Claus Michael Ringel

Bio: Claus Michael Ringel is an academic researcher from Bielefeld University. The author has contributed to research in topics: Indecomposable module & Quiver. The author has an hindex of 48, co-authored 211 publications receiving 9425 citations. Previous affiliations of Claus Michael Ringel include Shanghai Jiao Tong University & King Abdulaziz University.

Tame Algebras and Integral Quadratic Forms

Abstract: 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

Abstract: 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

Abstract: (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 representation theory of the symmetric group

Abstract: 1. Symmetric groups and their young subgroups 2. Ordinary irreducible representations and characters of symmetric and alternating groups 3. Ordinary irreducible matrix representations of symmetric groups 4. Representations of wreath products 5. Applications to combinatories and representation theory 6. Modular representations 7. Representation theory of Sn over an arbitrary field 8. Representations of general linear groups Appendices Index.

Canonical bases arising from quantized enveloping algebras

Abstract: 0.2. We are interested in the problem of constructing bases of U+ as a Q(v) vector space. One class of bases of U+ has been given in [DL]. We call them (or, rather, a slight modification of them, see ?2) bases of PBW type, since for v = 1, they specialize to bases of U+ of the type provided by the Poincare see however ? 12.)

Representation theory and complex geometry

Abstract: Preface.- Chapter 0. Introduction.- Chapter 1. Symplectic Geometry.- Chapter 2. Mosaic.- Chapter 3. Complex Semisimple Groups.- Chapter 4. Springer Theory.- Chapter 5. Equivariant K-Theory.- Chapter 6. Flag Varieties, K-Theory, and Harmonic Polynomials.- Chapter 7. Hecke Algebras and K-Theory.- Chapter 8. Representations of Convolution Algebras.- Bibliography.

Foundations of Module and Ring Theory : A Handbook for Study and Research

Abstract: This volume provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work. It is a useful and stimulating study for those new to the subject as well as for researchers and serves as a reference volume. Starting form a basic understanding of linear algebra, the theory is presented and accompanied by complete proofs. For a module M, the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category. Developing the techniques in o[M] is no more complicated than in full module categories and the higher generality yields significant advantages: for example, module theory may be developed for rings without units and also for non-associative rings. Numerous exercises are included in this volume to give further insight into the topics covered and to draw attention to related results in the literature.

Moduli of representations of finite dimensional algebras

Abstract: IN this paper, we present a framework for studying moduli spaces of finite dimensional representations of an arbitrary finite dimensional algebra A over an algebraically closed field k. (The abelian category of such representations is denoted by mod-A.) Our motivation is twofold. Firstly, such moduli spaces should play an important role in organising the representation theory of wild algebras. Secondly, such moduli spaces can be identified with moduli spaces of vector bundles on special projective varieties. This identification is somewhat hidden in earner work ([6], [7]) but has become more explicit recently ([4], [12]). It can now be seen to arise from a 'tilting equivalence' between the derived category of mod-A and the derived category of coherent sheaves on the variety. It is well-established that mod-A is equivalent to the category of representations of an arrow diagram, or 'quiver', Q by linear maps satisfying certain 'admissible' relations. Thus, the problem of classifying A -modules with a fixed class in the Grothendieck group K0(mod-A), represented by a 'dimension vector' a, is converted into one of classifying orbits for the action of a reductive algebraic group GL(a) on a subvariety VA(a) of the representation space 9t{Q, a) of the quiver. Now, the moduli spaces provided by classical invariant theory ([1], [18]) are not interesting in this context. This is because the classical theory only picks out the closed GL(a)-orbits in VA{a), which correspond to semisimple /4-modules, and the quiver Q is chosen so that there is only one semisimple A -module of each dimension vector. On the other hand, we can apply Mumford's geometric invariant theory, with the trivial linearisation twisted by a character x of GL(a), which restricts our attention to an open subset of VA(a), consisting of semistable representations. Within this open set there are more closed orbits and the corresponding algebraic quotient is then a more interesting moduli space. In fact, this approach also has a classical flavour, since it involves the relative (or semi-) invariants of the GL(a) action. The main purpose of this paper is to show that the notions of semistability and stability, that arise from the geometric invariant theory, coincide with more algebraic notions, expressed in the language of mod-A Indeed, the definition is formulated for an arbitrary abelian category as follows: