Dieter Happel

Triangulated categories in the representation theory of finite dimensional algebras

Preface 1. Triangulated categories 2. Repetitive algebras 3. Tilting theory 4. Piecewise hereditary algebras 5. Trivial extension algebras References Index.

Minimal algebras of infinite representation type with preprojective component

Let A be a finite dimensional, basic and connected algebra (associative, with 1) over an algebraically closed field k. Denote by e1,...,en a complete set of primitive orthogonal idempotents in A and by Ai= A/AeiA. A is called a minimal algebra of infinite representation type provided A is itself of infinite representation type,whereas all Ai, 1≤i≤n,are of finite representation type. The main result gives the classification of the minimal algebras having a preprojective component in their Auslander-Reiten quiver. The classification is obtained by realizing that these algebras are essentially given by preprojective tilting modules over tame hereditary algebras.

Binary polyhedral groups and Euclidean diagrams

Recently, J. McKay [7] has observed that the irreducible complex representations of the binary polyhedral groups can be arranged in order to form the vertices of a Euclidean diagram in such a way that the tensor product of any irreducible representation M with the standard two-dimensional representation is the direct sum of the irreducible representations which are the neighbors of M in the diagram, and he asked for an explanation. In this note, we will show that any self-dual two-dimensional representation gives rise to a generalized Euclidean diagram, and that this in fact can be used to give a proof of the classification theorem of the binary polyhedral groups which at the same time furnishes a list of the irreducible representations and also gives the minimal splitting field.

Directing projective modules

Let A be an Artin algebra. The A-modules which we consider are always left modules of finite length. If X, Y, Z are A-modules, the composition of maps f : X ~ Y and g : Y ~ Z is denoted by f 9 : X ~ Z. The category of (finite length) A-modules is denoted by A-mod. If X, Y are indecomposable A-modules, we denote by rad (X, Y) the set of non-invertible maps from X to Y. A path in A-mod is a sequence (X o . . . . . Xs) of (isomorphism classes of) indecomposable A-modules X i, 0 1, and X o = Xs, then the path (Xo, . . . , X~) is called a cycle. A indecomposable A-module is called directing if X does not occur in a cycle. Our first aim will be to extend the definition of a directing module to decomposable modules. We show that an indecomposable projective A-module P is directing if and only if the radical of P is directing. In case the top of P is injective it follows that P is directing if and only if the radical of P is directing as a module over the factor algebra of A by the trace ideal of P.

Representation-finite trivial extension algebras

Abstract Let A be a finite-dimensional basic connected associative algebra over an algebraically closed field, and T(A)=A ⋉ DA its trivial extension by its minimal injective cogenerator. We prove that T(A) is representation-finite of Cartan class Δ if and only if A is an iterated tilted algebra of Dynkin class Δ. The proof also yields a construction procedure for iterated tilted algebras of Dynkin type.

Finitistic dimension of standardly stratified algebras

We prove that the projectively and the injectively defined finitistic dimensions of a standardly stratified algebra are always finite by giving the optimal bound for these numbers in terms of the number of simple modules.

Hereditary categories with tilting object

Let k be an algebraically closed field and H a connected abelian k-category which is hereditary, that is Ext2( , ) vanishes on H. Assume also that Hom(X, Y ) and Ext1(X, Y ) are finite dimensional vector spaces over k for all X and Y in H. We consider such hereditary categories H which in addition have a tilting object T , that is, an object T such that {X; Ext1(T, X) = 0} = FacT , the factors of finite direct sums of copies of T . Hereditary categories H with a tilting object T are of special interest in connection with the construction of the class of algebras called quasitilted algbras, which was introduced in [HRS1]. They are by definition the algebras of the form EndH(T )op. An important property is that H and EndH(T )op have equivalent bounded derived categories. The main examples of such hereditary categories are the category modH of finitely generated modules over a finite dimensional hereditary k-algebra H and the category cohX of coherent sheaves on a weighted projective line in the sense of [GL1]. There are also others derived equivalent to them. Because of the simple description of the corresponding bounded derived category Db(H), it is possible to give a description of those in the same derived equivalence class. (See [LS] [H2]). It follows from [HRe] that they automatically have a tilting object. The quasitilted algebras provide common generalization for the tilted and the canonical finite dimensional algebras. Note that the tilted algebras are those coming from modH using an arbitrary

THE TRACE OF THE COXETER MATRIX AND HOCHSCHILD COHOMOLOGY

Abstract For a basic finite-dimensional k-algebra Λ of finite global dimension over an algebraically closed field k the Coxeter matrix ΦΛ is defined. The following formula is shown: trM Θ Λ = ∑ i⩾ (−1) i dim k H i (Λ) where tr is the trace and Hi(Λ) is the ith Hochschild cohomology space.

Reconstruction of Path Algebras from their Posets of Tilting Modules

Let A = kΔ be the path algebra of a finite quiver without orientedcycles. The set of isomorphism classes of multiplicity free tilting modules is in a natural way a partially ordered set. We will show here that T A uniquely determines Δ if Δ has no multiple arrows and no isolated vertices.

Auslander generators of iterated tilted algebras

Let A be an iterated tilted algebra. We will construct an Auslander generator M in order to show that the representation dimension of A is three in case A is representation infinite.