Induction and restriction functors for cellular categories Pei Wang 7 1 0 2 Abstract n Cellular categories are a generalization of cellular algebras, which include a a number of important categories such as (affine)Temperley-Lieb categories, J Brauerdiagramcategories,partitioncategories,the categoriesof invariantten- 5 sors for certain quantised enveloping algebras and their highest weight repre- 2 sentations, Hecke categories and so on. The common feather is that, for most ] of the examples, the endomorphism algebras of the categories form a tower of T algebras. In this paper, we give an axiomatic framework for the cellular cate- R gories relatedto the quasi-hereditarytower and then study the representations . h in terms of induction and restriction. In particular, a criteria for the semi- t simplicity of cellular categoriesis given by using the cohomology groups of cell a m modules. Moreover, we investigate the algebraic structures on Grothendieck groups of cellular categories and provide a diagrammatic approachto compute [ the multiplication in the Grothendieck groups of Temperley-Lieb categories. 3 v 4 7 2 6 1 Introduction 0 . 1 Cellular categories were defined by Westbury in [46] as a generalization of cellular 0 algebras,whichwerefirstintroducedbyGrahamandLehrerin[20]. Inacellularcat- 7 1 egory, thehom-spaceofanytwoobjectsisspannedbyadistinguishedbasis,so-called v: cellular basis. Therefore, an endomorphism algebra in a cellular category is cellular. i Inparticular, if wecan regard an algebra as acategory withoneobject, then acellu- X lar algebra is indeed a cellular category. It was shown that many important classes r a of algebras arising in representation theory, invariant theory, knot theory, subfac- tors and statistical mechanics are cellular (see e.g. [4, 17, 20, 24, 40, 41, 49, 50]), and most of their categorical analogues are also cellular, such as Temperley-Lieb categories [48], Brauer diagram categories [30], partition categories [23, 33], the categories of invariant tensors for certain quantised enveloping algebras and their highest weight representations [46, 47], the categories of Soegrel bimodules [16] and other more general Hecke categories (a strictly object-adapted cellular category due 2010 Mathematics Subject Classification: 16D90;16G10;16E30;16E20;18D10. Keywords: cellular category; cell module; quasi-hereditary algebra; Grothendieck group; Temperley-Lieb category. 1 to [15]). As is known, if an algebra admits a cellular structure, one will have a prac- ticable waytodescribetherepresentations andhomological propertiesofthealgebra [20, 8]. In this artical, we shall investigate the properties of cellular categories. Lots of important examples of cellular algebras actually occur in towers A ⊂ 0 A ⊂ A ⊂ ··· with an intense interplay between each other in terms of induction 1 2 and restriction. These include Temperley-Lieb algebras [31] and their cyclotomic analogues[43],Braueralgebras[6,7,38,39],blobalgebras[36,37],partitionalgebras [5, 21, 33, 34] and so on. The tower method was first developed by Jones [22] and Wenzl [45] for semi-simple case. Further, for the general case, Cox, Martin, Parker and Xi established a framework of towers of quasi-hereditary algebras by combining the ideas from the tower formalism in [18] with the notion of recollement in [9]. Then, influenced by the work of Ko¨nig and Xi [25] as well as by the work of Cox et al[12], the analogous for cellularity were given by Goodman and Graber[19]. Thereareinfactanumberofcellularcategories withtheendomorphismalgebras forming a tower. Further, note that the hom-space of two