c i , j = − | G || i | · | j | · |{ arrows between G -orbits i and j in ∆ }| for i = j . Denote by Φ + ( C ) the set of positive roots.There is a canonical projection between the root lattices f : Z ∆ −→ Z (∆ /G )given by f ( ǫ i ) = E π ( i ) , and called the folding projection ; see [24, Section 10.3]. Itdoes not preserve the bilinear forms. However, it sends positive roots to positiveroots. Moreover, by adapting the proof of [16, Lemma 5.3], [14, Proposition 15]proves that f restricts to a surjective map f : Φ + (∆) ։ Φ + ( C ) . We observe that the folding projection induces an isomorphism between the quo-tient group of G -coinvariants in Z ∆ and Z (∆ /G ).We mention that there is a folding inclusion from the dual root lattice of C into Z ∆ , which identifies the dual root lattice with the subgroup of G -invariants in Z ∆ ;see [14, Section 2]. Working with species over a finite field, one observes that theextension-of-scalars functor along a suitable field extension yields a categorificationof the folding inclusion; see [14, the proof of Theorem 24] and [6, Section 9].Recall that P ∆ is the path category of ∆. Then the G -action on ∆ induces a G -action on P ∆ . We have the corresponding skew group category P ∆ ⋊ G .We identify the path algebra K ∆ with the category algebra K P ∆ . By Proposi-tion 2.1, we have the following natural isomorphism of algebras ̟ : K ( P ∆ ⋊ G ) ∼ −→ K ∆ G, ( q, g ) q g, (7.4)for any path q in ∆ and g ∈ G .Set C = C ( C, D,
Ω) to be the EI category associated to (
C, D,
Ω); see Defini-tion 6.1. For each i ∈ Obj( C ) = ∆ /G , we haveAut C ( i ) = h η i | η c i i = Id i i , which is a cyclic group of order c i = p a i . By Theorem 6.5, we identify C with C (∆ , U tr ). Fix the choices (5.1) for the G -action on P ∆ . Then we obtain an equivalence of categories ι : C ∼ −→ P ∆ ⋊ G (7.5)which satisfies ι ( i ) = ι ( i ). The functor ι induces the following isomorphism ofgroups Aut C ( i ) ∼ −→ G ι ( i ) = Aut P ∆ ⋊ G ( ι ( i )) , η i σ p a − a i . (7.6)Recall from Definition 6.2 the algebra H = H ( C, D,
Ω). Each i corresponds toan idempotent e i of H . Moreover, we have e i He i = Span K { e i , ε i , · · · , ε c i − i } . By Proposition 6.4, there is an isomorphism of algebras θ : H ∼ −→ K C (7.7)which identifies e i He i with K Aut C ( i ). Indeed, we have θ ( e i ) = Id i and θ ( ε i ) = η i − Id i .We now combine (7.4), (7.5) and (7.7) into the following sequence of equivalences. K ∆ G -mod ̟ ∗ / / K ( P ∆ ⋊ G )-mod can ( K -mod) P ∆ ⋊ Gι ∗ (cid:15) (cid:15) H -mod K C -mod θ ∗ o o ( K -mod) C can Here, the two can’s mean the canonical equivalence in (2.2), and the upper starfunctors are given by restriction of scalars. For example, ι ∗ sends a functor X on P ⋊ G to the composite functor X ◦ ι . We compose the sequence into an equivalenceΨ : K ∆ G -mod ∼ −→ H -mod . The following terminology is introduced in [11, Definition 1.1 and Section 11].A left H -module Y is locally free , provided that each e i Y , as an e i He i -module, isfree. For such a module, its rank vector is defined as followsrank( Y ) = X i ∈ ∆ /G rank e i He i ( e i Y ) E i ∈ Z (∆ /G ) . A left H -module Y is called τ -locally free , provided that for any k ∈ Z , τ kH ( Y ) islocally free. Slightly different from [11], we do not require τ -locally free H -modulesto be indecomposable. Theorem 7.8.
Keep the assumptions in Setup ( ♣ ). Let M be a left K ∆ -module.Then Ψ( M G ) is a τ -locally free H -module satisfying rank Ψ( M G ) = f (dim M ) . (7.8) Assume further that
End K ∆ ( M ) is local and elementary. Then End H (Ψ( M G )) is local. If moreover K is perfect, then End H (Ψ( M G )) is local and elementary. Denote by H -mod τ -lf the full subcategory of H -mod consisting of τ -locally freemodules. The identity (7.8) might be visualized as a commutative diagram. K ∆-mod dim (cid:15) (cid:15) Ψ ◦ ( − G ) / / H -mod τ -lf rank (cid:15) (cid:15) Z ∆ f / / Z (∆ /G ) KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 31
The diagram indicates that the composite functor Ψ ◦ ( − G ) categorifies the foldingprojection f between the root lattices.It is natural to categorify the folding projection f : Φ + (∆) → Φ + ( C ) between thepositive roots using the same functor between indecomposable modules. However,we have to restrict to the Dynkin cases; see Proposition 7.9. Proof. Step 1.
