J.A. de la Peña

On the representation type of one point extensions of tame concealed algebras

Letk be an algebraically closed field and Λ a finite dimensionalk-algebra. LetqΛ be the quadratic Tits form associated with Λ. If Λ is tame we show thatqΛ is weakly semipositive. Let Λ be a one-point extension of a tame concealed algebra, then Λ is tame iffqΛ is weakly semipositive.

The connectivity index of a weighted graph

Abstract Let G be a simple graph and consider the m-connectivity index m X(G) = ∑ i 1 − i n hellip; i m+1 ,1 d i 1 d i m … d i m+1 , where i 1 − i 2 −⇝−i m+1 runs over all paths of length m in G and di denotes the degree of the vertex i. We find upper bounds for mξ(G) using the eigenvalues of the Laplacian matrix of an associated weighted graph. The method provides also lower bounds for 1ξ(G).

Comparing the simplicial and the hochschild cohomologies of a finite dimensional algebra

Abstract Let A be a finite dimensional, basic algebra over an algebraically closed field k. We compare the Hochschild cohomology group Hi(A) = Hi(A, A) with the simplicial cohomology groups SHi(A, Z) defined in [6] for any abelian group Z. We show that SH1 (A,k+) is always a subspace of H1(A) and in case A is schurian SHi(A, k+) is a subspace of Hi(A) for i ≥ 1. We show several consequences of these results.

On the corank of the Tits form of a tame algebra

Let Λ = k[Q]I be a finite-dimensional, directed k-algebra with k an algebraically closed field. Let qΛ be the Tits (quadratic) form of Λ. The isotropic corank of qΛ denoted by corank0 qgL, is the maximal dimension of a convex half-space over Q contained in Σ0(qΛ = {0 ≤ ν ϵ Qn: qΛ(ν) = 0}, where n is the number of vertices of Q. We show that a strongly simply connected cycle-finite algebra Λ, has corank0qΛ ≤ 2. A strongly simply connected algebra Λ is tame domestic if and only if qgL is weakly non-negative and corank0 qΛ ≤ 1. We also characterize polynomial growth algebras using invariants associated with the Tits form.

