Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 36 (1995), No. 1, 37-71.

Quasi–hereditary Algebras which are Twisted Double Incidence Algebras of Posets

Bangming Deng and Changchang Xi^{1}

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

Abstract. For each finite poset, we associate it with a family M of matrices and define the corresponding M–twisted double incidence algebra which is a generalization of the construction given by Dyer. In the paper we mainly study the quadratic dual and Ringel dual of the M–twisted double incidence algebra. We prove that if the given poset is a tree then its M-twisted double incidence algebra is a BGG–algebra and the Ringel dual can be determined in detail (with some natural restriction on matrices). Moreover, we show that in the tree case with all matrices non–zero the processes of forming Ringel duals and quadratic duals of M–twisted double incidence algebras, respectively, commute with each other.

Introduction

In his study of Kazhdan–Lusztig–Stanley polynomials with interpretations in the representation theory of finite dimensional quadratic algebras, Dyer introduced in [4] for certain class of posets, including Bruhat intervals or face lattices, families of finite dimensional algebras, and proved that these algebras have many nice properties, and in particular, satisfy the main conjecture mentioned in [4, 1.6]. The algebras constructed from posets satisfying the conditions in [4] are quasi–hereditary algebras with dualities which fix simple modules. Such algebras are called BGG–algebras (see [9], and [3]). It is well–known that each block of the category O introduced by Bernstein, Gelfand and Gelfand in [1] is equivalent to the module category of a finite dimensional BGG–algebra such that its quadratic dual algebra is also quasi–hereditary. Quasi–hereditary algebras have been introduced by Cline, Parshall and Scott in order to describe the structure of certain finite dimensional algebras which arise naturally in the representation theory of semisimple complex Lie algebras and of reductive algebraic groups, and then are ring–theoretically investigated by Dlab and Ringel in [5]. Recently, Ringel [11] associated with each quasi–hereditary algebra

Typeset by AMS-TEX

1Supported by the Fok Ying Tung Education Foundation 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1995 Heldermann Verlag, Berlin

A another quasi–hereditary algebra R(A) which we call the Ringel dual of A. Being the endomorphism algebra of a generalized tilting module, the Ringel dual has many applications in the study of symmetric groups and Schur algebras (see [8]).

The main objective of the present paper is the study of Ringel duals and quadratic duals for a class of quasi–hereditary algebras which are defined to be twisted double incidence algebras of posets as well as of the relation between these two dualities.

First we simplify the construction in [4] and construct for any finite poset a family of finite dimensional algebras, here we do not require any restriction on posets and labellings. Since the labellings in our paper are families of matrices, we call them matrix labellings. Of course, if the given poset and the labelling satisfy all conditions in [4, Sect. 3], then our construction yields the same algebra as in [4] whose quasi–heredity is proved by Dyer. (Note that the algebras in [4] are a special class of a much larger class of algebras studied by Dyer in a big program.) In general, the algebras we constructed are not always quasi– hereditary. However, we prove that if the Hasse diagram of the poset is a tree then the constructed algebra is quasi–hereditary. Since in this case all the matrices are symmetric, the defined algebra is even a BGG–algebra.

Let X be a finite poset with a matrix labelling M. We consider the defined algebra A_{(X,M)}, and we call it the M–twisted double incidence algebra of X. To describe the quadratic dual of A_{(X,M)}, we first give a sufficient and necessary condition for A_{(X,M)} to be a quadratic algebra, and then introduce a new finite dimensional algebra B_{(X,M)} which is again defined by using the labelling poset X. One of our main results, Theorem 2.9, says that if A_{(X,M)} is quadratic then the quadratic dual of A_{(X,M)} is of the form B_{(X^{op}, −M^{t})}. Usually, the algebra B_{(X,M)} is not a quasi–hereditary algebra. However, in case X is a tree (namely, the Hasse diagram of X is a tree) or X satisfies the conditions in [4, sect.3], the algebra B_{(X,M)} becomes quasi–hereditary. For Ringel duals as well as the relation between Ringel duals and quadratic duals we concentrate mainly on the case when X is a tree. The main results are 4.6 and 5.8. Namely, in good cases (if X is a tree and all matrices are invertible), the processes of forming the quadratic dual and the Ringel dual of A_{(X,M)} respectively, commute with each other.

We organize the paper as follows. In section 0 we recall some basic definitions and facts. In section 1 we generalize the construction in [4]. Section 2 is devoted to the description of quadratic dual of A_{(X,M)}. In order to prove 4.6 and 5.8, we discuss in section 3 the Borel subalgebra and Δ–subalgebra in the sense of König. As a byproduct, we have an upper bound for global dimension of A_{(X,M)} if it is quasi–hereditary. Using the results in section 3 we prove the main results in the last two sections.

0. Preliminaries

0.1 Let A be a finite dimensional algebra over an algebraically closed field k. We will consider finitely generated left A–modules, maps between A–modules will be written on the right side of the argument, thus the composition of maps f : M_{1} ⟶ M_{2} and g : M_{2} ⟶ M_{3} will be denoted by fg. The category of all finitely generated modules will be denoted by A–mod. Given a class Θ of A–modules, we denote by F(Θ) the class of all A–modules which have a Θ–filtration, that is, a filtration

0 = M_{t} ⊂ M_{t−1} ⊂ ⋯ ⊂ M_{1} ⊂ M_{0} = M

such that each factor M_{i−1}/M_{i} is isomorphic to an object in Θ for 1 ≤ i ≤ t. For a module M ∈ A–mod, we denote by add(M) the full additive subcategory of A–mod consisting of all finite direct sums of direct summands of M.

Let ∧ be a finite poset in bijective correspondence with the isomorphism classes of simple A–modules. For each λ ∈ ∧, let E(λ) be a simple module in the isomorphism class corresponding to λ and P (λ) (or P_{A}(λ)) a projective cover of E(λ) and denote by Δ(λ) the maximal factor modules of P (λ) with composition factors of the form E(μ), μ ≤ λ. Dually, let Q(λ) (or Q_{A}(λ)) be an injective hull of E(λ) and denote by ∇(λ) the maximal submodule of Q(λ) with the composition factors of the form E(μ), μ ≤ λ. Let Δ (respectively, ∇) be the full subcategory of all Δ(λ), λ ∈ ∧ (respectively, all ∇(λ), λ ∈ ∧). We call the modules in Δ the standard modules and the ones in ∇ the costandard modules.

The algebra A is said to be quasi–hereditary with respect to (∧, ≤) if for each λ ∈ ∧ we have

(i) End_{A}(Δ(λ)) ≅ k;

(ii) P (λ) ∈ F(Δ), and moreover, P (λ) has a Δ–filtration with quotient Δ(μ) for μ ≥ λ in which Δ(λ) occurs exactly once.

For a quasi–hereditary algebra A with respect to a poset ∧ we call the elements in ∧ weights and ∧ the weight poset of A. By (A, ∧) we denote a quasi–hereditary algebra A with the weight poset ∧.

If a quasi–hereditary algebra has a duality δ which fixes simple modules, we call it a BGG–algebra (see [9], [3]).

0.2 The usual (and equivalent) definition of quasi–hereditary algebras uses the notion of a heredity ideal. A heredity ideal of an algebra A is an idempotent ideal I, with I(rad(A))I = 0, and such that _{A}I is a projective module (or, equivalently, I_{A} is a projective right module). A chain of ideals

A = I_{0} ⊇ I_{1} ⊇ ⋯ ⊇ I_{n} = 0

is called a heredity chain provided I_{t−1}/I_{t} is a heredity ideal of A/I_{t}, for 1 ≤ t ≤ n. A finite dimensional algebra is quasi–hereditary if and only if it has a heredity chain. Note that a heredity chain may always be refined so that for every t, the indecomposable summands of a fixed module _{A}(I_{t−1}/I_{t}) are all isomorphic, say isomorphic to some Δ(t), and then these modules are just the standard modules. Conversely, given an algebra A which satisfies the conditions (i) and (ii), then for every t there exists a maximal left ideal I_{t} of A which belongs to F({Δ(t + 1), ⋯, Δ(s(A))}), where s(A) is the number of the isomorphism classes of simple A–modules, it clearly will be a two–sided ideal, and, in this way, we obtain a heredity chain (I_{t})_{t} for A. Note that the A/I_{t}–modules are just those A–modules which only have composition factors of the form E(i), with i ≤ t.

0.3 To check whether an ideal in A is a heredity ideal one may use the following result due to Dlab and Ringel.

Lemma [5]. Let A be a finite dimensional k–algebra and I an ideal generated by the idempotent e in A. If e(rad(A))e = 0 then _{A}I is projective if and only if the multiplication map

Ae ⊗_{eAe} eA ⟶ AeA

is bijective.

0.4 Now we recall the definition of quadratic algebras and their quadratic duals (see [2]). Consider a k–algebra A_{0} ≅ k^{n}. Let A_{1} be a finite dimensional A_{0}–bimodule, and form the tensor algebra T_{A_{0}}(A_{1}) := ⊕_{n∈ℕ}T^{n}(A_{1}) over A_{0}, where T^{n}(A_{1}) := A_{1} ⊗_{A_{0}} ⋯ ⊗_{A_{0}} A_{1}

denotes the n–fold tensor product (and T^{0}(A_{1}) = A_{0}). An algebra of the form A = T_{A_{0}}(A_{1})/ < R(A) >, where < R(A) > is the two–sided ideal of T_{A_{0}}(A_{1}) generated by some A_{0}–subbimodule R(A) of T^{2}(A_{1}), is called a (basic) quadratic algebra.

Regard the dual space A_{1}^{∗} = Hom_{k}(A_{1}, k) as an A_{0}–bimodule with the action (v)(afb) = (bva)f, where a, b ∈ A_{0} and f ∈ A_{1}^{∗}. The quadratic dual of A is defined by A^{!} := T_{A_{0}}(A_{1}^{∗})/ < R(A)^{⊥} >, where R(A)^{⊥} is the annihilator R(A)^{⊥} := {f ∈ A_{1}^{∗} ⊗_{A_{0}} A_{1}^{∗} | (R(A))f = 0} of R(A) (we identify (A_{1} ⊗_{A_{0}} A_{1})^{∗} with A_{1}^{∗} ⊗_{A_{0}} A_{1}^{∗} by sending (f ⊗ g)^{∗} to g^{∗} ⊗ f^{∗}).

0.5 We fix some notation and terminology used throughout. For a finite poset X we denote by |X| the cardinality of X, and write x <· y (or y> · x) to indicate that x < y and that there is no z ∈ X satisfying x < z < y. For x, y ∈ X with x ≤ y the closed subinterval [x, y] is defined to be the set of all z ∈ X with x ≤ z ≤ y. A chain from x to y of length n is a sequence x = x_{0} < x_{1} < ⋯ < x_{n} = y of elements in X. A maximal chain of length n from x to y is a sequence x = x_{0} <· x_{1} <⋯ <· x_{n} = y. The minimum (resp. maximum) of the lengths of all chains from x to y is called the minimal (resp. maximal) length of [x, y].

We denote by k^{n×m} the set of all n × m matrices over k and by M^{t} the transpose of a matrix M.

1. Definition of algebras A_{(X,M)}

In this section we define for any finite poset X a family of finite dimensional algebras over an algebraically closed field k. Our construction is a generalization of that given in [4] and includes some special cases in [12]. We prove that A_{(X,M)} is quasi–hereditary if the Hasse diagram of X is a tree.

1.1 Let k be an algebraically closed field and X a finite poset. We consider each closed subinterval [x, ω] of X with minimal length 2. Suppose y_{1}, ⋯, y_{n} are elements in [x, ω] such that x <· y_{i} <· ω, 1 ≤ i ≤ n, are all maximal chains from x to ω of length 2, i.e. the Hasse diagram of [x, ω] has a subdiagram of the following form

y_{1}
⋮
x  y_{i}  ω
⋮
y_{n}

and such a diagram is called a mesh diagram of x and ω. With such a mesh we associate a matrix M_{n}(x, ω) ∈ k^{n×n}, say

M_{n}(x, ω) = (a_{y_{i}y_{j}}^{(x,ω)})_{y_{i}y_{j}}.

Then we say that X is labelled by matrices, denoted by (X, M), where M is the set of all the labelling matrices, and call M a matrix labelling on X.

1.2 Let X be a finite poset . We first define an associative k–algebra A′_{X} with a k– basis consisting of all symbols x_{n} ⋯ x_{1}x_{0}, where n ≥ 0 and x_{i} ∈ X, 0 ≤ i ≤ n, satisfy either x_{i−1} <· x_{i} or x_{i−1}> · x_{i}. The multiplication is defined on the basis by setting (y_{m} ⋯ y_{1}y_{0})(x_{n} ⋯ x_{1}x_{0}) equal to y_{m} ⋯ y_{1}x_{n} ⋯ x_{1}x_{0} if y_{0} = x_{n} and 0 otherwise, then extended to A′_{X} by linearity. The algebra A′_{X} is already introduced by Dyer in [4]. Obviously, there is a k–algebra anti-involution ε of A′_{X} defined on the basis vectors by ε : x_{n} ⋯ x_{1}x_{0} ⟼ x_{0}x_{1} ⋯ x_{n}.

1.3 Suppose that we are in the situation of 1.1. For x <· y, x <· z in X, we define

r_{yxz} = yxz − ∑_{ω} a_{yz}^{(x,ω)}yωz,

where the sum is over all ω satisfying y <· ω and z <· ω. Note that we allow in the above definition that y = z and that if there is no ω satisfying y <· ω and z <· ω then the summation is zero.