objects in a cellular category is a natural nexus for their endomorphism algebras. Those motivate us to introduce a class of cellular categories by an axiomatic menner, so-called cellular tower category, or CTC (see Section 3.2), in which the endomorphismalgebrasarerequiredtobequasi-hereditary, andhereintheinduction and restriction behave well in the tower. In this setting, the homological aspects of representation theory are computed efficiently by using induction and restriction. Precisely, let K bea field, and A bea cellular tower category (see Definition 3.6) with its object set the set of natural numbers N. By definition, the endomorphism algebra A of an object n is a cellular algebra with an index set Λ . Suppose that n n ∆ (λ) denotes the cell module of A corresponding to the index λ ∈ Λ . n n n Our main result can be stated as the following theorem. Theorem 1.1 Let A be a cellular tower category. Suppose that for all n ∈ N and pairs of indices λ ∈ Λ \Λ and µ ∈ Λ \Λ we have n n−2 n n−4 Ext1 (∆ (λ),∆ (µ)) = 0, An n n Then each of the endomorphism algebras A in A is semi-simple. n In[8],Caoprovidedacriteriaofsemi-simplicity foracellular algebrabychecking thefirstcohomology groupsof cellmodulesforall indices. Rather, Theorem 1.1tells us that, the verification needs only some of the indices in a cellular tower category. The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with algebra and coalgebras structures. Many cases of interest, such as symmetric group algebras, Hecke algebras and other deformations, give rise to a dual pair of Hopf algebras, which are realization of some classical algebras in the theory of symmetric function [51, 28, 2, 44]. The common feature is that the examples admit Mackey’s formula, which implies that the comultiplication is an algebra homomorphism, that is, making both Grothendieck groups into bialgebra. In [3], Bergeron and Li gave an analogue of Mackey’s formula by an axiom and introduced a general notation of a tower of algebras, which ensures that the 2 Grothendieck groups of a tower of algebras can be a pair of graded dual Hopf al- gebras. For a cellular category with a quasi-hereditary tower, we shall see that the analogue of Mackey’s formula never holds. TheGrothendieck groups, however, have algebra and coalgebra structures under certain conditions. Further, we shall study the algebraic structures on the Grothendieck groups of Temperley-Lieb categories. This paper is organized as follows. In Section 2, we shall recall some nota- tions and basic facts. In Section 3, we first study Morita contexts of endomorphism algebras in a cellular category in 3.1 and then we introduce the cellular tower cat- egories and prove Theorem 1.1 in 3.2. Section 4 studies the algebraic structures on Grothendieck groups of cellular categories. In particular, we give a method to compute the multiplication in the Grothendieck group of a Temperley-Lieb category in 4.2. 2 Preliminaries In this section, we shall recall some basic definitions and facts needed in our later proofs. Throughout the paper, all algebras are finite-dimensional algebras over a fixed field K. All modules are finitely generated unitary left modules. For an algebra A, the category of A-modules is denoted by A-mod. Let A always be a small K-linear category with finite dimensional hom-spaces, that is, the class of objects is a set and every hom-set is a finite dimensional K-vector space and the composition map of morphisms is bilinear. For an object n in A, the endomorphism algebra End(n) of n is simply denoted by A . n 2.1 Cellular categories We now recall the definition of cellular