ON THE NUMBER OF TERMS IN THE MIDDLE OF ALMOST SPLIT SEQUENCES OVER TAME ALGEBRAS

Let A be a finite dimensional tame algebra over an algebraically closed field k. It has been conjectured that any almost split sequence 0 → X → ⊕i=1Yi → Z → 0 with Yi indecomposable modules has n ≤ 5 and in case n = 5, then exactly one of the Yi is a projective-injective module. In this work we show this conjecture in case all the Yi are directing modules, that is, there are no cycles of non-zero, non-iso maps Yi = M1 → M2 → · · · → Ms = Yi between indecomposable A-modules. In case, Y1 and Y2 are isomorphic, we show that n ≤ 3 and give precise information on the structure of A. Let A be a finite dimensional algebra over an algebraically closed field k. We denote by modA the category of finite dimensional left A-modules (an object in modA is simply called a module). For a non-projective indecomposable module X , there exist an indecomposable non-injective module τ A X called the Auslander-Reiten translate and an almost split sequence 0 → τAX → E → X → 0 (see [2], [18]). Since their introduction, almost split sequences have played a central role in the representation theory of algebras (see for example [2]). For an almost split sequence 0 → τ A X → E → X → 0, consider the indecomposable decomposition E = s(X) ⊕ i=1 Yi. There has been considerable attention paid to the relation between properties of the algebra A and the values s(X) for different modules X (and of course between properties of X and the value s(X)). Among other interesting results we recall that if A is representation finite, then s(X) ≤ 4, for every indecomposable non-projective module X [3], [7] (see also [10] and [12]). It has been conjectured by S. Brenner that, for A a tame algebra, s(X) ≤ 5 for every indecomposable non-projective module X . This is known to hold for many examples, in particular for the important case of hereditary tame algebras. To state the main results of this work, we recall some concepts. A cycle in modA is a sequence X0 f1 −→ X1 f2 −→ · · · fs −→ Xs = X0 of non-zero nonisomorphism maps between indecomposable modules; the cycle is said to be finite if fi / ∈ radA (Xi−1, Xi), for all i = 1, . . . , s. An indecomposable module X is said to be directing if it does not belong to a cycle in modA. The algebra A is said to be cycle-finite if all cycles in modA are finite. We recall that a cycle finite algebra is tame [1]. The Coxeter matrix φA of A and its spectral radius ρ(φA) = max {‖λ‖ : λ eigenvalue of φA} are important invariants (see for example [5], [14], [17], [20]). In case Received by the editors August 22, 1996 and, in revised form, April 25, 1997. 1991 Mathematics Subject Classification. Primary 16G60, 16G70. c ©1999 American Mathematical Society 3857 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3858 J. A. DE LA PEÑA AND M. TAKANE A is a triangular algebra, we denote by ρ̃(A) = max {ρ(φB) : B = A/AeA for some idempotent e ∈ A}. Theorem 1. Assume that A is a triangular algebra such that ρ̃(A) ≤ t− 2 + √ t2 − 4t 2 for some natural number t ≥ 4. Let 0 → τ A X → s(X) ⊕ i=1 Yi → X → 0 be an almost split sequence in modA such that Yi is directing (i = 1, . . . , s(X)). Then s(X) ≤ t + 1. Theorem 2. Assume A is a tame algebra and let 0 → τAX → s(X) ⊕ i=1 Yi → X → 0 be an almost split sequence in modA such that Yi is directing (i = 1, . . . , s(X)). Then the following holds: (a) s(X) ≤ 5. Moreover, if s(X) = 5, then for some j ∈ {1, . . . , s(X)}, the module Yj is projective and injective. (b) If s(X) ≥ 3, and Y1 ∼= Y2, then s(X) = 3 and the module Y3 is projective and injective. Theorem 3. Assume A is a cycle-finite algebra. Then for any almost split sequence 0 → τ A X → s(X) ⊕ i=1 Yi → X → 0 we have s(X) ≤ 5. If s(X) = 5, there is some Yj (1 ≤ j ≤ s) which is projective and injective. We prove the theorems in section 2 after some preliminary considerations. We gratefully acknowledge support of CONACYT and DGAPA, UNAM. 1. Cycles and almost split sequences 1.1. Let H = k∆ be the path algebra of a quiver ∆ without oriented cycles (see [6]). A tilting module T in modH satisfies: ExtH(T, T ) = 0 and there is an exact sequence 0 → H → T ′ → T ′′ → 0 with T ′, T ′′ ∈ add T (see [18]). For a tilting module HT , the algebra B = EndH(T ) is called a tilted algebra of type ∆. We recall that an indecomposable B-module X is sincere if HomB(P, X) 6= 0 for every projective B-module P . If X is a sincere directing indecomposable B-module, then B is a tilted algebra [18, p. 375]. 1.2. The Auslander-Reiten quiver ΓA of A has as vertices representatives of the iso-classes of indecomposable modules; there are as many arrows X → Y in ΓA as dimk radA(X, Y )/ radA(X, Y ). A path Y0 → Y1 → · · · → Ys in ΓA is said to be sectional if τ A Yi+1 6= Yi−1 for all i = 1, . . . , s − 1. In case Y0 = Ys, we have a sectional cycle if τAYi+1 6= Yi−1 for all i = 1, . . . , s mod s. A sectional path in ΓA contains no sectional cycle [4]. A component C of ΓA is directing if it is formed by directing modules. 