1.4 Definition. Let I(X, M) be the ideal of A′_{X} generated by elements of the following two types

(1.4.1)  r_{yxz}, x, y, z ∈ X with x <· y and x <· z,

(1.4.2)  u_{n} ⋯ u_{1}u_{0} − v_{m} ⋯ v_{1}v_{0}, v_{m} ⋯ v_{1}v_{0} − u_{n} ⋯ u_{1}u_{0}, for u < v in X,

where u = u_{n} <⋯ <· u_{1} <· u_{0} = v and u = v_{m} <⋯ <· v_{1} <· v_{0} = v are maximal chains from u to v.

We denote by A_{(X,M)} the quotient algebra of A′_{X} by I(X, M) and call it an M–twisted double incidence algebra of X (This name is suggested by S.König).

1.5 Remarks and examples. First note that if X satisfies the conditions in [4, 3.1–3.2], then the above definition yields the same algebra as in [4, 3.8]. Certainly, there are a lot of posets which do not satisfy the conditions in [4]. For example, let X be a poset with the following Hasse diagram

y_{1}  ω_{1}
x y_{2} ω_{2}
y_{3}  ω_{3}

Then it is clear that this poset does not fulfill the conditions in [4]. However, we can still get a quadratic algebra A_{(X,M)} by using 1.4. We take

M(x, ω_{1}) = [ \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \end{matrix} ] , M(x, ω_{2}) = [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ] , M(x, ω_{3}) = [^{0} ]

Then A_{(X,M)} is the path algebra of the quiver

y_{1}  ω_{1}
x  y_{2}  ω_{2}
y_{3}  ω_{3}

modulo the ideal I(X, M).

Second, if we take all the matrices to be zero matrices then we get the algebra A(I(X)), the dual extension of I(X), defined in [12] and discussed in [7], where I(X) is the incidence algebra of X. Recall that the algebra I(X) by definition is the quotient of I′_{X} by the ideal E(X), where I′_{X} denotes the subalgebra of A′_{X} with basis vectors x_{n} ⋯ x_{1}x_{0} satisfying x_{n} <⋯ <· x_{1} <· x_{0}, and where E(X) is the ideal of I′_{X} generated by the elements of the type u_{n} ⋯ u_{1}u_{0} − v_{m} ⋯ v_{1}v_{0} for u = u_{n} <⋯ <· u_{1} <· u_{0} = v and u = v_{m} <⋯ <· v_{1} <· v_{0} = v maximal chains from u to v in X.

Finally, if we take all the labelling matrices to be symmetric then the anti–involution ε induces a k–algebra anti–involution of A_{(X,M)} which fixes all primitive idempotents x with x ∈ X, and we still denote it by ε.

1.6 Proposition. For a finite poset X with an arbitrary matrix labelling M, the algebra A_{(X,M)} is finite dimensional.

Proof. For convenience, we write A′ = A′_{X}, A = A_{(X,M)} and I = I(X, M), and if there is no confusion in the context, we write sometimes x_{n} ⋯ x_{1}x_{0} for x_{n} ⋯ x_{1}x_{0} + I in A.

For the proof of this proposition, we show much more, namely the following three properties.

(1) Let x be a maximal element in X. Then Ax can be spanned by the set

{x_{n} ⋯ x_{1}x_{0}x + I | n ∈ ℕ, x_{n} <⋯ <· x_{0} <· x in X}

over k.

To prove (1) it is enough to show the following statement: For x_{n} ⋯ x_{1}x_{0} ∈ A′, if x_{n} ⋯ x_{1}x_{0}x ∉ I then x_{n} <⋯ <· x_{1} <· x_{0}. Indeed, if n = 0 or 1 then the statement follows directly from the fact that x is a maximal element and xyz ∈ I for y <· x and y <· z. Now suppose that the statement is true for n. Let us take an element x_{n+1}x_{n} ⋯ x_{1}x_{0} ∈ A′ such that x_{n+1}x_{n} ⋯ x_{1}x_{0}x ∉ I. Of course, we have x_{n} ⋯ x_{1}x_{0}x ∉ I. By induction hypothesis, we have x_{n} <⋯ <· x_{1} <· x_{0} <· x. If x_{n+1} <· x_{n} then we have what we want. Suppose x_{n} <· x_{n+1}, then

x_{n+1}x_{n}x_{n−1} = ∑_{ω} a_{x_{n+1}x_{n−1}}^{(x_{n},ω)}x_{n+1}ωx_{n−1} + r_{x_{n+1}x_{n}x_{n−1}}

in A′, where a_{x_{n+1}x_{n−1}}^{(x_{n},ω)} ∈ k and the sum is over all ω ∈ X satisfying x_{n+1} <· ω, x_{n−1} <· ω. Again by induction hypothesis, ωx_{n−1} ⋯ x_{1}x_{0}x ∈ I for all ω since ω> · x_{n−1}. Thus

\begin{aligned}x_{n+1}x_{n}x_{n−1} ⋯ x_{1}x_{0}x = & ∑_{ω} a_{x_{n+1}x_{n−1}}^{(x_{n},ω)}x_{n+1}ωx_{n−1} ⋯ x_{1}x_{0}x \\ & + r_{x_{n+1}x_{n}x_{n−1}}x_{n−2} ⋯ x_{1}x_{0}x \end{aligned}

lies in I. This contradiction shows that x_{n} <· x_{n+1} is impossible, and then finishes the proof of (1).

Dually, we have

(1′) Let x be a maximal element in X. Then xA can be spanned by the set

{xy_{m}y_{m−1} ⋯ y_{0} + I | m ∈ ℕ, x> · y_{m}> · y_{m−1}> ⋯> · y_{0} in X}

over k.

From (1) and (1′) we have

(2) xAx ≅ k and AxA is finite dimensional.

In fact, that Ax and xA are finite dimensional implies that AxA is also finite dimensional. Moreover, we have by (1) that Ax admits a k–basis {x, γ_{1}x, ⋯, γ_{m}x} with γ_{j} = x_{j1}x_{j2} ⋯ x_{jm_{j}} and x_{j1} <· x_{j2} <⋯ <· x_{jm_{j}} <· x, where m_{j} ≥ 1 is a natural number for j ≤ m. By (1′) xA has as k–basis the set {x, xδ_{1}, ⋯, xδ_{m}}, where δ_{j} = x_{jm_{j}} ⋯ x_{j2}x_{j1}. Since γ_{1}x, ⋯, γ_{m}x are k–linear independent, we know x_{i1} ≠ x_{j1} if i ≠ j. Note that

xγ_{j}x = 0 for all j according to (1). It is clear that AxA = ∑_{j=1}^{m} Axδ_{j}. Since the x_{ji} are

pairwise orthogonal idempotents, we have that AxA = Ax ⊕ Axδ_{1} ⊕ ⋯ ⊕ Axδ_{m}.

(3) Let x be a maximal element in X. We set X_{1} = X\{x}, equip X_{1} with the order relation induced by X and consider the algebra A_{(X_{1},M_{1})}, where the matrix labelling M_{1} of X_{1} is the restriction of the given labelling M of X on X_{1}. Let π_{X} and π_{1} denote the canonical projections from A′ = A′_{X} to A = A_{(X,M)} and from A to A/AxA, respectively. Note that the kernel of the composition of π_{X} and π_{1} is A′xA′ + I. Since X_{1} is a full convex subposet of X, the algebra A′_{X_{1}} is canonically embedded into A′. This embedding is denoted by μ.

The ideal I(X_{1}, M_{1}) of A′_{X_{1}} is mapped to zero under the morphism μπ_{X}π_{1}. In fact, if ρ ∈ A′_{X_{1}} is an element of type (1.4.1), then there exists an element ρ′ ∈ A′ of the same type such that ρ′ = ρ+a with a ∈ A′xA′. In this case we have (ρ)μπ_{X}π_{1} = (ρ′ −a)π_{X}π_{1} = 0 − ((a)π_{X})π_{1} = 0. If ρ ∈ A′ is of type (1.4.2), then ρ ∈ I = I(X, M), thus (ρ)μπ_{X}π_{1} = 0. Hence we deduce that (I(X_{1}, M_{1}))μπ_{X}π_{1} = 0. Therefore, μπ_{X}π_{1} induces an algebra homomorphism ν : A_{(X_{1}, M_{1})} ⟶ A/AxA, that is, we have the following commutative diagram:

A′_{X_{1}}  ⟶^{μ} A′
↓^{π_{X}}
A
π_{X_{1}} ↓  ↓^{π_{1}}
A_{(X_{1},M_{1})} ⟶^{ν} A/AxA

We shall prove that ν is an isomorphism of algebras. It is clear that ν is surjective. It remains to show that ν is injective. In order to do this, we prove first that for any a ∈ I with a = r + b, where r ∈ A′_{X_{1}} and b ∈ A′xA′, there holds r ∈ I(X_{1}, M_{1}). Since I is generated by the elements of types (1.4.1) and (1.4.2), it is enough to prove the statement for such generators. For the generators this becomes obvious.

With the above statement we can show that ν is injective. Taking an a + I(X_{1}, M_{1}) ∈ A_{(X_{1},M_{1})} with (a + I(X_{1}, M_{1}))ν = 0, we have 0 = (a + I(X_{1}, M_{1}))ν = (a)μπ_{X}π_{1}. that is, a ∈ I + A′xA′. Thus there are elements r′ ∈ I and b′ ∈ A′xA′ such that a = r′ + b′ ∈ A′_{X_{1}}. If we write r′ as a sum of r ∈ A′_{X_{1}} and b ∈ A′xA′, then r ∈ I(X_{1}, M_{1}). Hence a = r+(b+b′) implies that a = r ∈ I(X_{1}, M_{1}). Therefore, ν is injective.

Using induction on the number of elements of X, one can easily see that A_{(X,M)} is finite dimensional.

1.7 In general, the algebra A_{(X,M)} associated with a matrix labelling poset (X, M) may not be quasi–hereditary (see 1.8). However, Dyer proved that A_{(X,M)} is quasi–hereditary if (X, M) satisfies the conditions in [4, Sect. 3]. In the following we prove that if X is a tree, the algebra A_{(X,M)} is also quasi–hereditary. A further study of quasi–heredity of A_{(X,M)} will be given in a subsequent paper.

Theorem. If X is a tree then for any matrix labelling M, the algebra A_{(X,M)} is quasi– hereditary.

Proof. From the proof of 1.6 it suffices to prove that A_{(X,M)}xA_{(X,M)} is a heredity ideal in A_{(X,M)} for each maximal element x in X. To this goal we first prove the following fact.

(1) The elements of the form yxz := y_{0}y_{1} ⋯ y_{m}xz_{0}z_{1} ⋯ z_{n} with y = y_{0} <· y_{1} <⋯ <· y_{m} <· x> · z_{0}> · z_{1}> ⋯> · z_{n} = z in X form a basis of A_{(X,M)} over k.

Indeed, to each x ∈ X we attach a symbol α_{x} and denote by V_{x} the vector space over k with the basis set {α_{y} | x ≤ y, y ∈ X}. For x <· y in X, by i_{x}^{y} : V_{y} → V_{x} we denote the inclusion and by p_{y}^{x} : V_{x} → V_{y} the k–linear map such that

p_{y}^{x}(α_{z}) = \{ \begin{matrix}α_{y}, & if z = x, \\ ∑_{z<· z′}a_{zz′}α_{z′}, & if y ≤ z, \\ 0, & otherwise. \end{matrix}

where a_{zz′} = a_{y_{0}y_{0}}^{(x,y_{1})}a_{y_{1}y_{1}}^{(y_{0},y_{2})} ⋯ a_{y_{n}y_{n}}^{(y_{n−1},z)}a_{zz}^{(y_{n},z′)} and y = y_{0} <· y_{1} <⋯ <· y_{n} <· z is the unique maximal chain from y to z.

For x <· y in X, one then has

(1.7.1)  i_{x}^{y}p_{y}^{x} = ∑_{y<· w} a_{yy}^{(x,w)}p_{w}^{y}i_{x}^{w} ,

that is, V = ⊕_{x∈X}V_{x} becomes a module over A_{(X,M)}.

Further, for each x ∈ X, by j_{x} : V_{x} → V and π_{x} : V → V_{x} we denote the canonical inclusion and projection respectively. Then by (1.7.1) there is a unique algebra homomorphism ϕ : A_{(X,M)} → End_{k}(V) such that ϕ(x) = π_{x}j_{x} for x ∈ X and

ϕ(xy) = π_{x}p_{y}^{x}j_{y}, ϕ(yx) = π_{y}i_{x}^{y}j_{x}

if x <· y in X. It is easy to see that for y ≤ x ≥ z and for α_{y} ∈ V_{y}, one has (α_{y})(ϕ(yxz)) = α_{x} ∈ V_{z} ⊂ V. Hence the elements ϕ(yxz) with y ≤ x ≥ z are linearly independent over k, that is, the elements yxz ∈ A_{(X,M)} are linearly independent over k. On the other hand, by the definition of A_{(X,M)} the elements yxz with y ≤ x ≥ z span A_{(X,M)} over k. Therefore, the elements yxz form a basis of A_{(X,M)} over k.

(2) If x is a maximal element in X, the multiplication map

μ : A_{(X,M)}x ⊗_{k} xA_{(X,M)} ⟶ A_{(X,M)}xA_{(X,M)}

is bijective. This is equivalent to that for every y, z ∈ X, the induced map

μ_{yz} : yA_{(X,M)}x ⊗_{k} xA_{(X,M)}z ⟶ yA_{(X,M)}xA_{(X,M)}z

is bijective.

Indeed, the dimensions of yA_{(X,M)}x and xA_{(X,M)}z over k are at most 1, and that yA_{(X,M)}x ≠ 0 (resp. xA_{(X,M)}z ≠ 0) implies y ≤ x (resp. x ≥ z). If yA_{(X,M)}x = 0 or xA_{(X,M)}z = 0, it is obvious that μ_{yz} is bijective. If yA_{(X,M)}x ≠ 0 and xA_{(X,M)}z ≠ 0, then one has 0 ≠ yxz ∈ A_{(X,M)}, and thus μ_{yz} is bijective. Hence A_{(X,M)}xA_{(X,M)} is a heredity ideal of A_{(X,M)}.

