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The module theoretical approach to quasi-hereditary algebras

Quasi-hereditary algebras were introduced by L.Scott [S] in order to deal with highest weight categories as they arise in the representation theory of semi–simple complex Lie algebras and algebraic groups. Since then, also many other algebras arising naturally, such as the Auslander algebras, have been shown to be quasi-hereditary. It seems to be rather surprising that the class of quasi-hereditary algebras, defined in purely ring–theoretical terms, has not been studied before in ring theory. The central concept of the theory of quasi-hereditary algebras are the notions of a standard and a costandard module; these modules depend in an essential way, on a (partial) ordering of the set of all simple modules. So we start with a finite dimensional algebra A, and a partial ordering of the simple A–modules, in order to define the standard modules ∆(i) and the costandard modules ∇(i); we have to impose some additional conditions on their endomorphism rings, and on the existence of some filtrations, in order to deal with quasi-hereditary algebras. This is the content of the first chapter. The second chapter collects some properties of quasi-hereditary algebras, in particular those needed in later parts of the paper. The third chapter presents the process of standardization: here, we give a characterization of the categories of ∆–filtered modules over quasi-hereditary algebras. In fact, we show that given indecomposable A–modules Θ(1), . . . ,Θ(n) over a finite–dimensional algebra such that rad(Θ(i),Θ(j)) = 0, and Ext(Θ(i),Θ(j)) = 0 for all i ≥ j, the category F(Θ) of all A–modules with a Θ–filtration is equivalent (as an exact category) to the category of all ∆–filtered modules over a quasi-hereditary algebra. The forth and the fifth chapter consider cases when the standard modules over a quasi-hereditary algebra have special homological properties: first, we assume that any ∆(i) has projective dimension at most 1, then we deal with the case that the dominant

On algebras of finite representation type

Throughout the paper, K denotes a fixed commutative field. LetF be a field containing Kin its center such that FK is finite dimensional. A finite (partially) ordered set Y together with an order preserving mapping of .Y into the lattice of all subfields of F containing K is called a K-structure for F, thus, for i E Y, there is given a subfield Fi of F and, moreover, KC Fi C Fj for each i k < Z} with Fi = Fj = F,, = F, = G. Given a K-structure .4” for F, the weighted width of Y is defined as the maximum of all possible sums CjEJ dim F,? , where J is a subset of mutually unrelated elements of 9. An Y-space (W, WJ is a right vector space W over F together with an F,-subspace Wi for each ie 9, such that i < j implies Wi C Wj . The weighted dimension of (W, Wi) is the maximum of all dim WF, . For a given K-structure Y, the Y-spaces form an additive category in which the morphisms (W, Wi) --f (II”, W,‘) are F-linear mappings v: W + W satisfying ~JW~ C W,‘, i E 9. Therefore, the concepts of a direct sum and of an indecomposable Y-space are defined. A K-structure 9 is said to be of Jinite type if there is only a finite number of finite dimensional indecomposable Y-spaces. In the case of a classical K-structure, that is Fi = F for all i E Y, L. A. Nazarova and A. V. Roiter [ 151 and M. M. Kleiner [l l] have characterized the structures of finite type. Their results are extended in the following theorem.