We first show that the H -module Ψ( M G ) is locally free andsatisfies the required identity for the rank vector.Recall that the isomorphism θ identifies e i He i with K Aut C ( i ). Therefore, itsuffices to claim that for each i ∈ Obj( C ) = ∆ /G , ι ∗ ◦ can ◦ ̟ ∗ ( M G )( i )is a free module over K Aut C ( i ) with rank X i ∈ i dim K ( e i M ) . Here, we view ι ∗ ◦ can ◦ ̟ ∗ ( M G ) as a functor over C .For the claim, we observe the following identity. ι ∗ ◦ can ◦ ̟ ∗ ( M G )( i ) = can ◦ ̟ ∗ ( M G )( ι ( i ))= ( e ι ( i ) G ) . ( M G )= M g ∈ G e g ( ι ( i )) M g − = M i ∈ i e i M { g ∈ G | g − ( ι ( i )) = i } Here, for the second equality we recall that the trivial path e ι ( i ) is the identityendomorphism of ι ( i ) in P ∆ , and for the third one, we use the fact that g ( e ι ( i ) ) = e g ( ι ( i )) .By (7.6), we identify K Aut C ( i ) with K G ι ( i ) . As G is abelian, we have G ι ( i ) = G i for each i ∈ i . Then we observe that the left K G ι ( i ) -action on the above directsummand e i M { g ∈ G | g − ( ι ( i )) = i } is really only on the right side, that is, on the set { g ∈ G | g − ( ι ( i )) = i } viathe multiplication in G . The latter G ι ( i ) -action is free and transitive. Therefore,the G ι ( i ) -action on the above direct summand is free of rank dim K ( e i M ). Thisobservation implies the claim. Step 2.
Since Ψ is an equivalence, it commutes with Auslander-Reiten transla-tions. Then we have isomorphisms τ kH Ψ( M G ) ≃ Ψ τ k ( M G ) ≃ Ψ( τ k K ∆ ( M ) G ) , where the isomorphism on the right side follows from Lemma 7.3. Here, the un-adorned τ means the Auslander-Reiten translation of K ∆ G -modules. By Step 1,we infer that each H -module τ kH Ψ( M G ) is locally free, that is, Ψ( M G ) is τ -locally free.The equivalence Ψ induces an isomorphism of algebrasEnd H (Ψ( M G )) ≃ End K ∆ G ( M G ) . Then the last statement follows from Proposition 7.6. (cid:3)
Denote by K ∆-ind a complete set of representatives of indecomposable K ∆-modules. Similarly, H -ind τ -lf is a complete set of representatives of indecomposable τ -locally free H -modules. As G acts on K ∆-ind by twisting endofunctors, we havethe orbit set K ∆-ind /G . Proposition 7.9.
Keep the assumptions in Setup ( ♣ ). We assume further that ∆ is of Dynkin type. Then the following commutative diagram is well defined K ∆ - ind dim (cid:15) (cid:15) Ψ ◦ ( − G ) / / H - ind τ - lf rank (cid:15) (cid:15) Φ + (∆) f / / Φ + ( C ) , whose vertical arrows are bijections. In particular, the map Ψ ◦ ( − G ) induces abijection K ∆ - ind /G ∼ −→ H - ind τ - lf . Proof.