This finishes the proof.

1.8 Remark. If the poset X is not a tree, then the statement (1) in the proof of Theorem 1.7 may not be true. Hence in general the defined algebra A_{(X,M)} is not quasi–hereditary. For example, let X be the poset with the following Hasse diagram:

2

1 4 5

3

and with the labelling matrices M(1, 4) = [ \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ] , M(2, 5) = [1], M(3, 5) = [1]. Then

the algebra A_{(X,M)} is given by the following quiver

2

1  4  5

3

with the relations

\begin{matrix}αα′ = γ′γ, αβ′ = βα′ = 0, ββ′ = δ′δ, γγ′ = η′η, δδ′ = η′η, \\ ηη′ = 0, γα = δβ, α′γ′ = β′δ′. \end{matrix}

It is easy to check that A_{(X,M)} is not quasi–hereditary.

2. The quadratic dual of A_{(X,M)}

Let A_{(X,M)} be the algebra associated with a finite poset X labelled by M, a family of matrices (see section 1). In this section we describe the quadratic dual algebra A_{(X,M)}^{!} when A_{(X,M)} is quadratic. The first question is when A_{(X,M)} is quadratic, and we have the following

2.1 Lemma. The algebra A_{(X,M)} is quadratic if and only if the incidence algebra I(X) is quadratic.

This follows directly from the definition of A_{(X,M)}.

2.2 For x ≤ y in X, recall that by a maximal chain from x to y of length n we mean a sequence x = x_{0} <· x_{1} <⋯ <· x_{n} = y. We say that another chain x = x′_{0} <· x′_{1} <⋯ <· x′_{n′} = y is adjacent to the first one if n = n′ and there is a unique 1 ≤ j ≤ n − 1 such that x_{j} ≠ x′_{j}. The poset X is called adjacent if for every x ≤ y in X and for every two maximal chains C, D from x to y, there exists a sequence C = C_{0}, C_{1}, ⋯, C_{m} = D of maximal chains from x to y such that C_{i} is adjacent to C_{i+1} for 0 ≤ i ≤ m − 1.

Lemma. The incidence algebra I(X) is quadratic if and only if X is adjacent.

Proof. It is easy to see that the condition is sufficient. We now prove the necessity of the condition.

Assume that I(X) is quadratic, then the ideal E(X) is generated by the elements of the form

xy_{1}z − xy_{2}z for x, y_{1}, y_{2}, z ∈ X with x <· y_{1} <· z, x <· y_{2} <· z.

Suppose x ≤ y. Let C : x = x_{0} <· x_{1} <⋯ <· x_{n} = y and D : x = x′_{0} <· x′_{1} <⋯ <· x′_{n′} = y be maximal chains from x to y. We shall prove that C and D are pre–adjacent, that is, there is a sequence C = C_{0}, C_{1}, ⋯, C_{p} = D of maximal chains from x to y such that C_{i} is adjacent to C_{i+1} for 0 ≤ i ≤ p − 1.

From x_{0}x_{1} ⋯ x_{n} − x′_{0}x′_{1} ⋯ x′_{n′} ∈ E(X) it follows that

(2.2.1)  \begin{matrix}x_{0}x_{1} ⋯ x_{n} − x′_{0}x′_{1} ⋯ x′_{n′} \\ = ∑_{i=1}^{m} a_{i}x_{i0} ⋯ x_{is_{i}}(u_{i}v_{i}^{−}w_{i} − u_{i}v_{i}^{+}w_{i})x_{i,s_{i}+1} ⋯ x_{it_{i}} \end{matrix}

in I′_{X}, where a_{i} ∈ k and C_{i}^{σ} : x_{i0} <⋯ <· x_{is_{i}} <· u_{i} <· v_{i}^{σ} <· w_{i} <· x_{i,s_{i}+1} <⋯ <· x_{it_{i}} are maximal chains in X, i ∈ {1, ⋯ m}, σ ∈ {−, +}. It is easy to see that C_{i}^{−} is adjacent to C_{i}^{+} for all 1 ≤ i ≤ m. By c, d and c_{i}^{σ} we then denote the elements x_{0}x_{1} ⋯ x_{n}, x′_{0}x′_{1} ⋯ x′_{n′} and x_{i0} ⋯ x_{is_{i}}u_{i}v_{i}^{σ}w_{i}x_{i,s_{i}+1} ⋯ x_{it_{i}} in I′_{X}, respectively, where i = 1, ⋯, m and σ = −, +.

Without loss of generality, we may assume that x_{i0} = x, x_{it_{i}} = y for 1 ≤ i ≤ m, i.e. C_{i}^{σ}, i = 1, ⋯, m, σ = −, +, are maximal chains from x to y.

Suppose that C_{1}^{−}, C_{1}^{+}, ⋯, C_{t}^{−}, C_{t}^{+} are pre–adjacent to C and that C_{t+1}^{−}, C_{t+1}^{+}, ⋯, C_{m}^{−}, C_{m}^{+} are not pre–adjacent to C. We claim that D is pre–adjacent to C. Otherwise, each element in {c} ∪ {c_{j}^{σ} | 1 ≤ j ≤ t, σ = −, +} is not a k–linear combination of the elements in {d} ∪ {c_{j}^{σ} | t + 1 ≤ j ≤ m, σ = −, +} in I′_{X}. According to (2.2.1) we have in I′_{X} that

c − ∑_{j=1}^{t} a_{j}(c_{j}^{−} − c_{j}^{+}) = d + ∑_{j=t+1}^{m} a_{j}(c_{j}^{−} − c_{j}^{+}).

By linear independence, we get c − ∑_{j=1}^{t} a_{j}(c_{j}^{−} − c_{j}^{+}) = 0, that is,

x_{0}x_{1} ⋯ x_{n} = c = ∑_{j=1}^{t} a_{j}(c_{j}^{−} − c_{j}^{+}) ∈ E(X).

On the other hand, we know from the definition of an incidence algebra that for any maximal chain x_{0} <· x_{1} <⋯ <· x_{n}, there holds x_{0}x_{1} ⋯ x_{n} ∉ E(X). Thus we have a contradiction. This shows D is adjacent to C and finishes the proof.

2.3 In order to describe A_{(X,M)}^{!}, we define an algebra B_{(X,M)} associated with each matrix labelling poset (X, M). The algebra A′_{X} is defined as in section 1.

Definition. The algebra B_{(X,M)} is the quotient of A′_{X} by the two–sided ideal J(X, M) generated by the elements of the following two types:

(2.3.1)  r_{yxz} = yxz − ∑_{ω} a_{yz}^{(xω)}yωz for x <· y, x <· z,

where the sum is over all ω ∈ X satisfying y <· ω, z <· ω, and

(2.3.2)  q_{xy} = ∑_{i} xx_{i1} ⋯ x_{im_{i}}y, q′_{xy} = ∑_{i} yx_{im_{i}} ⋯ x_{i1}x, for x < y in X,

where the both sums are over all maximal chains x <· x_{i1} <⋯ <· x_{im_{i}} <· y from x to y of length ≥ 2. Note that if there is no maximal chain from x to y of length ≥ 2 in X, then q_{xy} = 0.

2.4 Proposition. For an arbitrary matrix labelling poset (X, M), the algebra B_{(X,M)} is quadratic.

Proof. Suppose x < y in X. We denote by q_{[x, y]} the sum of all length 2 maximal chains from x to y in X. If there is no such chain then q_{[x, y]} = 0. It is easy to see that

(2.4.1)  q_{xy} = q_{[x, y]} + ∑_{x<· u<y} xu · q_{uy}.

Further, we denote by J′(X, M) the ideal of B_{(X,M)} generated by the elements of type (2.3.1) and of types q_{[x, y]}, q′_{[x, y]} for x < y in X, where q′_{[x, y]} is the image of q_{[x, y]} under ε given in 1.2. We shall prove that J(X, M) = J′(X, M).

(1) To prove J(X, M) ⊆ J′(X, M) it is enough to show that the elements of type (2.3.2) lie in J′(X, M).

Suppose x < y in X. By l(x, y) we denote the maximum of lengths of maximal chains from x to y. We use induction on l(x, y) to prove q_{xy} ∈ J′(X, M). If l(x, y) = 1, then q_{xy} = 0 ∈ J′(X, M). Now suppose that for every x < y in X with 1 ≤ l(x, y) < n, the element q_{xy} lies in J′(X, M). Choose x < y in X with l(x, y) = n. Then for all x <· u < y, there holds l(u, y) < n = l(x, y). By induction hypothesis, q_{uy} ∈ J′(X, M) for all u. Then by (2.4.1) one has

q_{xy} = q_{[x, y]} + ∑_{x<· u<y} xu · q_{uy} ∈ J′(X, M).

Similarly, q′_{xy} ∈ J′(X, M). Therefore, J(X, M) ⊆ J′(X, M).

(2) The inverse inclusion J′(X, M) ⊆ J(X, M) follows from

q_{[x, y]} = q_{xy} − ∑_{x<· u<y} xu · q_{uy} ∈ J(X, M).

By (1) and (2) we have that J(X, M) = J′(X, M) is generated by quadratic relations, i.e. B_{(X,M)} is quadratic.

2.5 Let x ∈ X be maximal. We set X_{1} = X\{x} and equip X_{1} with the order relation induced by X. By M_{1} we denote the labelling of the restriction of M on X_{1}. Then we have the following

Proposition. There is a canonical algebra isomorphism from the algebra B_{(X_{1},M_{1})} to B_{(X,M)}/B_{(X,M)}xB_{(X,M)}, where B_{(X,M)}xB_{(X,M)} denotes the ideal of B_{(X,M)} generated by the idempotent x.

The proof of this proposition is similar to that of (4) in Proposition 1.6.

2.6 Corollary. For each (X, M), the dimension of B_{(X,M)} over k is finite.

Proof. Let x be a maximal element in X. Using a similar argument in the proof of (1) in Proposition 1.6, one can see that B_{(X,M)}x is spanned by {x_{n} ⋯ x_{1}x_{0}x + J(X, M) | n ∈ ℕ, x_{n} <⋯ <· x_{1} <· x_{0} <· x in X} over k and that xB_{(X,M)} is spanned by {xy_{m} ⋯ y_{1}y_{0} + J(X, M) | m ∈ ℕ, x> · y_{m}> ⋯> · y_{1}> · y_{0} in X} over k. Thus B_{(X,M)}x and xB_{(X,M)} are finite dimensional k–space. Therefore, B_{(X,M)}xB_{(X,M)} is a finite dimensional k–space. Using Proposition 2.5 and induction on the number of the elements in X, one can obtain that B_{(X,M)}xB_{(X,M)} is finite dimensional.

2.7 Suppose A_{(X,M)} is quadratic. Following Manin ( see [7] ), we write the quadratic algebra in the following form:

A_{(X,M)} ⟷ {A_{1}, R(A_{1}) ⊆ A_{1} ⊗_{A_{0}} A_{1}},

where A_{0} is the vector space spanned by x ∈ X and A_{1} is spanned by xy and yx for x <· y in X, and where R(A_{1}) is the vector space spanned by

{uv_{1}w − uv_{2}w | u, v_{1}, v_{2}, w ∈ X with u <· v_{1} <· w, u <· v_{2} <· w}
∪ {r_{yxz}|x, y, z ∈ X with x <· y, x <· z}.

In this case, one can also see that A_{1} ⊗_{A_{0}} A_{1} is the vector space spanned by

S = {xy ⊗ yz = xyz | x, y, z ∈X with x <· y and y <· z, or x <· y and y> · z
or x> · y and y <· z, or x> · y and y> · z}.

By definition, the quadratic dual algebra A_{(X,M)}^{!} is given by

A_{(X,M)}^{!} ⟷ {A_{1}^{∗}, R(A_{1})^{⊥} ⊆ (A_{1} ⊗_{A_{0}} A_{1})^{∗} = A_{1}^{∗} ⊗_{A_{0}} A_{1}^{∗}},

where A_{1}^{∗} = Hom_{k}(A_{1}, k), where R(A_{1})^{⊥} = {f ∈ A_{1}^{∗} ⊗_{A_{0}} A_{1}^{∗} | (R(A_{1}))f = 0} and where the bimodule structure of A_{1}^{∗} over A_{0} is defined in an obvious way. Here we identify (A_{1} ⊗_{A_{0}} A_{1})^{∗} with A_{1}^{∗} ⊗_{A_{0}} A_{1}^{∗} by sending (a ⊗ b)^{∗} to b^{∗} ⊗ a^{∗}. Then A_{1}^{∗} ⊗_{A_{0}} A_{1}^{∗} is the vector space spanned by

S^{∗} = {(yz)^{∗} ⊗ (xy)^{∗} = (xy ⊗ yz)^{∗} = (xyz)^{∗} | x, y, z ∈ X, xyz ∈ S}.

Lemma. As a vector space, R(A)^{⊥} is spanned by the elements of the following two types

(2.7.1)  q_{[u, w]}^{∗} = ∑_{v}(uvw)^{∗}, q′_{[u, w]}^{∗} = ∑_{v}(wvu)^{∗}, u, v, w ∈ X,

where the both sums are over all v such that u <· v <· w, and

(2.7.2)  r_{yωz}^{∗} = (yωz)^{∗} + ∑_{x} a_{yz}^{(x,ω)}(yxz)^{∗} for y <· ω, z <· ω in X,

where the sum is over all x satisfying x <· y, x <· z.