Finite dimensional algebras and related topics

Preface. Equivalences of Blocks of Group Algebras M. Broue. On the Endomorphism Algebras of Gelfand--Graev Representations C.W. Curtis. Harish--Chandra Vertices, Green Correspondence in Hecke Algebras, and Steinbergs Tensor Product Theorem in Nondescribing Characteristic R. Dipper. On Tilting Modules and Invariants for Algebraic Groups S. Donkin. Harish--Chandra Subalgebras and Gelfand--Zetlin Modules Yu.A. Drozd, V.M. Futorny, S.A. Ovsienko. Algebras Associated to Bruhat Intervals and Polyhedral Cones M.J. Dyer. Symmetric Groups and Quasi-Hereditary Algebras K. Erdmann. Quasitilted Algebras D. Happel, I. Reiten, S.O. Small. Tilting Theory and Differential Graded Algebras B. Keller. Wild Canonical Algebras and Rings of Automorphic Forms H. Lenzing. The Ext Algebra of a Highest Weight Category B. Parshall. Coxeter Transformations and the Representation Theory of Algebras J.A. de la Pena. Translation Functors and Equivalences of Derived Categories for Blocks of Algebraic Groups J. Rickard. Blocks with Cyclic Defect (Green Orders) K.W. Roggenkamp. Rigid and Exceptional Sheaves on a DelPezzo Surface A.N. Rudakov. Quasihereditary Algebras and Kazhdan--Lusztig Theory L.l. Scott. Cycles in Module Categories A. Skowronski. Relative Homology M. Auslander, O. Solberg. Tilting Theory and Selfinjective Algebras T. Wakamatsu.

Normal forms of real matrices with respect to complex similarity

This paper gives a complete classification of real linear transformations between two complex vector spaces in terms of matrices.

On modular representations of A4

Let k be a field of characteristic 2. The representation theory of the alternating group A4 over k has an essentially twofold character depending on whether k does or does not contain the cubic root of unity. The case when x2 + x + 1 is reducible over k is reflected in the case when k is algebraically closed, which has been treated in detail by several authors (see S. B. Conion [Z], E. Kern [9]). The case when x*+x + 1 is irreducible over k has been considered explicitly only recently (see U. Schoenwaelder [lo]). Of course, there is a well-known general procedure (see D. G. Higman [8]) to find the represenations of A, in both cases. This is due to the fact that A4 contains the Klein 4-group V, as a normal subgroup and the representations of V, are well-understood (see V. A. Bagev [ 11, A. Heller, I. Reiner [7)). It is this procedure that has been used by U. Schoenwaelder in the case where x2 + x + 1 is irreducible over k. An inherent diffl~ulty of this method appears in the explicit listing and full understanding of certain representations, viz. those belonging to the one-parameter family of kA,-modules which is induced from the one-parameter family of indecomposable kV,-modules. Whereas in the case of V, these representations are indexed by the irreducible polynomials over k, the corresponding index set appears to be much more involved in the case A,: One has to consider an action of the cyclic group of order 3 on the polynomials and determine the corresponding orbits. However, there is an alternative approach presented in our paper, which shows that also in the case of AA, the one-parameter family of representations may be indexed again by the set of all irreducible

Rings with the double centralizer property

Let M be a left unital module over a ring R; write %? = endM Nesbitt and Thrall [ 111). Recently, several authors proved the converse for commutative rings (Dickson and Fuller [2]; Camillo [l]) and Jans in [7] for finitedimensional algebras over algebraically closed fields. In the same paper, Jans conjectured that the converse was true in general. The aim of the present paper is to describe the structure of balanced rings and of certain local QF 1 rings. The first result in this direction is the following theorem proved previously by Camillo [l] for commutative rings.

Standardly Stratified Extension Algebras

Abstract The paper generalizes some of our previous results on quasi-hereditary Koszul algebras to graded standardly stratified Koszul algebras. The main result states that the class of standardly stratified algebras for which the left standard modules as well as the right proper standard modules possess a linear projective resolution – the so called linearly stratified algebras – is closed under forming their Yoneda extension algebras. This is proved using the technique of Hilbert and Poincare series of the corresponding modules. #Communicated by D. Happel.

Towers of semi-simple algebras

This is an extension of the work of Goodman-de la Harpe-Jones on pairs of multi-matrix algebras and the corresponding index. In particular, it is shown that the basic invariant of a pair of finite-dimensional semi-simple algebras is a bimodule together with an integral n-tuple, and that the entire theory is equivalent to the theory of finite-dimensional hereditary algebras whose square of the radical equals zero.

A class of bounded hereditary noetherian domains

Let k be a commutative field, F and G division rings containing k in the center and finite dimensional over k. Let

Constructions of Stratified Algebras

c be a bimodule, with k operating centrally, and such that dim