1.3. Let η : 0 → τ A X → s(X) ⊕ i=1 Yi → X → 0 be an almost split sequence in modA and let B(η) be a quotient A/AeA, with e idempotent in A, of minimal dimension such that η is formed by B(η)-modules. By [18], one of the modules τ A X , X or Yi (1 ≤ i ≤ s(X)) is sincere as B(η)-module. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE MIDDLE OF AN ALMOST SPLIT SEQUENCE 3859 Lemma. Let η : 0 → τ A X → s(X) ⊕ i=1 Yi → X → 0 be an almost split sequence in modA such that Yi is directing as B(η)-module, for i = 1, . . . , s(X). Then (a) X and τ A X are directing as B(η)-modules; (b) B(η) is a tilted algebra. Proof. Consider the almost split sequence η : 0 → τ A X (hi) −−→ s(X) ⊕ i=1 Yi (gi) −−→ X → 0. Let X = Z0 f1 −→ Z1 −→ · · · −→ Zs−1 fs −→ Zs = X be a cycle in modB(η). The map fs : Zs−1 → Zs = X factorizes through the sink map (gi)i : s(X) ⊕ i=1 Yi → X and for some j ∈ {1, . . . , s(X)} we get a non-zero map f ′ s : Zs−1 → Yj . Therefore we get a cycle Yj gj −→ X = Z0 f1 −→ Z1 −→ · · · −→ Zs−1 f ′ s −→ Yj . A contradiction. Hence X (and similarly τ A X) is directing. The algebra B(η) is tilted by (1.1). 1.4 Proposition. Let X be a directing indecomposable non-projective module and η : 0 → τ A X → s(X) ⊕ i=1 Yi → X → 0 be the corresponding almost split sequence. Assume s(X) ≥ 3. Then one of the following conditions holds: (a) B(η) is a tilted algebra and X belongs to a directing component of ΓB(η); (b) there exists a sectional path τAX = Z0 → Z1 → · · · → Zs in ΓB(η) with Zs an injective module. Proof. We shall denote B = B(η). Assume first that Yi is directing as B-module for i = 1, . . . , s(X). By (1.3), B is a tilted algebra. By [9], we know the structure of ΓB: there is a postprojective, a preinjective and a connecting component (some of these components may coincide) and components of type ZA∞ or ZA∞/(n), possibly with inserted ray modules or coinserted coray modules. Since s(X) ≥ 3, then X belongs to a postprojective, preinjective or connecting component of ΓB, all of which are directing. Hence (a) holds. Assume Y1 is not directing. Let Y1 = Z0 f1 −→ Z1 −→ · · · fs −→ Zs = Y1 be a cycle in modB. By (1.2), we may assume that one of the following situations occurs: (1) Y1 = Z0 f1 −→ Z1 −→ · · · fr −→ Zr fr+1 −−−→ Zr+1 −→ · · · −→ Y1 such that Z0 f1 −→ Z1 −→ · · · fr −→ Zr is a sectional path in ΓB and τB Zr+1 = Zr−1; (2) for all i ∈ N, there is a sectional path in ΓB, Y1 = Z0 f1 −→ Z1 −→ · · · fi −→ Zi and a path Zi −→ Z ′ i+1 −→ · · · −→ Y1 in modB. If some Zi is injective (1 ≤ i ≤ r in case (1) or 1 ≤ i in case (2)), then (b) holds. We assume that no Zi is injective in order to get a contradiction. First observe that situation (1) cannot happen. Otherwise, we get a cycle Y1 h1 −→ X −→ τ− B Y1 −→ τ− B Z1 −→ · · · τ− B Zr−1 = Zr+1 −→ · · · −→ Y1 h1 −→ X, where h1 : Y1 → X is an irreducible map. Contradicting that X is a directing A-module. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3860 J. A. DE LA PEÑA AND M. TAKANE Let n be the number of iso-classes of simple A-modules and consider a sectional path Y1 = Z0 f1 −→ Z1 f2 −→ Z2 −→ · · · fn −→ Zn in ΓB as given in situation (2). Moreover, there is a path Zn → Z ′ n+1 → · · · → Y1 in modB. We shall prove that n ⊕ i=0 Zi is a partial cotilting module, which yields the desired contradiction. Indeed, since the number of summands of the partial cotilting module is not bigger than n, then we get Zi ∼= Zj for some j > i. By (1.2), the cycle Zi → · · ·Zj−1 → Zj is not sectional and hence Zj−1 ∼= τBZi+1 or Zj−2 ∼= τBZi which yields a cycle through X as above. Let us first show that i dimB Zi ≤ 1, i = 0, . . . , n. Otherwise i dimB Zi > 1 and there are an indecomposable projective B-module P and a map 0 6= g ∈ HomB(τ− B Zi, P ) (see [18]). Since s(X) ⊕ i=1 Yi is a sincere B-module, there are some j ∈ {1, . . . , s(X)} and a map 0 6= g′ ∈ HomB(P, Yj). We get a cycle X −→ τ− B Yj −→ · · · −→ τ− B Zi g −→ P g ′ −→ Yj hj −→ X, again a contradiction. Let i, j ∈ {0, . . . , n}; we show that ExtB(Zi, Zj) = 0. Otherwise, there is a map 0 6= g ∈ HomB(τ− B Zj , Zi). Then we get a cycle X −→ τ− B Yj −→ · · · −→ τ− B Zj g −→ Zi fi+1 −−−→ Zi+1 −→ · · · · · · → Zn −→ · · · −→ Y1 hi −→ X, and a contradiction. This shows that n ⊕ i=0 Zi is a partial cotilting module, which completes the proof. 1.5. We say that X is a predecessor of Y in ΓA (and Y a successor of X) if there is a path X = Z0 → Z1 → · · · → Zs = Y in ΓA. Proposition. Let η : 0 → τ A X → s(X) ⊕ i=1 Yi → X → 0 be an almost split sequence such that Y1 ∼= Y2. Then (a) If dimk τAX < dimk Y1, then τ −n A X and τ−n A Yi (1 ≤ i ≤ s(X)) are well defined for all n ≥ 0. Moreover, X has no injective successors and does not belong to any oriented cycle in ΓA. (b) If dimk τAX > dimk Y1, τ n A X and τ A Y1 are well defined for all n ≥ 0. Moreover, τ A X has no projective predecessors and does n

The spectral radius of the Galois covering of a finite graph

Abstract Let π:Δ→Δ′ be a Galois covering of a finite graph Δ′ defined by the action of a group G . We study the problem of the relation between the spectral radius r (Δ) of Δ and that of Δ′. We show that r ( Δ )⩽ r ( Δ ′)⩽ r ( Δ ) 2 . We prove that in case the group G is amenable, then r ( Δ )= r ( Δ ′).

Algorithms for weakly nonnegative quadratic forms

Abstract Let q : Z n → Z with q ( v ) = ∑ n i = 1 v ( t ) 2 + ∑ i j a ij v ( i ) v ( j ) be a unit form. We present an algorithm that allows one to check if q is weakly nonnegative [i.e., q ( v ) ⩾ 0 for any vector v ∈ N n ]. The algorithm also calculates the set of critical vectors of q . We sketch the relation of this problem to the representation theory of finite-dimensional algebras.

Tree algebras with non-negative tits form

One of the seminal results of modern representation theory of algebras is Gabriels theorem saying that a quiver algebra A = kQ accepts only finitely many isoclasses of indecomposable modules if and only if the underlying quiver of Q is a Dynkin diagram. The proof may be done by introducing the Tits form q ~ of A and showing that the conditions above are equivalent to qa being a positive quadratic form. In fact, the isoclasses of indecomposable A-modules are in one-to-one correspondence with the positive roots of q ~ . Some years later it was shown that a quiver algebra A = kQ is of tame type if and only if the corresponding Tits form is non-negative. See [8]. We recall that a finite dimensional algebra over an algebraically closed field k is said to be tame if for each dimension, almost all the indecomposable modules occur in a finite number of one-parameter families. We are interested in the classification of tame algebras and their representations. Let A be a basic triangular algebra with a presentation A = kQ/I, where Q is a finite connected quiver without oriented cycles and I is an admissible ideal of the path algebra kQ, see [8]. Let Qo = (1,. . . , n) be the set of vertices of Q. The Tits form q ~ : Zn+ Z is defined by

Isotropic Vectors of Non-negative Integral Quadratic Forms

We say that an integral quadratic formq:Zn → Z of the shape q(x) = ∑ni=1qix(i)2+∑i<jqijx(i) × (j) is a semi-unit form ifqi ≤ 1 for every 1 ≤ i ≤ n. A deflation forqis the linear transformationT−ij:Zn → :Zndefined byes ↦ esifs ≠ jandej ↦ ej − eiifqij < 0. We show that for a non-negative semi-unit formq:Zn → Z with corankq = corank+q, there exists an iterated sequence of deflationsTsuch thatq = qTsatisfiesqij ≥ 0 for alli,j. Further we describe the lattice of positive isotropic vectors of the non-negative semi-unit formsqwith corankq = corank+ q = 2.

Derived Tubularity: a Computational Approach

Let A be a finite dimensional algebra over an algebraically closed field. We denote by mod A the category of finite dimensional left A-modules and by D b (A) the derived category of mod A (see for example [17]] for definitions). We say that two algebras A and B are derived equivalent if their derived categories D b (A) and D b (B) are equivalent as triangulated categories.