categories, which are generalised by West- burry [46] from the cellular algebras. Definition 2.1 [46] Let A be a K-linear category with an anti-involution ∗ ( that is, ∗ is a dual functor on A with (−)∗∗ = id ). Then cell datum for A consists of a A partially ordered set Λ, a finite set M(n,λ) for each λ ∈ Λ and each object n of A, and for λ ∈ Λ and m,n any two objects A we have an inclusion C : M(m,λ)×M(n,λ) → Hom (m,n) A C : (S,T) 7−→ Cλ . S,T The conditions for this datum are required to: (C1) For all objects m,n in A, the image of the map C : M(m,λ)×M(n,λ) → Hom (m,n) A λa∈Λ is a basis for Hom (m,n) as a K-space. A (C2) For all objects m,n, all λ ∈ Λ and S ∈ M(m,λ), T ∈M(n,λ) we have (Cλ )∗ = Cλ . S,T T,S 3 (C3)Forallobjectsp,m,n, allλ ∈ Λandalla ∈ Hom (p,m), S ∈ M(m,λ), T ∈ A M(n,λ) we have aCSλ,T ≡ ra(S,S′)CSλ′,T mod A(< λ), S′∈XM(p,λ) where r (S′,S) ∈ K is independent of T and A(< λ) is the K-span of a µ {C | µ < λ; S ∈ M(p,µ),T ∈ M(n,µ)}. S,T Remark 2.1 Ifwe regardany K-algebra as aK-linear category with oneobject, thenthisisageneralisation ofthedefinitionofacellularalgebra. Ontheotherhand, suppose A is a cellular category. For any object n, let Λ = { λ ∈ Λ | M(n,λ) 6= n ∅ } and M := M(n,λ), then the endomorphism algebra A of n is a cellular n n λ∈SΛn algebra with cell datum (Λ ,M ,C,∗), where C and ∗ are restrictions on A . The n n n basis {Cλ | S,T ∈ M ,λ ∈ Λ } is called a cellular basis for A . S,T n n n In [20], Graham andLehrer introducedthedefinition of cell modulesof acellular algebrabyusingthecellularbasis. Forcellularcategories, wecandefinecellmodules of the endomorphism algebra of an object in the same way. Definition 2.2 Let A be a cellular category. For each object n, let Λ be the index n set of the cellular algebra A (see Remark 2.1 above ). For each λ ∈ Λ define the n n left A -module ∆ (λ) as follows: ∆ (λ) is a K-space with basis C(n,λ) = {C(n,λ) | n n n X X ∈ M(n,λ)} and A -action defined by n (n,λ) (n,λ) aC = r (X,Y)C , (a ∈ A , X ∈M(n,λ)) X a Y n X Y∈M(n,λ) where r (X,Y) is the element of K defined in (C3). ∆ (λ) is called the cell module a n of A corresponding to λ. n Applying ‘∗’ on (C3) in Definition 2.1, we obtain (C3′) CTλ,Sa∗ ≡ ra(S,S′)CTλ,S′ mod A(< λ). S′∈XM(p,λ) where r (S′,S) ∈ K is independent of T. a The following lemma is a direct generalization of [20, Lemma 1.7] from cellular algebras to cellular categories. Lemma 2.3 Let A be a cellular category, m,n and q be objects in A, and let λ ∈ Λ. Then for any elements U ∈ M(m,λ), T,X ∈ M(n,λ) and Y ∈ M(p,λ), we have Cλ Cλ ≡ φ (T,X)Cλ mod A(< λ), U,T X,Y (n,λ) U,Y where φ is a map from M(n,λ)×M(n,λ) to K. (n,λ) 4 Proof. By (C3), we have Cλ Cλ ≡ r (X,Z)Cλ mod A(< λ), U,T X,Y Cλ Z,Y U,T X Z∈M(m,λ) and by (C3′) , it follows Cλ Cλ ≡ r (V,T)Cλ mod A(< λ). U,T X,Y Cλ U,V Y,X X V∈M(p,λ) Comparing the previous equations follows r (X,U)Cλ = r (Y,T)Cλ . Cλ U,Y Cλ U,Y U,T Y,X Because r (X,U) is independ of Y by (C3) and r (Y,T) is independ of U by Cλ Cλ U,T Y,X (C3′), we may write φ (T,X) := r (X,U) = r (Y,T). (n,λ) Cλ Cλ U,T Y,X Hence Cλ Cλ ≡ φ (T,X)Cλ mod A(< λ). U,T X,Y (n,λ) U,Y ✷ Thus we can define a bilinear form φ : ∆ (λ)×∆ (λ) → K by (n,λ) n n (n,λ) (n,λ) φ (C ,C )= φ (U,V), (n,λ) U V (n,λ) where C(n,λ),C(n,λ) ∈ C(n,λ) with U,V ∈ M(n,λ), extended φ bilinearly. U V (n,λ) The following lemma collects some known facts for the bilinear form φ . (n,λ) Lemma 2.4 [20, Prop 2.4] Keep the notation above. Then: (1) φ is symmetric, that is, x,y ∈ ∆ (λ), φ (x,y) = φ (y,x); (n,λ) n (n,λ) (n,λ) (2) For x,y ∈ ∆ (λ) and a ∈ A , we have φ (a∗x,y) = φ (x,ay); n n (n,λ) (n,λ) (3) For Cλ ∈ A and