Claus Michael Ringel

Tame algebras and integral quadratic forms

Integral quadratic forms.- Quivers, module categories, subspace categories (Notation, results, some proofs).- Construction of stable separating tubular families.- Tilting functors and tubular extensions (Notation, results, some proofs).- Tubular algebras.- Directed algebras.

Hall algebras and quantum groups

The Hall algebra of a finitary category encodes its extension structure. The story starts from the work of Steinitz on the module category of an abelian p-group, where the Hall algebra is the algebra of symmetric functions. The theory of Hall algebras is highlighted by Ringel around the 90’s in his seminal work realizing a half of a quantum group via the Hall algebra of quiver representations. Further developments include Lusztig’s canonical bases, cluster categories (Caldero-Keller), higher genus quantum algebras (Burban-Schiffmann, Schiffmann-Vasserot) – just to name a few. One of the goals of the seminar is to introduce Bridgeland’s construction of the quantized enveloping algebras (quantum groups) associated to a symmetric Kac-Moody Lie algebra via Hall algebras of Z/2-graded complexes of quiver representations and several recent progresses around it. In the last part of this seminar, we try to open the window to some applications of Hall algebras to mathematical physics via Hall algebra of curves: for example, they play an important rôle in the proof of AGT conjecture concerning pure N = 2 gauge theory for the group SU(r) ([SV]).

Auslander-reiten sequences with few middle terms and applications to string algebrass

(1987). Auslander-reiten sequences with few middle terms and applications to string algebrass. Communications in Algebra: Vol. 15, No. 1-2, pp. 145-179.

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

The indecomposable representations of the dihedral 2-groups

Let K be a field. We will give a complete list of the normal forms of pairs a, b of endomorphisms of a K-vector space such that a 2 b 2 = 0. Thus, we determine the modules over the ring R = K ( X , Y)/(X 2, y2) which are finite dimensional as K-vector spaces; here (X 2, y2) stands for the ideal generated by X 2 and y2 in the free associative algebra K (X, Y) in the variables X and Y. If G is the dihedral group of order 4q (where q is a power of 2) generated by the involutions 91 and 92, and if the characteristic of K is 2, then the group algebra K G is a factor ring of R, and the K G-modules KGM which have no non-zero projective submodule correspond to the K-vector spaces (take the underlying space of ~ M ) together with two endomorphisms a and b (namely multiplication by g ~ 1 and g 2 1 , respectively) such that, in addition to a Z b 2 -0 , also (ab) q = (ba) q = 0 is satisfied. We use the methods of Gelfand and Ponomarev developped in their joint paper on the representations of the Lorentz group, where they classify pairs of endomorphisms a, b such that ab = ba = O. The presentation given here follows closely the functorial interpretation of the Gelfand-Ponomarev result by Gabriel, which he exposed in a seminar at Bonn, and the author would like to thank him for many helpful conversations.

Finite dimensional hereditary algebras of wild representation type

Let R be a finite dimensional hereditary algebra. We are concerned with the problem of determining the indecomposable R-modules of finite length. This problem completely has been solved in the case when R is of finite or of tame representation type, but seems to be rather hopeless in the case of wild representation type. In this situation, the only known classes of modules are the socalled pre-projective and the pre-injective ones. The remaining indecomposable modules are called regular. In this paper, we want to initiate the study of the regular modules. The result we obtain seems to be rather surprising: we will show that the regular modules behave rather similar to modules over a serial algebra. In order to state the main theorem, we need the notion of an irreducible homomorphism, which was introduced by Auslander and Reiten [-3]. Let X and Y be two non-zero R-modules. A homomorphism f : X ~ Y is said to be irreducible, if it is neither a split monomorphism, nor a split epimorphism, and if for any factorisation X ~ I ~ Y o f f , either f is a split monomorphism o r f is a split epimorphism. Note that an irreducible homomorphism is always either a monomorphism or an epimorphism. A non-zero R-module S will be called quasisimple, ifS is regular, and there is no irreducible monomorphism of the form U ~ S with U non-zero. In this case, we will call the map 0 ~ S irreducible.

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.

Hall polynomials for the representation-finite hereditary algebras

Let k be a field. Let R be a finite-dimensional k-algebra with centre k which is representation-finite and hereditary; thus R is Morita equivalent to the tensor algebra of a k-species with underlying graph A a disjoint union of Dynkin diagrams, and the set of isomorphism classes of indecomposable R-modules corresponds bijectively to the set @+ of positive roots of the corresponding semisimple complex Lie algebra g (see [G] and [DRl ] ). Consequently, the Grothendieck group K( R mod) of all finitely generated R-modules modulo split exact sequences is the free abelian group with basis indexed by @ +. Let h be a Cartan subalgebra of g and g = n + @ h 0 n _ the corresponding triangular decomposition. Note that n+ is the direct sum of one-dimensional complex vectorspaces indexed by the elements of @ +, so we may identify K(R mod) Q @ and n + as vectorspaces, and we deal with the problem of how to recover the Lie multiplication of n, on K( R mod). We have shown in [R2] that the Grothendieck group K(R mod) may be considered in a natural way as a Lie algebra by using as structure constants the evaluations of Hall polynomials at 1. The aim of this paper is to show that this Lie algebra K(R mod) can be identified with a Chevalley H-form of n + ; in particular K( R mod) @ @ and n+ are isomorphic as Lie algebras. We are going to determine all possible polynomials which occur as Hall polynomials ~p;;~, where x, y, z E @ + . There are precisely 16 different polynomials (including the zero polynomial cpO), and the absolute value of their evaluations at 1 is bounded by 3. One easily observes that q;X = 0 in case y #z +x; thus let us assume y = z + x. In this case, precisely one of the two polynomials cp,‘, and cpzZ is non-zero. The non-zero polynomials cp;’ can be written in the form irqi, where [, = C’:i T’, with 1 d r 6 3, and (pi is one of the following 12 integral polynomials:

EXCEPTIONAL MODULES ARE TREE MODULES

Abstract We consider representations of a quiver over an arbitrary field. Recall that an indecomposable representation M without self-extensions is said to be exceptional. We are going to show that exceptional representations can be exhibited using matrices involving as coefficients just 0 and 1. Actually, if d is the dimension of M , there exists such a matrix presentation with precisely d − 1 non-zero coefficients: the corresponding “coefficient quiver” is a tree.

Invariant subspaces of nilpotent linear operators, I

Abstract Let k be a field. We consider triples (V, U, T), where V is a finite dimensional k-space, U a subspace of V and T : V → V a linear operator with Tn = 0 for some n, and such that T(U) U. Thus, T is a nilpotent operator on V, and U is an invariant subspace with respect to T. We will discuss the question whether it is possible to classify these triples. These triples (V, U, T) are the objects of a category with the Krull-Remak-Schmidt property, thus it will be suffcient to deal with indecomposable triples. Obviously, the classification problem depends on n, and it will turn out that the decisive case is n = 6. For n < 6, there are only finitely many isomorphism classes of indecomposable triples, whereas for n > 6 we deal with what is called “wild” representation type, so no complete classification can be expected. For n = 6, we will exhibit a complete description of all the indecomposable triples.