We observe that C is also of Dynkin type; compare [4, Proposition 6.5]. Themap dim is bijective by the well-known Gabriel’s theorem; see [9, 1.2 Satz] and [1,VIII.5]. By [11, Theorem 1.3], the map rank is bijective.It is well known that each indecomposable K ∆-module M satisfies End K ∆ ( M ) ≃ K ; for example, see [1, VIII, Lemma 6.1]. We infer from Theorem 7.8 that the H -module Ψ( M G ) is indecomposable. Then the above commutative diagram is welldefined. Since f : Φ + (∆) → Φ + ( C ) is surjective, we infer that the mapΨ ◦ ( − G ) : K ∆-ind −→ H -ind τ -lfis surjective. In view of Lemma 7.1, we have the induced bijection. (cid:3) Remark 7.10. (1) We mention that any non-symmetric Cartan matrix C ofDynkin type does appear in the situation of Proposition 7.9; see [19, 14.1.6] or[4, p.81, Table 1]. The representation theory related to the folding inclusion in theDynkin cases is studied in [26].(2) Since [11, Theorem 1.3] works currently only for Dynkin cases, we do notknow how to extend Proposition 7.9 to non-Dynkin quivers.In view of [15] and [7, 3.3 Theorem], the following open question, analogousto Kac’s theorem, is very natural: does the set of rank vectors of indecompos-able τ -locally free H -modules coincide with Φ + ( C )? We refer to [12] for relatedconsideration on rigid locally free H -modules.Assume that K is algebraically closed. By [15, Theorem 2], for any α ∈ Φ + (∆),there is an indecomposable K ∆-module M with dim( M ) = α . Combining thesurjectivity of f : Φ + (∆) → Φ + ( C ) and Theorem 7.8, we infer the following fact:for each β ∈ Φ + ( C ), there is an indecomposable τ -locally free H -module X withrank( X ) = β . This fact supports an affirmative answer to the above open question.We illustrate Proposition 7.9 with an explicit example. Example 7.11.
Let K be a field of characteristic two, and let ∆ be the followingquiver of type A . α / / ′ α ′ o o KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 33
The Auslander-Reiten quiver Γ K ∆ is as follows. ! ! ❇❇❇❇❇❇❇❇❇ ′ o o '&%$ !" @ @ ✁✁✁✁✁✁✁✁ (cid:30) (cid:30) ❁❁❁❁❁❁❁❁ ′ ❴ ❴ ❴✤✤ ✤✤❴ ❴ ❴ o o > > ⑤⑤⑤⑤⑤⑤⑤⑤ ❆❆❆❆❆❆❆❆❆ ′ = = ⑤⑤⑤⑤⑤⑤⑤⑤ o o Here, the dotted arrows denote the Auslander-Reiten translation. We visualize eachmodule using its radical layers, and represent composition factors by their corre-sponding vertices.Let G = { G , σ } be a cyclic group of order two, and let σ act on ∆ by inter-changing α and α ′ . The associated Cartan triple ( C, D, Ω) is of type B and givenas follows: C = (cid:18) − − (cid:19) , D = diag(2 , , and Ω = { (1 , } . The algebra H = H ( C, D, Ω) is given by the following quiver ε : : α o o ε d d subject to relations ε = 0 = ε . In practice, one simply deletes the loop ε .The Auslander-Reiten quiver Γ H is as follows; see [11, Subsection 13.6] . (cid:31) (cid:31) ❅❅❅❅❅❅❅❅❅ o o /.-,()*+ A A ✂✂✂✂✂✂✂✂✂ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄ ❴ ❴ ❴✤✤ ✤✤❴ ❴ ❴ o o @ @ ✁✁✁✁✁✁✁✁ ❆❆❆❆❆❆❆❆ (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ B B ☎☎☎☎☎☎☎☎ o o = = ④④④④④④④④ ●●●●●●●●●● o o = = ④④④④④④④④④ o o < < ②②②②②②②②②② Here, the leftmost and rightmost arrows in the bottom are identified. We haveframed all the indecomposable τ -locally free H -modules. The central three-dimensional H -module is locally free, but not τ -locally free.We apply Proposition 7.9 to obtain the bijection Θ = Ψ ◦ ( − G ) : K ∆ - ind /G ∼ −→ H - ind τ - lf . The twisting endofunctor on K ∆ - mod with respect to σ turns Γ K ∆ upside down. Bycomparing Γ K ∆ and Γ H , we observe that Θ preserves the frames of the modules, thatis, each indecomposable K ∆ -module M and Θ( M ) have the same kind of frames.By Lemma 7.3, the bijection Θ is compatible with Auslander-Reiten translations.The following observation might be compared with [22, Theorem 3.8] : by applying Θ to the square in Γ K ∆ , we infer that, in general, Θ does not preserve Auslander-Reiten sequences. Acknowledgements.
This paper is partly written when Wang visited Universityof Stuttgart in 2019; she is grateful to Steffen Koenig for inspiring comments and hishospitality. Chen thanks Jan Schr¨oer for a helpful email concerning Remark 7.10(2).We thank Bernhard Keller for enlightening discussions. This work is supported byNational Natural Science Foundation of China (No.s 11901551 and 11971449) andthe Fundamental Research Funds for the Central Universities.