Proof. It is easy to check that the elements of the types (2.7.1) and (2.7.2) lie in R(A)^{⊥}. Conversely, let Φ ∈ R(A)^{⊥}, and we write

\begin{aligned}Φ = & ∑_{x<· y<· z} a_{xyz}(xyz)^{∗} + ∑_{x> · y> · z} b_{xyz}(xyz)^{∗} + ∑_{\begin{matrix}x<· y \\ z<· y \end{matrix}} c_{xyz}(xyz)^{∗} \\ & + ∑_{\begin{matrix}x> · y \\ z> · y \end{matrix}} d_{xyz}(xyz)^{∗} =: Φ_{1} + Φ_{2} + Φ_{3} + Φ_{4} , \end{aligned}

where a_{xyz}, b_{xyz}, c_{xyz}, d_{xyz} ∈ k and Φ_{i} denotes the i–th term in the sum of the above equality.

Let [u, w] be a closed subinterval of minimal length 2 in X with the mesh diagram of u and w of the following form (s ≥ 1):

v_{1}
⋮
u  v_{i}  w
⋮
v_{s}

then uv_{i}w − uv_{j}w ∈ R(A) for 1 ≤ i, j ≤ s. Thus 0 = (uv_{i}w − uv_{j}w)Φ = (uv_{i}w)Φ − (uv_{j}w)Φ = a_{uv_{i}w} − a_{uv_{j}w}, i.e. a_{uv_{i}w} = a_{uv_{j}w} for 1 ≤ i, j ≤ s. Therefore, Φ_{1} = ∑_{x<· y<· z} a_{xyz}(xyz)^{∗} = ∑_{u,w} a_{uw}q_{[u, w]}^{∗}, where the sum is over all mesh diagrams of u and

w for u, w in X, and where a_{uw} = a_{uv_{i}w} for 1 ≤ i ≤ s.

Similarly, Φ_{2} = ∑_{x> · y> · z} b_{xyz}(xyz)^{∗} = ∑_{u,w} b_{uw}q′_{[u, w]}^{∗}.

Now consider a full subposet of X with the following Hasse diagram.

(1)  \begin{aligned}& & & & e \\ g_{1} & g_{2} & ⋯ & g_{s} & & h_{1} & h_{2} & ⋯ & h_{t} \\ & & & & f \end{aligned}

such that g_{1}, ⋯, g_{s} (resp. h_{1}, ⋯, h_{t}) are all the elements v in X satisfying v <· e, v <· f (resp. e <· v, f <· v) with s + t ≥ 1.

By definition, r_{eg_{i}f} = eg_{i}f − ∑_{j=1}^{t} a_{ef}^{(g_{i},h_{j})}eh_{j}f ∈ R(A), i = 1, ⋯, s. Then

0 = (r_{eg_{i}f})Φ = (eg_{i}f)Φ − ∑_{j=1}^{t} a_{ef}^{(g_{i},h_{j})}(eh_{j}f)Φ = c_{eg_{i}f} − ∑_{j=1}^{t} a_{ef}^{(g_{i},h_{j})}d_{eh_{j}f}

thus

Φ_{3} + Φ_{4} = ∑_{\begin{matrix}x<· y \\ z<· y \end{matrix}} c_{xyz}(xyz)^{∗} + ∑_{\begin{matrix}x> · y \\ z> · y \end{matrix}} d_{xyz}(xyz)^{∗}

= ∑_{e,f} ∑_{j} d_{eh_{j}f}((eh_{j}f)^{∗} + ∑_{i} a_{ef}^{(g_{i},h_{j})}(eg_{i}f)_{i}^{∗})

= ∑_{e,f} ∑_{j} a_{eh_{j}f}r_{(eh_{j}f)}^{∗} ,

where the first sum is over all pairs (e, f) such that X contains a full subposet of the form (1).

From the argument above we conclude that Φ = Φ_{1} + Φ_{2} + Φ_{3} + Φ_{4} is a k–linear combination of the elements of the types (2.7.1) and (2.7.2). This finishes the proof.

With the help of the lemma, we now can describe the quadratic dual algebra A_{(X,M)}^{!}.

2.8 Let X be a finite poset. By X^{op} we denote the opposite poset of X. With each matrix labelling M on X we associate a matrix labelling −M^{t} on X^{op} as follows: Let [x, z] be a closed subinterval of minimal length 2 in X with the following mesh diagram of x and z:

y_{1}
⋮
x  y_{i}  z
⋮
y_{n}

and denote by M(x, z) ∈ k^{n×n} the labelling matrix to the mesh diagram of x and z. Then [z, x] is a closed subinterval of minimal length 2 in X^{op} with the mesh diagram of z and x of the form:

y_{1}
⋮
z  y_{i}  x
⋮
y_{n}

Now we label the mesh diagram of z and x in X^{op} by the matrix −M(x, z)^{t}, where M(x, z)^{t} denotes the transpose of M(x, z). In such a way, X^{op} becomes a matrix labelling poset, denoted by (X^{op}, −M^{t}).

As a consequence of Lemma 2.7, Proposition 2.4 and the construction of the algebra B_{(X^{op}, −M^{t})}, we have the following

Theorem. For any matrix labelling poset (X, M), if the poset X is adjacent, then

A_{(X,M)}^{!} ≅ B_{(X^{op}, −M^{t})}.

2.9 Remark. The description of the quadratic dual A_{(X,M)}^{!} in Theorem 2.8 is a generalization of that given in [4]. Namely, if the poset and the labelling M satisfy the conditions in [4], then 2.8 is just [4, 3.6(a)].

2.10 One interesting question is when the quadratic dual A_{(X,M)}^{!} is quasi–hereditary, i.e. when B_{(X^{op}, −M^{t})} is quasi–hereditary. The following propositions give a partial answer to this question.

Proposition 1. If the Hasse diagram of X is a tree then A_{(X,M)}^{!} is quasi–hereditary.

Proof. We shall prove that B_{(X,M)} is quasi–hereditary for any tree X with arbitrary matrix labelling M. Set B = B_{(X,M)}. As in the proof of Proposition 1.6 we can show that for each maximal element x in X there holds End_{B}(Bx) ≅ k. Now we prove that the multiplication map Bx ⊗_{k} xB ⟶ BxB is injective. It is equivalent to showing that for each pair y, z in X the induced map

μ : yBx ⊗_{k} xBz ⟶ yBxBz

is injective. If y = x or y = z then μ is clearly bijective. Suppose both y ≠ x and z ≠ x. Since X is a tree, we have that x_{1}x_{2}x_{3}, x_{3}x_{2}x_{1} ∈ J(X, M) for all x_{1} <· x_{2} <· x_{3} in X. Then the space yBx is at most of dimension 1. Similarly, the space xBz is at most of dimension 1. If yBx = 0 or xBy = 0, then μ is injective. Suppose that yBx ≠ 0 and xBz ≠ 0. Then yBx = kyx with y <· x and xBz = kxz with z <· x. In this case, yxz ∉ J(X, M) according to the definition of B_{(X,M)}. Thus μ is injective. Hence if x is maximal in X, then BxB is a heredity ideal in B_{(X,M)}.

Now applying induction on the number of elements of X and 2.5, we can see immediately that B_{(X,M)} is quasi–hereditary for any poset X whose Hasse diagram is a tree.

Let us remark that if X = {1 < 2 < ⋯ < n} and if all matrices in the labelling M are [1], then the algebra B_{(X,M)} is just a finite type block of some q–Schur algebra (see [13]).

Proposition 2. If X is a poset with the following Hasse diagram (n ≥ 1):

y_{1}
⋮
x  y_{i}  z
⋮
y_{n}

Then for an arbitrary labelling matrix M ∈ k^{n×n}, the algebra B_{(X,M)} is quasi–hereditary.

Proof. For simplicity, we write B = B_{(X,M)} and J = J(X, M). By Proposition 2.5, it is enough to prove that the following map

\begin{matrix}Ψ Bz ⊗_{zBz} zB & ⟶ BzB \\ a ⊗ b & ⟼ ab \end{matrix}

is bijective. It is obvious that Ψ is surjective. To see the injectivity we compare the dimensions of both sides over k. Since zBz ≅ k, dim_{k}Bz⊗_{zBz} zB = dim_{k}Bz×dim_{k}zB. One can easily see, as k–spaces, that Bz has a basis S_{1} = {z, y_{1}z, ⋯, y_{n}z, xy_{1}z, ⋯, xy_{n−2}z}, that zB has a basis S_{2} = {z, zy_{1}, ⋯, zy_{n}, zy_{1}x, ⋯, zy_{n−2}x} and that BzB has a basis {ab : a∈ S_{1}, b∈ S_{2}}, i.e. dim_{k}BzB = dim_{k}Bz × dim_{k}zB = 4n^{2}. This finishes the proof.

3. Δ–subalgebras and global dimensions

In the paper [10] S.König defines Δ–subalgebras and Borel subalgebras for quasi–hereditary algebras which might serve as finite–dimensional associative substitutes of Lie theoretic Borel subalgebras. In this section we prove that if the algebra A_{(X,M)} is quasi–hereditary, it always has a Δ–subalgebra and a Borel subalgebra. The results of this section will be used in the next two sections to calculate the dimensions of Ext–groups.

Let us recall from [10] some definitions.

3.1 Definition. Let (A, ∧) be a quasi–hereditary algebra with the weight poset ∧ and standard modules Δ(λ), λ ∈ ∧. Let B be a subalgebra of A (both algebras have the same unit element). Then B is called an exact Borel subalgebra of (A, ∧) if and only if there exists a bijection between the set of weights of B and the set of weights of A such that the following conditions are satisfied:

(D) The algebra (B, ∧) is directed (i.e. quasi–hereditary with simple standard modules and injective costandard modules) with respect to the partial order induced from the partial order of the set of weights of A;

(T) Tensor induction A ⊗_{B} − is an exact functor, thus A_{B} is projective as a right B–module;

(W) For each λ ∈ ∧, the following holds:

A ⊗_{B} E_{B}(λ) ≅ Δ_{A}(λ),

where E_{B}(λ) denotes the simple B–module corresponding to the weight λ.

The algebra B is called a strong exact Borel subalgebra of (A, ∧) if and only if both the following condition (S) and the above conditions (T) and (W) are satisfied.

(S) There is a maximal semisimple subalgebra S(A) of A which is also a maximal semisimple subalgebra of B, thus simple A–modules and simple B-modules coincide.

It is noted by S.König that if a quasi–hereditary algebra has a strong exact Borel subalgebra B then gl.dim.B ≤ gl.dim.A (see 3.5 below). There is a dual notion of Δ–subalgebra (see [10]).

3.2 Definition. Let (A, ∧) be a quasi–herditary algebra and C a subalgebra of A, having the same number of weights. Suppose the weights of C are identified (via certain bijection) with the weights of A, and the set of weights of C in this way is made into a poset. Assume that the algebra (C^{op}, ∧) is directed (i.e. (C, ∧) is quasi–hereditary with projective standard modules). Assume that for each weight i the projective A–module A ⊗_{C} P_{C}(i) (where P_{C}(i) is the indecomposable projective C–module corresponding to the weight i) decomposes into a direct sum of exactly one copy P_{A}(i) (which is the indecomposable projective A–module corresponding to the weight i) and some indecomposable projective A–modules having weights different from i. Fix for each weight i an epimorphism κ(i) : A ⊗_{C} P_{C}(i) → Δ_{A}(i).

Then C is called a Δ–subalgebra of (A, ∧) if and only if for each weight i the restriction of κ(i) to P_{C}(i) ⊂ A ⊗_{C} P_{C}(i) is an isomorphism of C–modules:

κ(i) |_{P_{C}(i)}: P_{C}(i) ≅ Δ_{A}(i).

If, in addition, C contains a maximal semisimple subalgebra of A, then C is called a strong Δ–subalgebra of (A, ∧).

3.3 Remark. A special case in the definition is that one takes C to be a subalgebra of A with the same complete set of orthogonal primitive idempotents, that is, 1 = e_{1} + ⋯ + e_{n} with e_{i} ∈ C. In this case A ⊗_{C} P_{C}(i) = A ⊗_{C} Ce_{i} ≅ Ae_{i} and κ(i) can be chosen to be the canonical surjective map Ae_{i} → Δ_{A}(i).

As pointed out in [10] the relation between a Borel subalgebra and a Δ–subalgebra is as follows.

3.4 Theorem [10]. Let (A, ∧) be a quasi–hereditary algebra and B a subalgebra of A. Then B is an exact Borel subalgebra of (A, ∧) if and only if B^{op} is a Δ–subalgebra of (A^{op}, ∧).

3.5 Lemma. Let A be a quasi–hereditary algebra which has a strong exact Borel subalgebra B. Then for each natural number l, for each B–module M, and for each A–module N there is an isomorphism of vector spaces:

Ext_{A}^{l}(A ⊗_{B} M, N) ≅ Ext_{B}^{l}(M, N|_{B}),

where N|_{B} denotes the restriction of the A–module N to B.

Proof. The lemma is called Eckmann–Shapiro lemma. For the convenience of readers, we include a proof here. It is enough to show the above isomorphism of Ext–groups. If l = 0, this is just the adjunction formula. If M is a projective B–module, there is nothing to be shown. If M is not projective, then we choose an exact sequence

0 ⟶ K ⟶ P ⟶ M ⟶ 0

with P a projective B–module. This yields an exact sequence

0 ⟶ A ⊗_{B} K ⟶ A ⊗_{B} P ⟶ A ⊗_{B} M ⟶ 0.

Thus we have the following commutative diagram:

0⟶ (A ⊗_{B} M,_{A}N)⟶ (A ⊗_{B} P,_{A}N)⟶ (A ⊗_{B} K,_{A}N)⟶ Ext_{A}^{1}(A ⊗_{B} M,_{A}N)⟶ 0
↓  ↓  ↓
0 ⟶ (M, N|_{B}) ⟶ (P, N|_{B}) ⟶ (K, N|_{B}) ⟶ Ext_{B}^{1}(M, N|_{B}) ⟶ 0

where all vertical maps are adjunctions and where (−, −) denotes the homomorphism group. This shows that for l = 1, the lemma is true. For large l, the isomorphism follows by dimension shift.