C(n,λ) ∈ C(n,λ), we have Cλ ·C(n,λ) = φ (T,U)C(n,λ). S,T n U S,T U (n,λ) S Define rad (λ):= {x ∈ ∆ (λ) | φ (x,y) = 0 for all y ∈ ∆ (λ)}. n n (n,λ) n If φ 6= 0, then rad (λ) is the radical of the A -module ∆ (λ). (n,λ) n n n Let Λ0 = {λ ∈ Λ | φ 6= 0}. The following result shows that this set n n (n,λ) parameterizes the simple modules, which was proved by Graham and Lehrer for cellular algebras. Theorem 2.5 [20, Prop 3.4] Let A be a cellular K-algebra with the cell datum (Λ,M,C,∗). Suppose ∆(λ) and φ are the cell module and the bilinear form, re- λ spectively, corresponding to λ ∈ Λ. Let Λ0 = {λ ∈ Λ | φ 6= 0}. Then the set λ {L(λ) := ∆(λ)/red(λ) | φ 6= 0} is a complete set of non-isomorphic absolutely λ simple A-module. 5 The following theorem says that the issue of semi-simplicity reduces to the com- putation of the discriminants of bilinear forms associated to cell modules. Theorem 2.6 [20, Prop 3.8] Let A be a cellular K-algebra as above. Then the following are equivalent: (1) The algebra A is semi-simple; (2) All cell modules are simple and pairwise non-isomorphic; (3) The bilinear form φ is non-degenerate (that is, rad(λ) = 0) for each λ ∈ Λ λ In [26, 27], Konig and Xi investigated the relationships between cellular algebras and quasi-hereditary algebras in terms of comparing the so- called cell chains with hereditary chains (for quasi-hereditary algebras we refer to [9]), they gave some criterions for a cellular algebra to be quasi-hereditary. The following is equivalent to [26, Theorem 3.1]. Theorem 2.7 Let A be a cellular algebra with the cell datum (Λ,M,C,∗), and Λ0 be as above. Then A is quasi-heredity if and only if Λ = Λ0. This lemma says that a cellular algebra is quasi-heredity if and only if the poset Λ coincides with Λ0, that is, φ 6= 0 for all λ ∈ Λ. Thus in the case the set Λ λ parameterizes the simple modules. For our purpose in this paper, we need that the endomorphism algebras of a cellular category are quasi-hereditary, so we define: Definition 2.8 Let A be a cellular category. Suppose that (1) the object set of A is the set of natural numbers N, (2) for all m,n ∈ N satisfying m 6 n, if Hom (m,n) 6= 0 (equivalently, A Hom (n,m) 6= 0 by anti-involution ∗), then Λ is a saturated ordered subset of A m Λ , that is, Λ ⊂ Λ preserving ordering, and if λ < µ with µ ∈ Λ , λ ∈ Λ , then n m n m n λ ∈Λ . m (3) each endomorphism algebra A is quasi-hereditary. n Then A is called a hereditary cellular category. Remark2.2(A) InDefinition2.8,condition(2)saysthatindexsetsΛ preserve n the ordering of natural numbers. (B) Let A be a hereditary cellular category with cell datum (Λ,M,C,∗). For all m,n ∈ N, denote by Λ := { λ ∈ Λ | M(m,λ) 6= ∅ and M(n,λ) 6= ∅}, and (m,n) thus Λ is just Λ . (n,n) n By condition (2), if Hom (m,n) 6= 0 with m 6 n and M(m,λ) 6= ∅, then λ ∈ A Λ ⊆ Λ . Therefore this imples M(n,λ) 6= ∅ and Λ = Λ . m n (m,n) m 3 Induction and restriction functors for cellular cate- gories Let A be a K-linear category, and let m,n be objects in A. As is known, the hom- space Hom(m,n) is a natural left A -right A -bimodule, and hence we can consider m n 6 the functors Hom (m,n)⊗ − and −⊗ Hom (m,n). In this section, we first A An Am A study those functors in Section 3.1. In Section 3.2, We first give the definition of cellular tower category by an ax- iomatic menner and then give a proof of our main result Theorem 1.1. 