References [1]
M. Auslander, I. Reiten, and S.O. Smalø , Representation Theory of Artin Algebras,Cambridge Studies in Adv. Math. , Cambridge Univ. Press, Cambridge, 1995.[2] H. Bass , Covering theory for graphs of groups , J. Pure Appl. Algebra (1-2) (1993),3–47.[3] X.W. Chen , Equivariantization and Serre duality I , Appl. Categor. Struct. (2017),539–568.[4] X.W. Chen, and R. Wang , The finite EI categories of Cartan type , J. Algebra (2020),62–84.[5]
C. Cibils, and E.N. Marcos , Skew category, Galois covering and smash product of a k -category , Proc. Amer. Math. Soc. (1) (2005), 39–50.[6] B. Deng, and J. Du , Frobenius morphisms and representations of algebras , Trans. Amer.Math. Soc. (8) (2006), 3591–3622.[7]
B. Deng, and J. Xiao , A new approach to Kac’s theorem on representations of valuedquivers , Math. Z. (2003), 183–199.[8]
V. Dlab, and C.M. Ringel , Indecomposable representations of graphs and algebras , Mem.Amer. Math. Soc. , 1976.[9]
P. Gabriel , Unzerlegbare Darstellungen I , Math. Manu. (1972), 71–103.[10] C. Geiss , Quiver with relations for symmetrizable Catan matrices and algebraic Lie theory ,Proc. Int. Cong. Math. (2018), Rio de Janeiro , 117–142.[11] C. Geiss, B. Leclerc, and J. Schr¨oer , Quivers with relations for symmetrizable Cartanmatrices I: foundations , Invent. Math. (2017), 61–158.[12]
C. Geiss, B. Leclerc, and J. Schr¨oer , Rigid modules and Schur roots , Math. Z. (2020), 1245–1277.[13]
A. Grothendieck , Revˆetements ´etales et groupe fondamental (SGA 1) , Seminaire de Ge-ometrie Algebrique , Institut des Hautes ´Etudes Scientifiques, Paris, 1963.[14]
A. Hubery , Quiver representations respecting a quiver automorphism: a generalisation ofa theorem of Kac , J. London Math. Soc. (1) (2004), 79–96.[15] V. Kac , Infinite root systems, representations of graphs and invariant theory , Invent.Math. (1980), 57–92.[16] V. Kac , Infinite Dimensional Lie Algebras, Third Edition, Cambridge Univ. Press, Cam-bridge, 1990.[17]
L. Li , A characterization of finite EI categories with hereditary category algebras,
J. Alge-bra (2011), 213–241.[18]
M. Linckelmann , A version of Alperin’s weight conjecture for finite category algebras , J.Algebra (2014), 386–395.[19]
G. Lusztig , Introduction to Quantum Groups, Progress in Math. , Birkh¨auser, BostonBasel Berlin, 1993.[20]
W. L¨uck , Transformation Groups and Algebraic K-Theory, Lecture Notes Math. ,Springer-Verlag, 1989.[21]
C. Nastasescu, and F. Van Oystaeyen , Methods of Graded Rings, Lecture Notes Math. , Springer-Verlag, Berlin Heidelberg, 2004.[22]
I. Reiten, and Ch. Riedtmann,
Skew group algebras in the representation theory of artinalgebras , J. Algebra (1985), 224–282.[23] J.P. Serre , Trees, Translated from the French by John Stillwell, Springer-Verlag, BerlinHeidelberg New York, 1980.[24]
T.A. Springer , Linear Algebraic Groups, Second Edition, Progress in Math. , Birkh¨auser,Boston Basel Berlin, 1998.[25] R. Steinberg , Lectures on Chevalley Groups, Yale University, 1967.[26]
T. Tanisaki , Foldings of root systems and Gabriel’s theorem , Tsukuba J. Math. (1980),89–97.[27] R. Wang , Gorenstein triangular matrix rings and category algebras , J. Pure Appl. Algebra (2) (2016), 666–682.
KEW GROUP CATEGORIES, CARTAN MATRICES AND FOLDING 35 [28]
P. Webb , An introduction to the representations and cohomology of categories , in: GroupRepresentation Theory, 149–173, EPFL Press, Lausanne, 2007.[29]
P. Webb , Standard stratifications of EI categories and Alperin’s weight conjecture , J.Algebra (2008), 4073–4091.[30]
M. Wemyss , Lectures on noncommutative resolutions , arXiv:1210.2564v2, 2012.[31]
F. Xu , Support varieties for transporter category algebras , J. Pure Appl. Algebra218