A self–injective cellular algebra is weakly symmetric

by

Steffen König and Changchang Xi

1 Introduction

Textbooks on representation theory of finite groups or related topics often use the following hierarchy of finite dimensional algebras:

{semisimple}
∩
{symmetric}
∩
{weakly symmetric}
∩
{quasi − F robenius = self − injective}

Here self–injective means that each projective module is injective as well, whereas weakly symmetric says that the projective cover of any given simple module is the injective envelope of the same simple module. That is, the permutation top(P) ↦ soc(P) (for P indecomposable projective–injective) is the identity. It is well–known that in general all these inclusions are proper.

The main result of this note states that inside the class of cellular algebras (which contains group algebras of symmetric groups, various kinds of Hecke algebras, Brauer algebras, Temperley–Lieb algebras, ..., see [2]) the third one of these inclusions is an identity.

Theorem 1.1 Let A be a cellular algebra over a field. If A is self–injective, then it is weakly symmetric.

1991 Mathematics Subject Classification: Primary 16E10, 16G10, 18G20. Secondary 17B10, 19A31, 20C05, 20G05, 81R05.

C.C.Xi obtained support from NSFC (Grant No. 19831070).

We show by examples that the other inclusions given above still are proper, if one restricts attention to cellular algebras.

As a consequence of the theorem we obtain:

Corollary 1.2 Let A be a self–injective cellular algebra over a field. Let P be an indecomposable projective A–module. Then P either is simple or it has a cell filtration of length at least two. If a cell (=standard) module Δ is projective, then it must be simple.

In section two we recall the definition of cellular algebras and then we prove theorem 1.1. Moreover we show by examples, that the other inclusions given above are proper. In section three we discuss some other properties of self–injective cellular algebras. This includes a proof of corollary 1.2.

2 Proof of theorem, and some examples

We first have to recall the two equivalent definitions of cellular algebras given in [2] and (later) in [3].

For simplicity we stick to the ground ring being an (arbitrary) field k. By algebra we always mean a finite dimensional associative algebra with unit.

Definition 2.1 (Graham and Lehrer, [2]) An associative k–algebra A is called a cellular algebra with cell datum (I, M, C, i) if the following conditions are satisfied:

(C1) The finite set I is partially ordered. Associated with each λ ∈ I there is a finite set M(λ). The algebra A has a k–basis C_{S,T}^{λ} where (S, T) runs through all elements of M(λ) × M(λ) for all λ ∈ I.

(C2) The map i is a k–linear anti–automorphism of A with i^{2} = id which sends C_{S,T}^{λ} to C_{T,S}^{λ}.

(C3) For each λ ∈ I and S, T ∈ M(λ) and each a ∈ A the product aC_{S,T}^{λ} can be written as (∑_{U∈M(λ)} r_{a}(U, S)C_{U,T}^{λ}) + r′ where r′ is a linear combination of basis elements with upper index μ strictly smaller than λ, and where the coefficients r_{a}(U, S) ∈ k do not depend on T.

In the following we shall call a k–linear anti–automorphism i of A with i^{2} = id an involution of A. In [3] it has been shown that the previous definition is equivalent to the following one.

Definition 2.2 ([3]) Let A be a k–algebra. Assume there is an antiautomorphism i on A with i^{2} = id. A two–sided ideal J in A is called a cell ideal if and only if i(J) = J and there exists a left ideal Δ ⊂ J such that Δ has finite k–dimension and that there is an isomorphism of A–bimodules α : J ≃ Δ⊗_{k}i(Δ) (where i(Δ) ⊂ J is the i–image of Δ) making the following diagram commutative:

[figure: J; α; Δ ⊗_{k} i(Δ); J; i; α; Δ ⊗_{k} i(Δ); x ⊗ y ↦ i(y) ⊗ i(x)]

The algebra A (with the involution i) is called cellular if and only if there is a vector space decomposition A = J′_{1} ⊕ J′_{2} ⊕ … ⊕ J′_{n} (for some n) with i(J′_{j}) = J′_{j} for each j and such that setting J_{j} = ⊕_{l=1}^{j}J′_{l} gives a chain of two sided ideals of A: 0 = J_{0} ⊂ J_{1} ⊂ J_{2} ⊂ … ⊂ J_{n} = A (each of them fixed by i) and for each j (j = 1, …, n) the quotient J′_{j} = J_{j}/J_{j−1} is a cell ideal (with respect to the involution induced by i on the quotient) of A/J_{j−1}.