3.6 Now we consider the algebra A_{(X,M)} constructed in section 1. Suppose that A_{(X,M)} is quasi–hereditary with respect to X. From the proof of 1.6, the standard A_{(X,M)}–modules are just the indecomposable projective I(X)–modules, where I(X) is the incidence algebra of the poset X. Hence we have by definition the following Theorem. Let X be a finite poset with a matrix labelling M such that the algebra A_{(X,M)} is quasi–hereditary. Then the incidence algebra I(X) of X is a strong Δ–subalgebra of A_{(X,M)}.

If the labelling is symmetric, that is, all matrices are symmetric, then we have an anti– involution ε of the algebra A_{(X,M)} which fixes the complete set of primitive idempotents. This together with 3.4 yields the following

3.7 Corollary. Let X be a finite poset with a symmetric matrix labelling such that A_{(X,M)} is quasi–hereditary. Then A_{(X,M)} has a strong exact Borel subalgebra.

Now we shall prove that 3.7 holds more generally. Suppose A_{(X,M)} is quasi–hereditary.

Let B be the subalgebra of A_{(X,M)} generated by all elements x ∈ X and zy with y <· z in X. Then B and A_{(X,M)} have the same maximal semisimple subalgebra ∑_{x∈X} kx. If x

is a maximal element, then A_{(X,M)}xA_{(X,M)} is a heredity ideal by our assumption and xA_{(X,M)} can be spanned as a vector space by all elements of the form x_{0}x_{1} ⋯ x_{n} with x_{n} <⋯ <· x_{1} <· x_{0} = x. Hence the costandard module ∇_{A_{(X,M)}}(x) is just the injective A_{(X,M)}–module D(xA_{(X,M)}), where D = Hom_{k}(−, k). Note that the element x_{0}x_{1} ⋯ x_{n} with x_{n} <⋯ <· x_{1} <· x_{0} = x belongs to B. Therefore xB = xA_{(X,M)}. Hence the induction procedure of the proof of 1.6 shows that ∇_{A_{(X,M)}}(x) ≅ D(xB) as B–modules for every element x ∈ X. It follows then from [10, Theorem A] that B is an exact Borel subalgebra of A_{(X,M)}.

3.8 Theorem. Let X be a finite poset with a matrix labelling M such that A_{(X,M)} is quasi–hereditary. Then A_{(X,M)} has a strong exact Borel subalgebra.

As a first application of the above results we get an estimate of global dimension of A_{(X,M)}.

Let I^{+} be the ideal of A := A_{(X,M)} generated by all elements xy with x, y ∈ X and y <· x. Then A/I^{+} ≅ I(X). Similarly, let I^{−} be the ideal of A generated by all elements xy with x, y ∈ X and x <· y. Then B ≅ A/I^{−}. Hence every B–module can be regarded as an A–module.

Note that the anti–involution ε of A′_{X} induces an anti–isomorphism from I(X) to B. This implies that dim_{k}Δ(x) = dim_{k}∇(x) for all x ∈ X.

3.9 Theorem. Suppose X is a finite poset with a matrix labelling M such that A_{(X,M)} is quasi–hereditary. If the incidence algebra I(X) of X has global dimension m, then m ≤ gl.dim.A_{(X,M)} ≤ 2m.

Proof. For every simple I(X)–module E(x), we have a miminal projective resolution in I(X)–mod

0 ⟶ P_{l_{x}} ⟶ ⋯ ⟶ P_{1} ⟶ P_{0} ⟶ E(x) ⟶ 0,

where l_{x} is a natural number and l_{x} ≤ m. Since P_{i} is projective I(X)–module and every indecomposable projective I(X)–module can be considered as a standard A_{(X,M)}–module, it follows from P_{I(X)}(x) ≅ Δ_{A_{(X,M)}}(x) = A_{(X,M)} ⊗_{B} E(x) and proj.dim._{B}E(x) ≤ m that the projective dimension of P_{i} as A_{(X,M)}–module is bounded by m. Using the fact that if 0 ⟶ M_{n} ⟶ ⋯ ⟶ M_{1} ⟶ M_{0} ⟶ M ⟶ 0 is a long exact sequence and proj.dim.M_{i} ≤ k for all i, then proj.dim.M ≤ n+k, one obtains that gl.dim.A_{(X,M)} ≤ 2m. The inequality gl.dim.A_{(X,M)} ≥ m follows from 3.5.

As a second application we point out the k–dimension of A_{(X,M)}.

3.10 Theorem. Let X = {x_{1}, ⋯, x_{n}} be a finite poset with n elements. Suppose M is an arbitrary matrix labelling of X. If A_{(X,M)} is quasi–hereditary, then

dim_{k}A_{(X,M)} = ∑_{i=1}^{n}(dim_{k}P_{I(X)}(x_{i}))^{2},

where P_{I(X)}(x_{i}) is the indecomposable projective I(X)–module corresponding to the weight x_{i}.

Proof. Let C = (c_{ij}) be the Cartan–matrix of A_{(X,M)}. Then C = (dimΔ)^{t}(dim∇) by [6], where dimΔ is an n × n matrix with the i–th row the dimension vector \underline{dim}Δ(x_{i}), and where dim∇ is defined in a similar way. Since A_{(X,M)} is a basic algebra and dim_{k}Δ(x_{i}) = dim_{k}∇(x_{i}), we have

dim_{k}A_{(X,M)} = ∑_{1≤i, j≤n} c_{ij} = ∑_{i=1}^{n} dim_{k}Δ(x_{i}) · dim_{k}∇(x_{i})

= ∑_{i=1}^{n}(dimΔ(x_{i}))^{2} = ∑_{i=1}^{n}(dim_{k}P_{I(X)}(x_{i}))^{2}.

To end this section let us mention the following conjecture.

3.11 Conjecture. Let X be a finite poset with an arbitrary matrix labelling M. Suppose that A_{(X,M)} is quasi–hereditary, then gl.dim.A_{(X,M)} = 2 gl.dim.I(X), where I(X) is the incidence algebra of X.

4. Ringel duality of A_{(X,M)}

For each quasi–hereditary algebra A, C.M.Ringel has defined in [11] another quasi–hereditary algebra R(A) which is the endomorphism algebra of the characteristic module for A. His construction R behaves like a duality on the category of basic quasi–hereditary algebras, namely R(R(A)) ≅ A. Now we call the algebra R(A) the Ringel dual of A. In this section we study the Ringel dual of A_{(X,M)} in case the Hasse diagram of X is a tree and M is a symmetric, invertible matrix labelling. In this case we show that R(A_{(X,M)}) is again of the form A_{(Y, N)} for some matrix labelling poset (Y, N).

4.1 Let A be a quasi–hereditary algebra with the weight poset ∧. Then we have the following

(1) The intersection F(Δ_{A}) ∩ F(∇_{A}) contains exactly s(A) indecomposable modules, where s(A) is the cardinality of ∧. They may be parametrized as T (λ), λ ∈ ∧ such that the following holds. There are exact sequences

\begin{matrix}(a)  0 & ⟶ Δ(λ) ⟶ T (λ) ⟶ X(λ) ⟶ 0 \\ (b)  0 & ⟶ Y (λ) ⟶ T (λ) ⟶ ∇(λ) ⟶ 0 \end{matrix}

where X(λ) is filtered by Δ(μ)′s for certain μ < λ and Y (λ) by ∇(μ)’s for certain μ < λ. In particular, T (λ) has a unique composition factor isomorphic to E(λ) and all other composition factors are of the form E(μ) with μ < λ, where E(x) denotes the simple A–module corresponding to the weight x ∈ ∧. The modules T (λ) are called canonical modules.

(2) The module T := ⊕_{λ∈∧}T (λ) which is a tilting–cotilting module is called the charac-

teristic module for (A, ∧) and R(A) = End_{A}(T) is called the Ringel dual of A, it is also quasi–hereditary, with standard modules Δ_{R(A)}(λ) = Hom_{A}(T, ∇(λ)), where the weight poset of R(A) is ∧^{op}.

(3)

Ext_{A}^{n}(Δ(λ), ∇(μ)) = \{ \begin{matrix}k if n = 0 and λ = μ , \\ 0 otherwise. \end{matrix}

For the proofs of the above facts one may see [11].

4.2 Now we consider the algebra A_{(X,M)}. From now on, we suppose that X is a tree. In this case any matrix labelling of X is symmetric. It follows from the definition of the relations of type (1.4.1) that if all the matrices are invertible, that is, they are non–zero elements from k, then we can take all these non–zero elements to be 1, since such two labellings yield two isomorphic algebras. Hence in the following we always assume that all labelling matrices are [1] ∈ k^{1×1} and simply write A_{X} for A_{(X,M)} and I(X) for I(X, M).

Proposition. Let X be a tree. Then there is an exact sequence

0 ⟶ Δ(λ) ⟶ T (λ) ⟶ ⊕_{μ<· λ}T (μ) ⟶ 0

in A_{X}–mod for all λ ∈ X.

Proof. Use induction on the number |X| of elements in X. If |X| = 1 there is nothing to prove. Suppose the statement is true for all trees X with |X| ≤ n − 1. Let X be a tree with |X| = n and λ ∈ X. If λ is not a maximal element in X then we pick up a maximal element s ∈ X such that λ < s. Set X_{1} = X\{s}, then there is an exact sequence in A_{X_{1}}–mod

0 ⟶ Δ_{A_{X_{1}}} (λ) ⟶ T_{A_{X_{1}}} (λ) ⟶ ⊕_{μ<· λ}T_{A_{X_{1}}} (μ) ⟶ 0,

where A_{X_{1}} denotes the algebra A_{(X_{1}, M_{1})} with the matrix labelling M_{1} of M restricting on X_{1} = X\{s}. Note that A_{X_{1}} is isomorphic to the factor algebra of A_{X} by the ideal A_{X}sA_{X}.

Since T (λ) ∈ F({Δ_{A_{X}} (γ)|γ ≠ s}) = F(Δ_{A_{X_{1}}} ), the above exact sequence yields a desired one in A_{X}–mod. Hence we may assume that λ is a maximal element in X. Moreover, with a similar argument as above, we may assume that λ is the largest element in X and that the proposition holds for each x < λ.

Let γ ∈ X with γ < λ. Then dim_{k}Ext_{A_{X}}^{1} (Δ(γ), Δ(λ)) = 1 by Lemma 3.5. Let μ be an element in X with μ <· λ. Then we have a non–split exact sequence

η_{λμ} 0 ⟶ Δ(λ) ⟶ T_{λμ} ⟶ Δ(μ) ⟶ 0

with T_{λμ} obviously being indecomposable.

For each ν with ν < μ, using the canonical inclusion map Δ(ν) ⟶ Δ(μ), we can form the following pullback diagram:

η_{λν}  0 ⟶ Δ(λ) ⟶ T_{λν} ⟶ Δ(ν) ⟶ 0
∥  ↓  ↓
η_{λμ}  0 ⟶ Δ(λ) ⟶ T_{λμ} ⟶ Δ(μ) ⟶ 0

We shall prove that the upper exact sequence does not split.

Since λ is the largest element in X and μ <· λ, we see from the definition of Δ(μ) that T_{λμ} is isomorphic to P (μ), the projective A_{X}–module corresponding to the weight μ, and that dim_{k}Hom_{A_{X}} (P (ν), P (μ)) = 2. In order to show that the exact sequence does not split, it suffices to prove that Hom_{A_{X}} (Δ(ν), P (μ)) ≅ k.

On one hand, there holds

1 ≤ dim_{k}Hom_{A_{X}} (Δ(ν), P (μ)) ≤ dim_{k}Hom_{A_{X}} (P (ν), P (μ)) = 2.

On the other hand, by Lemma 3.5, there holds

dim_{k}Hom_{A_{X}} (Δ(ν), P (μ)) = dim_{k}Hom_{B}(E(ν), P (μ)|_{B}),

where B denotes the Borel subalgebra of A_{X} (see section 3), where E(ν) is the simple module corresponding to the weight ν, and where P (μ)|_{B} denotes the restriction of P (μ) to B. By the construction of A_{X}, we have that νP (μ) = νA_{X}μ = k(νx_{1}x_{2} ⋯ x_{n}μ + I(X)) ⊕ k(νx_{1} ⋯ x_{n}μλμ + I(X)), where ν <· x_{1} <· x_{2} <⋯ <· x_{n} <· μ is the unique maximal chain from ν to μ. For x_{1}ν ∈ B, one has

x_{1}ν · (νx_{1}x_{2} ⋯ x_{n}μ + I(X)) = x_{1}νx_{1}x_{2} ⋯ x_{n}μ + I(X).

Since λ is the largest element in X and all labelling matrices of X are [1], one has by definition that r_{x_{1}νx_{1}} = x_{1}νx_{1} − x_{1}x_{2}x_{1} ∈ I(X). Then

x_{1}νx_{1}x_{2} ⋯ x_{n}μ + I(X) = x_{1}x_{2}x_{1}x_{2}x_{3} ⋯ x_{n}μ + I(X).

Analogously, one has that r_{x_{2}x_{1}x_{2}} = x_{2}x_{1}x_{2} − x_{2}x_{3}x_{2} ∈ I(X), then

x_{1}x_{2}x_{1}x_{2}x_{3} ⋯ x_{n}μ + I(X) = x_{1}x_{2}x_{3}x_{2}x_{3} ⋯ x_{n}μ + I(X).