3.1 Morita contexts LetAbeahereditarycellularcategory withcelldatum(Λ,M,C,∗). Foreach m,n ∈ N, Hom (m,n)⊗ − is a functor from category A -mod to category A -mod. A An n m For each λ ∈ Λ = { λ ∈ Λ | M(m,λ) 6= ∅ and M(n,λ) 6= ∅}, let ∆ (λ) and (m,n) m ∆ (λ) be the cell modules of A and A corresponding to λ, respectively. n m n It is easy to see that the following map is an A -module homomorphism: m α: Hom (m,n)⊗ ∆ (λ) → ∆ (λ) A An n m (n,λ) (m,λ) a⊗C 7→ r (Z,X)C X a Z X Z∈M(m,λ) where a ∈ Hom (m,n),C(n,λ) ∈ C(n,λ), r (Z,X) is given by definition 2.1(C3). A X a The following lemma show that it is further an A -module isomorphism. m Lemma 3.1 Let A be a hereditary cellular category. For each m,n ∈ N and λ ∈ Λ , we have (m,n) Hom (m,n)⊗ ∆ (λ) ≃ ∆ (λ), A An n m as an A -isomorphism. m Proof. Because A is quasi-hereditary, it follows φ 6= 0 by Theorem 2.7, hence n (n,λ) there exist U ,T ∈ M(n,λ) such that φ (U ,T ) 6= 0. We then fix U ,T . 0 0 (n,λ) 0 0 0 0 For any C(m,λ) ∈ C(m,λ) with Y ∈ M(m,λ), we have 1 α(Cλ ⊗ Y φ(n,λ)(U0,T0) Y,U0 (n,λ) (m,λ) C ) = C . It follows that α is surjective and T0 Y dim (Hom (m,n)⊗ ∆ (λ)) > dim (∆ (λ)) = #M(m,λ). K A An n K m To prove previous inequality is actually a equality, fixed U ,T as above, it is 0 0 sufficient to show that {Cλ ⊗C(n,λ) | S ∈ M(m,λ)} is a spanning set of the K- S,U0 T0 space Hom (m,n)⊗ ∆ (λ). Thus, for any a ∈ Hom (m,n) and C(n,λ) ∈ C(n,λ), A An n A X we have 1 a⊗C(n,λ) = a⊗Cλ ·C(n,λ) X φ (U ,T ) X,U0 T0 (n,λ) 0 0 1 = aCλ ⊗C(n,λ) φ (U ,T ) X,U0 T0 (n,λ) 0 0 1 = ( r (X,S)Cλ ⊗C(n,λ)+b⊗C(n,λ)), φ (U ,T ) a S,U0 T0 T0 (n,λ) 0 0 X S∈M(m,λ) (n,λ) where b ∈ A(< λ). We next show b⊗C = 0. T0 7 µ Without loss of generality, consider a basis element b = C with µ < λ, W ∈ W,V M(m,µ) and V ∈ M(n,µ). According to the quasi-heredity, we have φ 6= 0. (n,µ) Hence, there exist U′,T′ ∈ M(n,µ) such that φ (U′,T′) 6= 0. It follows (n,µ) 1 µ (n,λ) µ µ (n,λ) C ⊗C = C C ⊗C W,V T0 φ (U′,T′) W,U′ T′,V T0 (n,µ) 1 µ µ (n,λ) = C ⊗C ·C = 0. φ (U′,T′) W,U′ T′,V T0 (n,µ) Hence α is injective. This finishes the proof. ✷ This lemma says that the functor Hom (m,n)⊗ − sends a cellular module to A An a cellular module with the same index. In fact, we shall show that this functor is an idempotent embedding functors [1]. We first recall the definition of Morita context. Definition 3.2 [29] Let A,B be two K-algebras, and let P , Q be bimodules, B A A B and θ,φ be a pair of bimodule homomorphisms θ : P ⊗ Q → B , φ:Q⊗ P → A A B B B A A such that for all x,y ∈P and f,g ∈ Q, θ(x⊗f)y = xφ(f ⊗y), fθ(x⊗g) = φ(f ⊗x)g. Then the tuple (A, P , Q ,B,θ,φ) is called a Morita context. B A A B If θ is surjective in the previous definition, we have the following lemma. Lemma 3.3 [29, Prop 18.17] Keep the notation above. Suppose θ is surjective. Then (1) θ is an isomorphism; (2) P and Q are finitely generated projective. A A (3) There are K-algebra isomorphism B ≃ End(P ) ≃ End( Q). A A Let A be a hereditary cellular category. For objects m,n ∈ N, the tuple (A , Hom (m,n) , Hom (n,m) ,A , η, ρ) m Am A An An A Am m is a Morita context, where Hom(m,n) is a A -A -bimodule, Hom(n,m) is a A - m n n A -module. The map m η : Hom (m,n)⊗ Hom (n,m) → A A An A m defined by the composition of morphisms, is an A -bimodule homomorphism; m Similarly, the map ρ: Hom (n,m)⊗ Hom (m,n) → A A Am A n defined by the composition of morphisms, is an A -bimodule homomorphism. n 8 Lemma 3.4 Keep the notation above. Let m,n ∈ N satisfying m 6 n. Suppose Hom (m,n) 6= 0. Then ρ is surjective. A Proof. LetCλ beacellular basiselement of A withλ ∈ Λ andS,T ∈ M(n,λ). S,T n n By Definition 2.8 and Remark 2.2(B), we have Λ = Λ ⊆ Λ . Hence λ ∈ Λ . n (n,m) m m Since A is quasi-hereditary, it follows φ 6= 0 by Theorem 2.7, therefore there m (m,λ) exist U ,V ∈ M(m,λ) such that φ (U ,V ) 6= 0. This implies 0 0 (m,λ) 0 0 1 ρ( Cλ ⊗Cλ )= Cλ , φ (U ,V ) S,U0 