The modules Δ(j), 1 ≤ j ≤ n, are called standard modules of the cellular algebra A, and the above chain in A is called a cell chain. (Standard modules are called cell modules in [2]).

Before we can prove the theorem 1.1 we have to demonstrate a lemma.

Lemma 2.1 Let A be an algebra (over a field) with cell ideal J (with respect to some involution i). Then all indecomposable projective–injective left A– ideals intersecting non–trivially with J are isomorphic.

Proof. Let P_{a} be an indecomposable projective–injective left A–ideal which intersects J non–trivially. We can write P as Ae for some idempotent e = e^{2} ∈ A. By definition, J has the form J ≃ Δ⊗_{k} i(Δ). Hence the intersection Ae∩J is isomorphic to Δ⊗_{k} i(Δ)e which is a direct sum of copies of Δ. But the socle L of P is simple. Hence J ∩ Ae has precisely one direct summand Δ, and this module must have simple socle which is isomorphic to L as well. Thus if Δ is a submodule of an injective module, say I, then I must be isomorphic to P.

Proof of 1.1.

We fix a self–injective cellular algebra A over a field k. We fix a cell chain 0 = J_{0} ⊂ J_{1} ⊂ … ⊂ J_{n} = A and we fix the involution i. We write the identity as a complete sum 1 = ∑_{j} e_{j} of pairwise orthogonal non–equivalent idempotents, and we decompose the algebra into A = ⊕_{j,l}e_{j}Ae_{l}. (Note that the idempotents e_{j} are not required to be primitive. We only have to separate the equivalence classes of idempotents from each other.)

We denote the indecomposable projective left A–ideals by P_{1}, …, P_{n}, where the ordering is compatible with the partial order in the cell structure in the following way: Let P_{a} and P_{b} be two projective left ideals, and let J be the smallest ideal in the cell chain intersecting P_{a} non–trivially. Then if J intersects P_{b} non–trivially, we must have b ≤ a. In fact, let J = J_{s} be the smallest ideal in the cell chain with the property that J ∩ P_{a} ≠ 0 and consider the factor algebra Ā := A/J_{s−1}. We denote by J̄ (respectively, ā) the image of J (respectively, a ∈ A) under the canonical surjective mapping from A to A/J_{s−1}. Thus J̄ is a cell ideal in Ā, and it gives us a standard module Δ. Since P_{a} ∩ J_{s−1} = 0, the projective A-module P_{a} = Ae_{a} is also a projective-injective Ā-module, where e_{a} is a primitive idempotent element in A. Thus we are in the situation of lemma 2.1 which implies Δ cannot be a submodule of any other injective Ā-module. This implies that if P_{b}, b ≠ a, is another indecomposable projective–injective A-module with P_{b} ∩ J ≠ 0, then P_{b} ∩ J_{s−1} ≠ 0, and hence the required compatibility is possible.

By assumption, each P_{a} is injective, i.e. P_{a} ≃ I_{σ(a)} where the index σ(a) is given by some permution σ and I_{σ(a)} is the injective hull of the simple module top(P_{a}). We want to show: a = σ(a) for every a. We proceed inductively.