Repeating the above argument, one finally gets

x_{1}ν · (νx_{1}x_{2} ⋯ x_{n}μ + I(X)) = x_{1}νx_{1}x_{2} ⋯ x_{n}μ + I(X)
=x_{1}x_{2}x_{1}x_{2}x_{3} ⋯ x_{n}μ + I(X) = · · · · · ·
=x_{1}x_{2} ⋯ x_{n−1}x_{n}μx_{n}μ + I(X) = x_{1}x_{2} ⋯ x_{n}μλμ + I(X) ≠ 0.

Therefore, νx_{1}x_{2} ⋯ x_{n}μ + I(X) does not lie in soc(P (μ)|_{B}) since x_{1}ν ∈ rad(B). From the simplicity of E(ν) it follows that ∑_{f} Im(f) ⊆ soc(P (μ)|_{B}), where the sum is over all f ∈ Hom_{B}(E(ν), P (μ)|_{B}). Then

Hom_{B}(E(ν), (P (μ)|_{B})) ⊆Hom_{B}(E(ν), soc(P (μ)|_{B})) ≅ ν · soc(P (μ)|_{B})
⊆k(νx_{1}x_{2} ⋯ x_{n}μλμ + I(X))

Thus dim_{k}Hom_{B}(E(ν), (P (μ)|_{B})) ≤ 1. As a result, Hom_{A_{X}} (Δ(ν), P (μ)) ≅ k. We then conclude that the exact sequence η_{λμ} does not split.

Further, let μ_{1}, ⋯, μ_{s} be all the elements in X satisfying μ_{i} <· λ, i = 1, ⋯, s. Applying Hom_{A_{X}} (−, Δ(λ)) to the exact sequence

0 ⟶ Δ(μ_{i}) ⟶ T (μ_{i}) ⟶ ⊕_{γ<· μ_{i}}T (γ) ⟶ 0,

one obtains a long exact sequence

0⟶Hom_{A_{X}}( ⊕_{γ<· μ_{i}}T (γ), Δ(λ))⟶Hom_{A_{X}}(T (μ_{i}), Δ(λ))⟶Hom_{A_{X}}(Δ(μ_{i}), Δ(λ))

⟶Ext_{A_{X}}^{1} ( ⊕_{γ<· μ_{i}}T (γ), Δ(λ)) ⟶Ext_{A_{X}}^{1} (T (μ_{i}), Δ(λ))⟶^{ϕ_{i}}Ext_{A_{X}}^{1} (Δ(μ_{i}), Δ(λ))

⟶Ext_{A_{X}}^{2} ( ⊕_{γ<· μ_{i}}T (γ), Δ(λ))

Since proj.dim.T (γ) ≤ 1, we have that Ext_{A_{X}}^{2} ( ⊕_{γ<· μ_{i}}T (γ), Δ(λ)) = 0, i.e. ϕ_{i} is surjective.

Thus we obtain the following commutative diagram with ξ_{λμ_{i}} a non–split exact sequence.

η_{λμ_{i}}  0 ⟶ Δ(λ) ⟶ T_{λμ_{i}} ⟶ Δ(μ_{i}) ⟶ 0
∥  ↓  ↓
ξ_{λμ_{i}}  0 ⟶ Δ(λ) ⟶ T′_{λμ_{i}} ⟶ T (μ_{i}) ⟶ 0

Now we take the sum of all ξ_{λμ_{i}} for all i = 1, ⋯, s, and then get the following exact sequence

⊕_{i}ξ_{λμ_{i}}  0 ⟶ ⊕_{i}Δ(λ) ⟶ ⊕_{i}T′_{λμ_{i}} ⟶ ⊕_{i}T (μ_{i}) ⟶ 0
↓^{σ}  ↓  ∥
ξ_{λ}  0 ⟶ Δ(λ) ⟶ T_{λ} ⟶ ⊕_{i}T (μ_{i}) ⟶ 0

Let σ : ⊕_{i}Δ(λ) ⟶ Δ(λ) be the map sending (a_{1}, ⋯, a_{s}) to ∑_{i} a_{i}. By constructing the

pushout diagram, one obtains an exact sequence ξ_{λ} ( see the above diagram ).

We shall prove that T_{λ} ∈ F(Δ) ∩ (F(∇)). Obviously, T_{λ} ∈ F(Δ). It remains to show T_{λ} ∈ F(∇), and by [11], it suffices to prove

Ext_{A_{X}}^{1} (Δ(ν), T_{λ}) = 0 for all ν ≤ λ

since proj.dim._{A_{X}} Δ(ν) = proj.dim._{B}E(ν) ≤ 1 for all ν ≤ λ.

If ν = λ, then Δ(ν) = P (λ) is projective, thus Ext_{A_{X}}^{1} (Δ(ν), T_{λ}) = 0. Now we assume that ν < λ. Apply Hom_{A_{X}} (Δ(ν), −) to ξ_{λ}, we get the long exact sequence

0 ⟶ Hom_{A_{X}} (Δ(ν), Δ(λ)) ⟶ Hom_{A_{X}} (Δ(ν), T_{λ}) ⟶ Hom_{A_{X}} (Δ(ν), ⊕_{i}T (μ_{i}))

⟶^{δ} Ext_{A_{X}}^{1} (Δ(ν), Δ(λ)) ⟶ Ext_{A_{X}}^{1} (Δ(ν), T_{λ}) ⟶ Ext_{A_{X}}^{1} (Δ(ν_{i}), ⊕_{i}T (μ_{i})).

Since T (μ_{i}) ∈ F(∇) for all 1 ≤ i ≤ s, one has Ext_{A_{X}}^{1} (Δ(ν), ⊕_{i}T (μ_{i})) = 0 by 4.2 (3). Note

that dim_{k}Ext_{A_{X}}^{1} (Δ(ν), Δ(λ)) = 1. To prove Ext_{A_{X}}^{1} (Δ(ν), T_{λ}) = 0, it is enough to prove that the connecting map δ is non–zero. Since X is a tree, there is a unique 1 ≤ j ≤ s such that ν ≤ μ_{j}. We denote by f_{j} the composition of the canonical indusion Δ(ν) ⟶ T (μ_{j}) and the canonical embedding T (μ_{j}) ⟶ ⊕_{i}T (μ_{i}). It is easy to see that (f_{j})δ = η_{λ}ν ≠ 0,

i.e. δ is non–zero. Therefore,

Ext_{A_{X}}^{1} (Δ(ν), T_{λ}) = 0 for all ν ≤ λ.

Thus

T_{λ} ∈ F(Δ) ∩ F(∇).

It remains to prove T_{λ} = T (λ), or equivalently, to prove that T_{λ} is indecomposable. This follows from the fact T_{λ} ∈ F(Δ) and the fact that all sequences ξ_{λμ_{i}} do not split. This finishes the proof.

4.3 The anti–isomorphism ε : A_{X} ⟶ A_{X} induces a duality E : A_{X} − mod ⟶ A_{X} − mod which fixes all E(λ), T (λ), λ ∈ X. Dually, by Proposition 4.2, one has an exact sequence

0 ⟶ ⊕_{μ<· λ}T (μ) ⟶ T (λ) ⟶ E(Δ(λ)) = ∇(λ) ⟶ 0

for each λ ∈ X.

Corollary. For every x ≤ y in X, there holds

dim_{k}Hom_{A_{X}} (Δ(x), T (y)) = 1 = dim_{k}Hom_{A_{X}} (T (y), ∇(x)).

Proof. Let x = x_{0} <· x_{1} <⋯ <· x_{n} = y be the unique maximal chain from x to y. We shall use induction on n to prove

(4.3.1)  dim_{k}Hom_{A_{X}} (Δ(x), T (y)) = 1.

If n = 0, i.e. x = y, this follows from 4.1(1). Suppose that for all x ≤ y in X, if the maximal chain from x to y is of length < n then (4.3.1) holds. Now assume that x < y and that x = x_{0} <· x_{1} <⋯ <· x_{n} = y is the maximal chain from x to y. Now apply Hom_{A_{X}} (Δ(x), −) to the exact sequence

0 ⟶ ⊕_{z<· y}T (z) ⟶ T (y) ⟶ ∇(y) ⟶ 0,

we obtain the following exact sequence:

0 ⟶ Hom_{A_{X}} (Δ(x), ⊕_{z<· y}T (z)) ⟶ Hom_{A_{X}} (Δ(x), T (y)) ⟶ Hom_{A_{X}} (Δ(x), ∇(y)).

Since x < y, there holds Hom_{A_{X}} (Δ(x), ∇(y)) = 0. Thus

dim_{k}Hom_{A_{X}} (Δ(x), T (y)) = dim_{k}Hom_{A_{X}} (Δ(x), ⊕_{z<· y}T (z)).

Because X is a tree, we have dim_{k}Hom_{A_{X}} (Δ(z), T (x)) = 0 for z <· y, z ≠ x_{n−1}. Thus

dim_{k}Hom_{A_{X}} (Δ(x), T (y)) = dim_{k}Hom_{A_{X}} (Δ(x), T (x_{n−1})).

The maximal chain from x to x_{n−1} is then of the length n − 1 < n. By induction hypothesis, dim_{k}Hom_{A_{X}} (Δ(x), T (x_{n−1})) = 1. Hence dim_{k}Hom_{A_{X}} (Δ(x), T (y)) = 1. Dually, dim_{k}Hom_{A_{X}} (T (y), ∇(x)) = 1.

4.4 For all μ <· λ, by i_{μλ} and π_{λμ} we denote the canonical immersion from T (μ) to T (λ) and the canonical projection from T (λ) to T (μ), respectively (see 4.2 and its dual). Further, if x = x_{0} <· x_{1} <⋯ <· x_{n} = y is the unique maximal chain from x to y, then we set i_{xy} = i_{x_{0}x_{1}}i_{x_{1}x_{2}} ⋯ i_{x_{n−1}x_{n}} and π_{yx} = π_{x_{n}x_{n−1}} ⋯ π_{x_{2}x_{1}}π_{x_{1}x_{0}}.

Consider a full additive subcategory C of A–mod and the objects N_{1}, N_{2} ∈ C, and let Irr_{C}(N_{1}, N_{2}) := rad_{C}(N_{1}, N_{2})/rad_{C}^{2}N_{1}, N_{2}) be the bimodule of irreducible maps from N_{1} to N_{2} in C.

Proposition. The two morphisms i_{μλ} and π_{λμ} for μ <· λ in X are irreducible in T := add( ⊕_{x∈X}T (x)). Moreover,

dim_{k}Irr_{T} (T (x), T (y)) = \{ \begin{matrix}1 if x <· y or y <· x, \\ 0 otherwise. \end{matrix}

Proof (a) If we have x < y in X, then each f ∈ Hom_{A_{X}} (T (x), T (y)) factors through the canonical inclusion i_{xy} : T (x) ⟶ T (y). Indeed, this follows from the fact that Hom_{A_{X}} (T (u), ∇(v)) = 0 for all u < v in X.

(b) For x ≤ y in X, if f ∈ Hom_{A_{X}} (T (x), T (y)) is not injective, then f factors through the inclusion (i_{zy})_{z}^{t} : ⊕_{z<· x}T (z) ⟶ T (y), where (i_{zy})_{z}^{t} denotes the transpose of (i_{zy})_{z} ∈ k^{1×n(x)} and where n(x) is the number of z ∈ X satisfying z <· x. Indeed, in case x = y, we have the following exact sequence

0 ⟶ ⊕_{z<· x}T (z) ⟶^{(i_{zx})_{z}^{t}} T (x) ⟶^{p_{x}} ∇(x) ⟶ 0.

Since dim_{k}Hom_{A_{X}} (T (x), ∇(x)) = 1 and f is not injective, we get fp_{x} = 0, i.e. f factors through (i_{zx})_{z}^{t} : ⊕_{z<· x}T (z) ⟶ T (x). Now suppose x < y. By (a), there is an f′ :

T (x) ⟶ T (x) such that f = f′i_{xy}. Since f is not injective, neither is f′. By the above argument, f′ factors through (i_{zx})_{z} : ⊕_{z<· x}T (z) ⟶ T (x). Therefore, f factors through

(i_{zx})_{z}^{t} i_{xy} = (i_{zy})_{z}^{t} : ⊕_{z<· x}T (z) ⟶ T (y).

(c) Now suppose that μ <· λ and that i_{μλ} admits a decomposition i_{μλ} = fg with f =

(f_{μδ_{i}})_{i} : T (μ) ⟶ ⊕^{m}_{i=1}T (δ_{i}) and g = (g_{δ_{i}λ})_{i}^{t} : ⊕^{m}_{i=1}T (δ_{i}) ⟶ T (λ) for some δ_{i} ∈ X, 1 ≤ i ≤ m,

that is, i_{μλ} = ∑_{i=1}^{m} f_{μδ_{i}}g_{δ_{i}λ}. We shall prove that f is a split monomorphism or that g is a

split epimorphism.

From i_{μλ} = ∑_{i=1}^{m} f_{μδ_{i}}g_{δ_{i}λ} and (a), we have

T (λ) ⊃ T (μ) = Im(i_{μλ}) ⊆ ∑_{i=1}^{m} Im(f_{μδ_{i}}g_{δ_{i}λ}) ⊆ T (μ).

Thus

∑_{i=1}^{m} Im(f_{μδ_{i}}g_{δ_{i}λ}) = T (μ) ⊂ T (λ).