V0,T S,T (n,λ) 0 0 Because Cλ is arbitrary, ρ is surjective. S,T ✷ As an immediate consequence of lemma 3.3 and 3.4 we get the following. Proposition 3.5 Let A be a hereditary cellular category. Suppose m,n ∈ N satis- fying m 6 n and Hom (m,n) 6= 0. Then A (1) ρ is an A -bimodule isomorphism; n (2) There exist idempotents e and f of A such that Hom (m,n) ≃ (A )e as m A m left A -module isomorphism, and Hom (n,m) ≃ f(A ) as right A -module iso- m A m m morphism, that is, Hom (m,n)⊗ − and −⊗ Hom (n,m) are idempotent em- A An An A bedding functors; (3) There are K-algebra isomorphism A ≃ e(A )e ≃ f(A )f. n m m ✷ 3.2 Cellular tower categories Inspired by the theory of quasi-heredity towers of recollement studied by Cox, Mar- tin, Parker andXi, wefirstintroducethedefinitionofcellular tower category. Under the framework, the homological aspects of representation theory are computed ef- ficiently by using induction and restriction. We then give a criteria for a cellular tower category to be semi-simple. Definition 3.6 A hereditary cellular category is called a cellular tower category (or CTC) if it satisfies (A1)– (A4) as follow: (A1) For each n > 0 the endomorphism algebra A can be identified n with a subalgebra of A which preserves the identities. n+1 For an A -module M, it has a natural A -module structure. Furthermore, n+1 n we have the restriction functor: Res : A -mod → A -mod n+1 n+1 n M 7→ M = Hom (A ,M). An+1 An An+1 n+1An We also have the induction functor: Ind : A -mod → A -mod n n n+1 9 M 7→ A ⊗ M. An n+1 An Our next three axioms ensure that induction and restriction behave well. (A2) For all n > 2 we have the following A -A -bimodule isomor- n−1 n−2 phism: Hom (n,n−2) ≃ A . A An−1 n−1An−2 By our assumption that A is quasi-hereditary, due to lemma 3.1, for any λ ∈ Λ n we have Hom (n+2,n)⊗∆ (λ) ≃ ∆ (λ). Furthermore, by using(A1) and (A2), A n n+2 we have Ind (∆ (λ)) = Res (Hom (n+2,n)⊗ ∆ (λ)). (1) n n (n−1) A An n If an A -module M in A -mod has a ∆ -filtration, that is, a filtration with n n n successive quotients isomorphic to some cell modules ∆ (λ)’s, then we define the n support of M, denoted by supp (M), to be the set of labels λ for which ∆ (λ) n n occurs in this filtration. (A3) For all m,n ∈ N satisfying m 6 n and that n−m is even, and for all λ ∈ Λ \Λ , we have that Res (∆ (λ)) has a ∆ -filtration and m m−2 n n (n−1) supp (Res (∆ (λ))) ⊆ (Λ \Λ )∪(Λ \Λ ) ⊆Λ . n−1 n n m−1 m−3 m+1 m−1 n−1 For (A3), usingHom (n+2,n)⊗∆ (λ) ≃ ∆ (λ) and equation (1), we deduce A n n+2 that for all m,n ∈ N satisfying m 6 n and that n − m is even, and for all λ ∈ Λ \Λ , we have that Ind (∆ (λ)) has a ∆ -filtration and m m−2 n n (n+1) supp (Ind (∆ (λ))) ⊆ (Λ \Λ )∪(Λ \Λ ) ⊆ Λ . n+1 n n m−1 m−3 m+1 m−1 n+1 (A4) Let n ∈ N. For each λ ∈ Λ \Λ there exists µ ∈ Λ \Λ such n n−2 n−1 n−3 that λ ∈ supp (Ind (∆ (µ))). n n−1 n−1 By using (A3), (A4) is equivalent to (A4′) Let n∈ N. For each λ ∈ Λ \Λ , there exists µ ∈ Λ \Λ such that n n−2 n−1 n−3 λ ∈supp (Res (∆ (µ))). n n+1 n+1 For a quasi-hereditary algebra we have that Ext1(∆(λ),∆(µ)) 6= 0 implies that µ < λ. Therefore (A4) is also equivalent to: Let n ∈ N. For each λ ∈ Λ there exists µ ∈ Λ such that there is a surjection n n−1 Ind (∆ (µ)) → ∆ (λ) → 0. n−1 n−1 n Due to Cao [8], the following theorem provides some homological characteriza- tions of the semi-simplicity of cellular algebra. Theorem 3.7 [8, Thm. 1.2] Let A be a cellular K-algebra with respect to an in- volution i and a poset (Λ,6). Let Λ be the subset of Λ, which parametrizes the 0 isomorphism classes of simple A-modules. Then the following statements are equiv- alent: 10