Induction start: a = 1. By assumption J = J_{1} intersects P_{1} non– trivially. Thus Δ_{1} has simple socle L_{σ(1)} and this socle is generated by an element x_{1} which is a path from 1 to σ(1). This means, x_{1} can be written as a product y_{σ(1),l_{1}} ⋯ y_{l_{j},l_{j+1}} ⋯ y_{l_{m},1} where each element y_{l_{j},l_{j+1}} is in

e_{l_{j}} · (rad(A) − rad^{2}(A)) · e_{l_{j+1}}

(Note that the ’arrows’ y_{l_{j},l_{j+1}} in such a path start and end at equivalence classes of primitive idempotents, not at primitive idempotents. Note, moreover, that the order in the product is chosen in such a way that the starting point 1 is at the very right and the ending point σ(1) is at the very left.) It is known ([3], proposition 5.1) that the involution i sends an idempotent to an idempotent in the same equivalence class. Thus the i–image i(x_{1}) is a path from σ(1) to 1 in the same sense. But J intersects all other projective (left or right) ideals trivially, hence fJ = Jf = 0 for all primitive idempotents f with Af ≄ P_{1}, since for example fJ ≠ 0 would imply J ∩ fA ≠ 0, a contradiction. Since J is fixed by i, the element i(x_{1}) must be in J again. This shows that the socle of J has a direct summand isomorphic to L_{1}, the simple quotient of P_{1}. But we know already that the socle of J is a direct sum of copies of L_{σ(1)} since P_{1} ≃ I_{σ(1)}. This implies 1 = σ(1).

Induction step: We have to consider the case of P_{j}. We know already that top(P − i) = soc(P_{i}) for 1 ≤ i < j. Hence σ(j) must be greater or equal j. Let J_{s} be the smallest ideal in the cell chain with P_{j} ∩ J_{s} ≠ 0. Let Ā = A/J_{s−1}. As above we get that the standard module Δ corresponding to the cell ideal J̄=J_{s}/J_{s−1} can be considered as a submodule of P_{j} with a simple socle L_{σ(j)} and that J̄ intersects the (projective–injective) modules P_{j+1}, …, P_{n} trivially. (Note that all the P_{t}, with t ≥ j + 1, have trivial intersection with J_{s}, so they are Ā-modules as well.) The socle L of P_{j} is generated by an element x which is a path (as defined above) from j to σ(j).

We claim that multiplying x on the right by any radical element gives zero. In fact, such a right multiplication defines a homomorphism φ from P_{j} to Ā which has image in rad(Ā). If φ is an injective map, then its image (being isomorphic to the injective module P_{j}) splits off. This contradicts the choice of φ. Hence φ cannot be injective and its kernel must contain the socle of P_{j}, which proves the claim. (Of course, multiplication on the left also gives zero, since x is in the socle.)

Hence multiplying i(x) on the left by any radical element of Ā also gives zero, which means that it must be in the socle of Ā. Since J̄ is fixed by i, the element i(x) stays inside J, the socle of which is a direct sum of copies of L. But this implies that i(x) is a path ending at σ(j) whereas by construction the path should end at j. Therefore j equals σ(j).

The following examples show that the theorem is optimal. More precisely, the other two inclusions discussed in the introduction are proper. Furthermore, the class of self–injective cellular algebras, which are not semisimple, is properly contained in the class of cellular algebras of infinite global dimension.

Group algebras of symmetric groups provide example of cellular algebras, which are self–injective, but not semisimple.

Example 1. Let k be a field and λ ∈ k. Let A = k < a, b, c, d > /I, where I is generated by a^{2}, b^{2}, c^{2}, d^{2}, ab, ac, ba, bd, ca, cd, db, dc, cb−λbc, ad− bc, da − bc. As a left module over itself this algebra looks as follows:

[figure: 1; a; b; c; d; bc]

Then this algebra is symmetric if and only if λ = 1. Indeed, suppose f : A ⟶ k is a non-zero symmetric linear form whose kernel does not contain any non-zero left ideal in A. Since the socle of A is one-dimensional and spanned by bc, we have 0 ≠ f(bc) = f(cb) = λf(bc) and λ = 1 . Conversely, if λ = 1, then we define a non-degenerate linear form f by f(1) = f(a) = f(b)=f(c) = f(d) = 0 and f(bc) = 1, and this makes A a symmetric algebra for λ = 1.