We claim that there is a δ_{j} such that f_{μδ_{j}}g_{δ_{j}λ} is injective. Otherwise, by (b), Im(f_{μδ_{i}}g_{δ_{i}λ}) ⊆

⊕_{γ<· μ}T (γ) for all δ_{i}. Hence ∑_{i=1}^{m} Im(f_{μδ_{i}}g_{δ_{i}λ}) ⊆ ⊕_{γ<· μ}T (γ) ⊂ T (μ), a contradiction. Thus with

f_{μδ_{j}}g_{δ_{j}λ} also f_{μδ_{j}} is injective. This implies μ ≤ δ_{j} by 4.1(1). If μ = δ_{j}, then f_{μδ_{j}} is invertible, i.e. f is a split monomorphism. In case μ < δ_{j}, we shall show that δ_{j} = λ. To this purpose, it is enough to prove that g_{δ_{j}λ} is injective. Suppose that g_{δ_{j}λ} is not injective. Using the dual argument of (b) and 4.1(1) we have that T (δ_{j}) ⊃ Δ(δ_{j}) ⊆ Ker(g_{δ_{j}λ}). Thus (Δ(μ))f_{μδ_{j}} ⊆ Δ(δ_{j}) ⊆ Ker(g_{δ_{j}λ}) since dim_{k}Hom_{A_{X}} (Δ(μ), T (δ_{j})) = 1 by Corollary 4.3. Hence Δ(μ) ⊆ Ker(f_{μδ_{j}}g_{δ_{j}λ}) and f_{μδ_{j}}g_{δ_{j}λ} is not injective, which is a contradiction. Therefore, g_{δ_{j}λ} is injective. This implies δ_{j} ≤ λ. Since μ <· λ and μ < δ_{j}, we must have δ_{j} = λ. Then g_{δ_{j}λ} is invertible, that is, g is a split epimorphism. Hence i_{μλ} is irreducible. Dually, π_{λμ} is irreducible for μ <· λ in X. The rest of the proposition becomes now obvious.

4.5 Proposition. (1) If μ <· λ and γ_{1}, ⋯, γ_{s} are all the elements in X satisfying γ_{i} <· μ, s ≥ 1, then

i_{μλ}π_{λμ} = ∑_{i=1}^{s} π_{μγ_{i}}i_{γ_{i}μ},

(2) If μ_{1} <· λ, μ_{2} <· λ and μ_{1} ≠ μ_{2}, then i_{μ_{1}λ}π_{λμ_{2}} = 0,

(3) If μ is minimal in X and μ <· λ, then i_{μλ}π_{λμ} = 0.

Proof. (1) The morphism π_{λμ} is the composition of the following morphisms

T (λ) ⟶^{π_{1}} T (λ)/Δ(λ) = ⊕_{δ<· λ}T (δ) ⟶^{π_{2}} T (μ),

where π_{1} is the canonical projection and π_{2} maps the direct summand T (μ) of ⊕_{δ<· λ}T (δ)

identically onto T (μ) and other summands to zero.

It is easy to see that Ker(i_{μλ}π_{1}) = T (μ) ∩ Δ(λ) = Δ(μ). Further, Hom_{A_{X}} (T (μ), T (δ)) = 0 for all δ <· λ, δ ≠ μ. Therefore, i_{μλ}π_{λμ} is the composition of the morphisms

T (μ) ⟶^{[π_{μγ_{1}} ,··· ,π_{μγs}]} T (μ)/Δ(μ) = ⊕_{i=1}^{s}T (γ_{i}) \underrightarrow{[ ⋮_{i_{γsμ}}^{i_{γ_{1}μ}} ]} T (μ),

that is, i_{μλ}π_{λμ} = ∑_{i=1}^{s} π_{μγ_{i}}i_{γ_{i}μ}.

(2) Since X is a tree, T (μ_{1}) and T (μ_{2}) admit no common composition factor by 4.1(1). Thus Hom_{A_{X}} (T (μ_{1}), T (μ_{2})) = 0 and i_{μ_{1}λ}π_{λμ_{2}} = 0.

(3) follows from the fact that T (μ) is simple for each minimal element μ in X.

4.6 We denote by X^{op} the opposite poset of X. If x ≤ y in X, we write y ≤^{op} x in X^{op}. We equip X^{op} with the matrix labelling M consisting only of [1] ∈ k^{1×1}. We then simply write A_{X^{op}} for A_{(X^{op}, M)} and I(X^{op}) for I(X^{op}, M).

As in 4.2, T = ⊕_{x∈X}T (x) denotes the characteristic module of A_{X}. For each x ∈ X,

we denote by e_{x} the composition of the canonical projection from T onto T (x) and the canonical immersion from T (x) into T. Then e_{x}, x ∈ X, form a complete set of orthogonal primitive idempotents of R(A_{X}) = End_{A_{X}} (T).

The correspondence x ⟼ e_{x}, xy ⟼ e_{x}i_{xy}e_{y}, yx ⟼ e_{y}π_{yx}e_{x}, for x <· y in X, induces an algebra homomorphism

Ψ′ : A′_{X^{op}} ⟶ R(A_{X}) = End_{A_{X}} (T)

which is clearly surjective by Corollary 4.4.

By Proposition 4.5, I(X^{op}) ⊆ Ker(Ψ′), thus Ψ′ gives rise to a surjective algebra homomorphism

Ψ : A_{X^{op}} ⟶ R(A_{X})

We shall prove that Ψ is an isomorphism by comparing the dimensions of A_{X^{op}} and R(A_{X}).

Since all the labelling matrices of X are symmetric, A_{X} is a BGG–algebra. This implies that R(A_{X}) is also a BGG–algebra. Similarly, A_{X^{op}} is a BGG–algebra,too. From 3.10 (see also [7, Lemma 1.4]), we have

dim_{k}A_{X^{op}} = ∑_{x∈X}(dim_{k}Δ_{A_{X^{op}}} (x))^{2}

and

dim_{k}R(A_{X}) = ∑_{x∈X}(dim_{k}Δ_{R(A_{X})}(x))^{2}.

On one hand, by the definition of A_{X^{op}}, there holds dim_{k}Δ_{A_{X^{op}}} (x) = n_{x}, the number of the elements y in X^{op} satisfying y ≤^{op} x, or equivalently, x ≤ y.

On the other hand, there holds Δ_{R(A_{X})}(x) = Hom_{A_{X}} (T, ∇(x)) by 4.1(2). Then it follows from 4.1(a) and Corollary 4.3 that

dim_{k}Δ_{R(A_{X})}(x) = ∑_{y∈X} dim_{k}Hom_{A_{X}} (T (y), ∇(x)) = n_{x} = dim_{k}Δ_{A_{X^{op}}} (x).

Therefore, dim_{k}A_{X^{op}} = dim_{k}R(A_{X}), i.e. Ψ is an isomorphism.

In fact, we have proved the following

Theorem. Let X be a tree labelled by matrices [1] ∈ k^{1×1}. Then

R(A_{X}) ≅ A_{X^{op}}.

4.7 Remark. If the matrix labelling on a tree X is not invertible, then Theorem 4.6 may be false. Let us consider the poset X = {1 < 2 < 3} with M(1, 3) = [0]. Then the Ringel dual R(A_{(X,M)}) is given by the quiver

1 ⟶^{β} 3 ⟶^{δ} 2

with the relation βα = γδ = βδγα = 0. Since X ≅ X^{op} as posets, A_{(X,M)} ≅ A_{(X^{op}, M)}. Clearly, R(A_{(X,M)}) ≇ A_{(X^{op}, M)}.

A further discussion of the Ringel dual in case all labellings are zero will appear elsewhere.

5. A relation between the Ringel duality and the quadratic duality

This section is devoted to the study of a relation between the Ringel duality and the quadratic duality of A_{(X,M)} in case X is a tree with an invertible matrix labelling. In this case we shall show that the processes of forming the dualities commute. To this goal, we first study the Ringel dual of B_{(X,M)}. Note that the algebra B_{(X,M)} is different from A_{(X,M)} and, comparing with A_{(X,M)}, may have arbitrary finite global dimension.

The main idea is similar to the one in section 4. For convenience of the reader, we give here proofs but omit some same arguments and refer to section 4.

5.1 From now on, we suppose again that X is a tree with a symmetric, invertible matrix labelling M. The algebra B_{(X, M)} is defined as in section 2. Using a similar argument in 4.2, we may assume that all labelling matrices in M are [1] ∈ k^{1×1}. We have seen in section 2 that B_{(X,M)} is a quadratic, quasi–hereditary algebra and that the anti–involution ε of A′_{X} induces an anti–involution ε′ of B_{(X,M)}. Thus B_{(X,M)} is even a BGG–algebra.

Further, by C′ (resp. B′) we denote the subalgebra of B_{(X,M)} generated by x ∈ X and xy (resp. yx) with x <· y in X. Notice that C′ and B′ are algebras with radical–square zero. As in section 3 one can show the following

Theorem. The algebra B′ is a strong exact Borel subalgebra of B_{(X,M)} and the algebra C′ a strong Δ–subalgebra of B_{(X,M)}.

5.2 For simplicity, we write B for B_{(X,M)} and J for J(X, M). For each λ ∈ X, by T_{B}(λ) we denote the canonical module of B corresponding to the weight λ and by _{B}T = ⊕_{λ∈X}T_{B}(λ)

the characteristic module of B. Then we have

Proposition. For each λ ∈ X, there is an exact sequence

0 ⟶ Δ_{B}(λ) ⟶ T_{B}(λ) ⟶ ⊕_{μ<· λ}Δ_{B}(μ) ⟶ 0.

Dually, there is an exact sequence

0 ⟶ ⊕_{μ<· λ}∇_{B}(μ) ⟶ T_{B}(λ) ⟶ ∇_{B}(λ) ⟶ 0.

Proof. First note that by the construction of B and the assumption on X, one can see that rad^{3}(B) = 0 and that there is an exact sequence

0 ⟶ ⊕_{λ<· γ}Δ_{B}(γ) ⟶ P_{B}(λ) ⟶ Δ_{B}(λ) ⟶ 0

for each λ ∈ X, where P_{B}(λ) is the projective B–module corresponding to the weight λ.

Now we use induction on the number |X| of the elements of X to prove that there is an exact sequence

0 ⟶ Δ_{B}(λ) ⟶ T_{B}(λ) ⟶ ⊕_{μ<· λ}Δ_{B}(μ) ⟶ 0.

for each λ ∈ X.

If |X| = 1, the proposition is clear. Suppose |X| = n > 1, and λ ∈ X. Using a similar argument in the proof of Proposition 4.2, we may assume that λ is the largest element in X.

For each μ <· λ in X, one has by Lemma 3.5 the equalities

dim_{k}Hom_{B}(Δ_{B}(μ), Δ_{B}(λ)) = dim_{k}Hom_{B′}(E(μ), Δ_{B}(λ)|_{B′}) = 1

and

dim_{k}Ext_{B}^{1}(Δ_{B}(μ), Δ_{B}(λ)) = dim_{k}Ext_{B′}^{1}(E(μ), Δ_{B}(λ)|_{B′}) = 1.

Hence there is a non–split exact sequence

η_{λμ} 0 ⟶ Δ_{B}(λ) ⟶ T_{λμ} ⟶ Δ_{B}(μ) ⟶ 0.

Using a similar argument in the proof of Proposition 4.2, one has that T_{λμ} = P_{B}(μ) and that for each ν <· μ, we can choose a non–zero morphism f_{νμ} : Δ_{B}(ν) → Δ_{B}(μ) which gives rise to the following commutative diagram

η_{λν}  0 ⟶ Δ_{B}(λ) ⟶ T_{νμ} ⟶ Δ_{B}(ν) ⟶ 0
∥  ↓  ↓^{f_{νμ}}
η_{λμ}  0 ⟶ Δ_{B}(λ) ⟶ P_{B}(μ) ⟶ Δ_{B}(μ) ⟶ 0

with η_{λν} a non-split exact sequence (Note that in this situation one should show that Hom_{B}(Δ_{B}(ν), P_{B}(μ)) = 0. However, this follows from the structure of P_{B}(μ) (see the defnition of B)).

By forming pushout diagram, we obtain the following commutative diagram:

⊕_{μ<· λ}η_{λμ}  0 ⟶ ⊕_{μ<· λ}Δ_{B}(λ) ⟶ ⊕_{μ<· λ}T_{λμ} ⟶ ⊕_{μ<· λ}Δ_{B}(μ) ⟶ 0
↓^{σ}  ↓  ∥
η_{λ}  0 ⟶ Δ_{B}(λ) ⟶ T_{λ} ⟶ ⊕_{μ<· λ}Δ_{B}(μ) ⟶ 0

where σ denotes the map defined by (a_{μ})_{μ} ⟼ ∑_{μ} a_{μ}.

We shall prove that T_{λ} = T_{B}(λ). This will follow from the facts (1) and (2) below.

(1) The module T_{λ} lies in F(Δ_{B}) ∩ F(∇_{B}). Indeed, T_{λ} ∈ F(Δ_{B}) follows from the above construction. To prove T_{λ} ∈ F(∇_{B}) we show that

Ext_{B}^{i}(Δ_{B}(ν), T_{λ}) = 0

for all i ≥ 1 and for all ν in X.

We use induction on i. In the following we prove that Ext_{B}^{1}(Δ_{B}(ν), T_{λ}) = 0 for all ν ≤ λ in X.

If ν = λ, then Δ_{B}(ν) = P_{B}(λ). Thus Ext_{B}^{1}(Δ_{B}(ν), T_{λ}) = 0. If ν < λ, we apply Hom_{B}(Δ_{B}(ν), −) to the exact sequence η_{λ} and obtain the following long exact sequence (5.2.1)

0 ⟶ Hom_{B}(Δ_{B}(ν), Δ_{B}(λ)) ⟶ Hom_{B}(Δ_{B}(ν), T_{λ}) ⟶ Hom_{B}(Δ_{B}(ν), ⊕_{μ<· λ}Δ_{B}(μ))

⟶^{δ_{ν}} Ext_{B}^{1}(Δ_{B}(ν), Δ_{B}(λ)) ⟶ Ext_{B}^{1}(Δ_{B}(ν), T_{λ}) ⟶ Ext_{B}^{1}(Δ_{B}(ν), ⊕_{μ<· λ}Δ_{B}(μ)).