Moreover, there is an involution i on A, namely, i fixes a and d, but interchanges b and c. With respect to this involution the algebra A is a cellular algebra. Let us exhibit a cell chain J_{1} ⊆ J_{2} ⊆ J_{3} = A of A by giving a basis in square form:

J_{1} : bc;  J_{2}/J_{1} : \begin{matrix}a, & b, \\ c, & d; \end{matrix}  J_{3}/J_{2} : 1

Note that we can use a result in [4] to verify that this chain is really a cell chain of A.

If we choose λ different from 1, then we get a cellular algebra which is weakly symmetric but not symmetric.

Example 2. Now we give an example of a cellular algebra of infinite global dimension, but not self–injective.

Consider the algebra A which is given by the following quiver

[figure: 1; 2; α; β; γ]

with relations γ^{2} = αβ, βα = 0, γα = 0, βγ = 0.

Thus A has the following decomposition as a left module over itself:

AA = \begin{aligned}& 1 \\ 1 & & 2 & ⊕ & 2 \\ & 1 & & & 1 \end{aligned}

Clearly, this algebra has infinite global dimension and is cellular, but it is not a self–injective algebra.

3 Properties of self–injective cellular algebras

Let us first prove corollary 1.2.

Proposition 3.1 Let A be a cellular algebra over a field k (with respect to an involution i), P an indecomposable projective–injective A–ideal with simple top L and J a cell ideal with associated standard module Δ. Suppose the intersection J ∩ P is not zero. Then the composition multiplicity [Δ : L] equals one and End_{A}(L) = k.

Proof. We can write the bimodule J as J ≃ Δ⊗_{k} i(Δ) and P = Ae for some primitive idempotent e ∈ A. Then J ∩ P equals Je which is isomorphic (as left module) to Δ⊗_{k} i(Δ)e. Denote by l the k–dimension of eΔ which equals that of i(Δ)e (by [3], Proposition 5.1). Then J ∩P is a direct sum of l copies of Δ. But being contained in the injective module P, it must have simple socle. Hence l equals one. By definition l also equals [Δ : L]dim_{k}(End_{A}(L)).

Proof of 1.2. Let A and P be as in the statement of 1.2. Denote by L the simple top of P. By theorem 1.1, L is isomorphic to the socle of P. Since A is cellular, its projective module P has a cell filtration. Assume this filtration has length one, that is, P equals a standard module, say Δ. Then by proposition 3.1, the composition multiplicity [Δ : L] equals one. But L occurs both as top and as socle, thus P must be simple.

For a cellular self–injective algebra, we also have the following property.

Proposition 3.2 Let {C_{s,t}^{λ} | s, t ∈ M(λ)} be a cell basis of a cellular algebra A. If A is self–injective, then AC_{s,t}^{λ} is indecomposable for λ a minimal element and s, t ∈ M(λ).

Proof. Since A is self–injective, we have that Hom_{A}(Ax, I) = xA∩I for any left ideal I in A and x ∈ A. In particular, End_{A}(AC_{s,s}^{λ}) ≃ C_{s,s}^{λ}A ∩ AC_{s,s}^{λ} ≃ kC_{s,s}^{λ} since λ is minimal. This shows that AC_{s,s}^{λ} is indecomposable. Now our proposition follows immediately from the following fact which holds true for all cell ideals.

Lemma 3.3 Let A be a k-algebra with an involution i. Suppose J is a cell ideal in A with a cell basis {C_{s,t} | 1 ≤ s, t ≤ n}. Then AC_{j,s} ≃ AC_{j,t} for all j, s and t.

Here, a cell basis of J is that subset of the basis C_{s,t}^{λ} given by definition 2.1 which spans J. That is, λ ∈ I is chosen minimal. Proof. We define r_{j} = (C_{j,1}, C_{j,2}, ..., C_{j,n})^{tr} , the column vector of the elements C_{j,1}, C_{j,2}, ..., C_{j,n}. Then for any element a ∈ A we have

ar_{j} = ( aC_{j,1} ; aC_{j,2} ; ⋮ ; aC_{j,n} ) = r_{a}(1, j)r_{1} + r_{a}(2, j)r_{2} + ... + r_{a}(n, j)r_{n},

where r_{a}(j, l) is an element in k for all j, l. Now we define a mapping

φ : AC_{j,s} ⟶ AC_{j,t}, ∑_{l} r_{a}(l, j)C_{l,s} ↦ ∑_{l} r_{a}(l, j)C_{j,t}

Since J is a cell ideal, this mapping is well-defined. Clearly, it is injective and surjective. Moreover φ is also an A-homomorphism by the definition of a cell ideal.

Under the assumption of A being self–injective, we can say more about this situation.

Proposition 3.4 If A is a self–injective algebra with an involution i and J is a cell ideal with a cell basis {C_{s,t}} such that i(C_{s,t}) = C_{t,s}, then each non–zero homomorphism from AC_{s,s} to AC_{t,t} is an isomorphism.

Proof. Since A is self–injective, we have that Hom_{A}(AC_{s,s}, AC_{t,t}) ≅ C_{s,s}A ∩ AC_{t,t} ⊂ kC_{s,t}. Let f_{0} be a non-zero homomorphism from AC_{s,s} to AC_{t,t}. Then C_{s,s}A ∩ AC_{t,t} = kC_{s,t} and there are two elements a, b ∈ A such that C_{s,s}a = C_{s,t} = bC_{t,t}. We define a A-homomorphism f : AC_{s,s} ⟶ AC_{t,t} by xC_{s,s} ⟼ xC_{s,s}a. It is clear that f is an A–homomorphism and that f_{0} is a scalar multiple of f. Thus it is sufficient to show that f is an isomorphism. Now we define g : AC_{t,t} ⟶ AC_{s,s} by yC_{t,t} ⟼ yC_{t,t}i(b). One can check that g is well–defined and an A–homomorphism obviously. Since C_{s,t} = bC_{t,t} = ∑_{l} r_{b}(l, t)C_{l,t}, we have that r_{b}(s, t) = 1, r_{b}(l, t) = 0 for l ≠ s and bC_{t,t}i(b) = C_{s,t}i(b) = ∑_{j} r_{b}(j, t)C_{s,j} = r_{b}(s, t)C_{s,s}. This implies that the composition of f and g is a non–zero endomorphism of AC_{s,s}. Thus the statement follows from End_{A}(AC_{s,s}) ≅ k.

Many known examples of quasi–hereditary algebras arise as endomorphism rings of sums of modules satisfying certain ordering conditions. This leads us to the following problem.

Question. Let A be a weakly symmetric cellular algebra. When is the endomorphism algebra End(⊕_{λ∈Λ,S∈M(λ)}AC_{S,S}^{λ}) a quasi-hereditary algebra?

References

[1] C.W.Curtis and I.Reiner, Representation Theorey of finite groups and associative algebras. John Wiley & Sons (1962).

[2] J.J.Graham and G.I.Lehrer, Cellular algebras. Invent. Math. 123, 1–34 (1996).

[3] S.König and C.C.Xi, On the structure of cellular algebras. In: I.Reiten, S.Smalø and Ø.Solberg (Eds.): Algebras and Modules II. Canadian Mathematical Society Conference Proceedings Vol. 24 (1998), 365–386.

[4] S.König and C.C.Xi, On the number of cells of a cellular algebra. Preprint 97-126, SFB 343 Bielefeld (1997). To appear in Communications in Algebra.

[5] S.König and C.C.Xi, When is a cellular algebra quasi-hereditary? Preprint 98-089, SFB 343 Bielefeld (1998). To appear in Mathematische Annalen.

Steffen König, Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany,

   koenig@mathematik.uni-bielefeld.de

Changchang Xi, Department of Mathematics, Beijing Normal University, 100875 Beijing, P.R. China,

   xicc@bnu.edu.cn