Now we consider the following cases:

(i) Case ν <· λ. By Lemma 3.5, we have

Ext_{B}^{1}(Δ_{B}(ν), Δ_{B}(λ)) ≅ Ext_{B′}^{1}(E_{B}(ν), Δ_{B}(λ)|_{B′}) ≅ k

and

Ext_{B}^{1}(Δ_{B}(ν), ⊕_{μ<· λ}Δ_{B}(μ)) ≅ Ext_{B′}^{1}(E_{B}(ν), ⊕_{μ<· λ}Δ_{B}(μ)|_{B′}) = 0.

where E_{B}(ν) denotes the simple B–module corresponding to the weight ν.

On the other hand, by f_{ν} we denote the canonical embedding from Δ_{B}(ν) to ⊕_{μ<· λ}Δ_{B}(μ).

Then (f_{ν})δ_{ν} = η_{λν} ≠ 0. Therefore, δ_{ν} is surjective, i.e. Ext_{B}^{1}(Δ_{B}(ν), T_{λ}) = 0. Moreover, from (5.2.1) we have that

Hom_{B}(Δ_{B}(ν), T_{λ}) ≅ k

for all ν <· λ.

(ii) Case ν <· μ_{0} <· λ for some μ_{0}. Again by Lemma 3.5, we have

Hom_{B}(Δ_{B}(ν), Δ_{B}(λ)) = 0 and Hom_{B}(Δ_{B}(ν), ⊕_{μ<· λ}Δ_{B}(λ)) ≅ k.

In this case δ_{ν} is non–zero since (f_{νμ_{0}})δ_{ν} = η_{λν} ≠ 0. This implies that

Hom_{B}(Δ_{B}(ν), T_{λ}) = 0.

Applying Hom_{B}(−, T_{λ}) to the exact sequence

0 ⟶ Δ_{B}(μ_{0}) ⟶ P_{B}(ν) ⟶ Δ_{B}(ν) ⟶ 0,

we get the following exact sequence

(5.2.2)

0 ⟶ Hom_{B}(Δ_{B}(ν), T_{λ}) ⟶ Hom_{B}(P_{B}(ν), T_{λ}) ⟶ Hom_{B}(Δ_{B}(μ_{0}), T_{λ})
⟶ Ext_{B}^{1}(Δ_{B}(ν), T_{λ}) ⟶ 0.

Obviously, Hom_{B}(P_{B}(ν), T_{λ}) ≅ k. By caculating the dimensions of each term in (5.2.2), one can easily see that Ext_{B}^{1}(Δ_{B}(ν), T_{λ}) = 0.

(iii) Case ν <· δ <· μ_{0} <· λ for some μ_{0} and δ in X. Applying Hom_{B}(−, T_{λ}) to the exact sequence (here the assumption that λ is the largest element is used)

0 ⟶ Δ_{B}(δ) ⟶ P_{B}(ν) ⟶ Δ_{B}(ν) ⟶ 0,

one has the exact sequence

0 ⟶ Hom_{B}(Δ_{B}(ν), T_{λ}) ⟶ Hom_{B}(P_{B}(ν), T_{λ}) ⟶ Hom_{B}(Δ_{B}(δ), T_{λ})
⟶ Ext_{B}^{1}(Δ_{B}(ν), T_{λ}) ⟶ 0.

In the second case we have seen that Hom_{B}(Δ_{B}(δ), T_{λ}) = 0. Therefore,

Ext_{B}^{1}(Δ_{B}(ν), T_{λ}) = 0.

In all remaining cases, the equality Ext_{B}^{1}(Δ_{B}(ν), T_{λ}) = 0 follows directly from the equality Ext_{B}^{1}(Δ_{B}(ν), Δ_{B}(λ)) = 0 = Ext_{B}^{1}(Δ_{B}(ν), ⊕_{μ<· λ}Δ_{B}(μ)), according to Lemma 3.5.

Thus we have proved that Ext_{B}^{i}(Δ_{B}(ν), T_{λ}) = 0 for all ν in X in case i = 1. Now suppose that Ext_{B}^{i}(Δ_{B}(ν), T_{λ}) = 0 for all ν in X. Using the exact sequence at the beginning of our proof, we then have

Ext_{B}^{i+1}(Δ_{B}(ν), T_{λ}) = Ext_{B}^{i}( ⊕_{ν<· δ}Δ_{B}(δ), T_{λ}) = 0.

Therefore, T_{λ} ∈ F(∇_{B}).

(2) The module T_{λ} is indecomposable. Indeed, this is an implication of the fact that T_{λ} lies in F(Δ_{B}) and the fact that each sequence η_{λμ} does not split for every μ <· λ.

This finishes the proof.

5.3 Corollary. (1) For μ <· λ in X, there holds

dim_{k}Hom_{B}(Δ_{B}(μ), T_{B}(λ)) = 1 = dim_{k}Hom_{B}(T_{B}(λ), Δ_{B}(μ)).

Dually,

dim_{k}Hom_{B}(∇_{B}(μ), T_{B}(λ)) = 1 = dim_{k}Hom_{B}(T_{B}(λ), ∇_{B}(μ)).

(2) For x ≠ y in X, there holds Hom_{B}(T_{B}(x), T_{B}(y)) ≠ 0 if and only if one of the following cases occurs:

i) x <· y,

ii) y <· x,

iii) There is an element z ∈ X such that z <· x and z <· y. Moreover, in all three cases, we have that dim_{k}Hom_{B}(T_{B}(x), T_{B}(y)) = 1.

Proof. This follows directly from Proposition 5.2.

5.4 For λ ∈ X, by i_{λ} we denote the canonical inclusion from Δ_{B}(λ) to T_{B}(λ) and by π_{λ} the canonical projection from T_{B}(λ) to ∇_{B}(λ). Further, for μ <· λ in X, we denote by i_{μλ} the canonical inclusion from ∇_{B}(μ) to T_{B}(λ) and by π_{λμ} the canonical projection from T_{B}(λ) to Δ_{B}(μ). Finally, for μ <· λ in X, we set ϕ_{μλ} = π_{μ}i_{μλ} and ψ_{λμ} = π_{λμ}i_{μ}.

Proposition. For μ <· λ in X, the morphisms ϕ_{μλ} and ψ_{λμ} are irreducible in T_{B} := add( ⊕_{x∈X}T_{B}(x)). Moreover,

dim_{k}Irr_{T_{B}}(T_{B}(x), T_{B}(y)) = \{ \begin{matrix}1 if x <· y or y <· x, \\ 0 otherwise. \end{matrix}

Proof. We only prove that ϕ_{μλ} is irreducible. The rest can be proved in a similar way or is obvious. Let ϕ_{μλ} admit a decomposition ϕ_{μλ} = gh with g = (g_{μs_{i}})_{i} : T_{B}(μ) ⟶

⊕^{m}_{i=1}T_{B}(s_{i}) and h = (h_{s_{i}λ})_{i}^{t} : ⊕^{m}_{i=1}T_{B}(s_{i}) ⟶ T_{B}(λ) for some s_{i} in X, 1 ≤ i ≤ m, i.e. ϕ_{μλ} =

∑_{i=1}^{m} g_{μs_{i}}h_{s_{i}λ}. By Corollary 5.3, one has that Hom_{B}(T_{B}(μ), T_{B}(λ)) = kϕ_{μλ}. Therefore, there is an s_{j} for 1 ≤ j ≤ m such that g_{μs_{j}}h_{s_{j}λ} = aϕ_{μλ} for an a ∈ k\{0}. Then g_{μs_{j}} ≠ 0 ≠ h_{s_{j}λ}. Again by Corollary 5.3, the inequality g_{μs_{j}} ≠ 0 then implies that s_{j} <· μ or μ <· s_{j} or that there is an x ∈ X such that x <· μ and x <· s_{j}, and the inequality h_{s_{j}λ} ≠ 0 implies that s_{j} <· λ or λ <· s_{j} or that there is an element y ∈ X such that y <· s_{j} and y <· λ. By checking each case, one can see that the cases s_{j} = μ and s_{j} = λ are the only possible cases such that the equality g_{μs_{j}}h_{s_{j}λ} = aϕ_{μλ} holds. In case s_{j} = μ, then h_{s_{j}λ} = h_{μλ} = bϕ_{μλ} for some b ∈ k\{0}. Thus g_{μs_{j}} is invertible since End(T_{B}(μ)) is a local ring. Similarly, in case s_{j} = λ, one has that h_{s_{j}λ} is invertible. Hence either g is a split monomorphism or h is a split epimorphism, i.e. ϕ_{μλ} is irreducible.

5.5 Lemma. For each λ ∈ X, there hold soc(T_{B}(λ)) = soc(Δ_{B}(λ)) and top(T_{B}(λ)) = top(∇_{B}(λ)).

Proof. If λ is a minimal element in X, then Δ_{B}(λ) = E_{B}(λ) = ∇_{B}(λ) and T_{B}(λ) = E_{B}(λ). Now suppose λ is not a minimal element. By Proposition 5.2, we have exact sequences

η_{λ} 0 ⟶ Δ_{B}(λ) ⟶^{i_{λ}} T_{B}(λ) ⟶^{(π_{λμ})_{μ}} ⊕_{μ<· λ}Δ_{B}(μ) ⟶ 0.

and

η′_{λ} 0 ⟶ ⊕_{μ<· λ}∇_{B}(μ) ⟶^{(i_{μλ})_{μ}^{t}} T_{B}(λ) ⟶^{π_{λ}} ∇_{B}(λ) ⟶ 0.

The morphism π_{λ} induces a morphism π′_{λ} : soc(T_{B}(λ)) → soc(∇_{B}(λ)) = E_{B}(λ). Since T_{B}(λ) has a unique composition factor isomorphic to E_{B}(λ) and contains a submodule Δ_{B}(λ), the morphism π′_{λ} is zero. Thus soc(T_{B}(λ)) is contained in ⊕_{μ<· λ}∇_{B}(μ). Therefore,

soc(T_{B}(λ)) = soc( ⊕_{μ<· λ}∇_{B}(μ)) = ⊕_{μ<· λ}E_{B}(μ) = soc(Δ_{B}(λ)).

Dually, top(∇_{B}(λ)) = top(T_{B}(λ)).

5.6 Corollary. (1) For μ <· λ, there holds

ϕ_{μλ}ψ_{λμ} = ∑_{γ<· μ} ψ_{μγ}ϕ_{γμ}

(2) For γ <· μ <· λ, there hold ϕ_{γμ}ϕ_{μλ} = 0 and ψ_{λμ}ψ_{μγ} = 0.

Proof. (1) By definition, ϕ_{μλ}ψ_{λμ} is the composition of the following morphisms

T_{B}(μ) ⟶^{π_{μ}} ∇_{B}(μ) ⟶^{i_{μλ}} T_{B}(λ) ⟶^{π_{λμ}} Δ_{B}(μ) ⟶^{i_{μ}} T_{B}(μ)

and ∑_{γ<· μ} ψ_{μγ}ϕ_{γμ} is the composition of the morphisms

T_{B}(μ) ⟶^{(π_{μγ})_{γ}} ⊕_{γ<· μ}Δ_{B}(γ) ⟶^{⊕_{γ}i_{γ}} ⊕_{γ<· μ}T_{B}(γ) ⟶^{⊕_{γ}π_{γ}} ⊕_{γ<· μ}∇_{B}(γ) ⟶^{(i_{γμ})_{γ}^{t}} T_{B}(μ).

On one hand, we consider the composition of the morphisms j_{λ} := (i_{νλ})_{ν}^{t} and p_{λ} := (π_{λν})_{ν}

⊕_{ν<· λ}∇_{B}(ν) ⟶^{j_{λ}} T_{B}(λ) ⟶^{p_{λ}} ⊕_{ν<· λ}Δ_{B}(ν).

Since soc(Δ_{B}(λ)) = ⊕_{ν<· λ}E_{B}(ν) = soc( ⊕_{ν<· λ}∇_{B}(ν)), we have that

Ker(j_{λ}p_{λ}) = Δ_{B}(λ) ∩ ( ⊕_{ν<· λ}∇_{B}(ν)) = ⊕_{ν<· λ}E_{B}(ν).

Thus Ker(i_{μλ}π_{λμ}) = ∇_{B}(μ) ∩ Ker(j_{λ}p_{λ}) = E_{B}(μ), i.e. the morphism i_{μλ}π_{μλ} is the composition of the following canonical projection and inclusion

∇_{B}(μ) ⟶ top(∇_{B}(μ)) = ⊕_{γ<· μ}E_{B}(γ) = soc(Δ_{B}(μ)) ⟶ Δ_{B}(μ).

On the other hand, for each γ <· μ, the morphism i_{γ}π_{γ} is the composition of the following canonical projection and inclusion

Δ_{B}(γ) ⟶ top(Δ_{B}(γ)) = E_{B}(γ) = soc(∇_{B}(γ)) ⟶ ∇_{B}(γ).

As a result, we have that

ϕ_{μλ}ψ_{λμ} = ∑_{γ<· μ} ψ_{μγ}ϕ_{γμ}

(2) This follows from Corollary 5.3.

5.7 Using a similar discussion in 4.6, we have the following

Theorem. Let X be a tree labelled by matrices [1] ∈ k^{1×1}. Then

R(B_{X}) ≅ B_{X^{op}},

where the opposite poset X^{op} of X is also labelled by matrices [1] ∈ k^{1×1}.

5.8 As a consequence of 2.9, 4.6 and 5.7 we have the following relation between the quadratic dual and the Ringel dual.

Theorem. Let X be a tree with an arbitrary invertible matrix labelling. Then

R(A_{(X,M)}^{!}) ≅ (R(A_{(X,M)}))^{!},

that is, the Ringel dual and the quadratic dual of A_{(X,M)} are commutative.

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Received December 16, 1993; revised version August 25, 1994.