Semi-invariants of canonical algebras

We describe the algebras of semi-invariants on the varieties of regular representations of canonical algebras. In particular, we show that these algebras are polynomial algebras or complete intersections. Received: 29 March 1999     

A characterization of endotrivial modules over p-groups

Let P be a finite p-group, and let k a field of characteristic p>0. We prove that a finietly generated kG-module M is endotrivial if and only if

Morita equivalences of Ariki–Koike algebras

We prove that every Ariki–Koike algebra is Morita equivalent to a direct sum of tensor products of smaller Ariki–Koike algebras which have q–connected parameter sets. A similar result is proved for the cyclotomic q–Schur algebras. Combining our results with work of Ariki and Uglov, the decomposition numbers for the Ariki–Koike algebras defined over fields of characteristic zero are now known in principle. Received: 22 March 2000; in final form: 19 September 2001 / Published online: 29 April 2002     

Automorphisms of blocks of group algebras and character values

Let G be a finite group, χ an irreducible complex character of G and A(χ) the block ideal of the group algebra ℚG relatedℴ χ. The aim of this paper is to study the group Aut (A(χ)) of all ring (or ℚ-algebra) automorphisms of A(χ). Especially we are interested in the existence of subgroups of Aut (A(χ)), which are isomorphic to a given subgroup Γ of the Galois group of the field of character values ℚ(χ) over the rationals. In this context we prove some results related to character values.     

Forms of pointed Hopf algebras

Using descent theory, we study Hopf algebra forms of pointed Hopf algebras. It turns out that the set of isomorphism classes of such forms are in one-to-one correspondence to other known invariants, for example the set of isomorphism classes of Galois extensions with a certain group F, or the set of isometry classes of m-ary quadratic forms. Our theory leads to a classification of all Hopf algebras over a field of characteristic zero that become pointed after a base extension, in dimension p, p ² and p ³, with p odd. Received: 22 November 1998     

Decompositions of degenerations

We develop a method to refine a given degeneration of modules. This enables us to give a new proof of the equivalence of the partial orders ≤_deg and ≤_ext in the case that the algebra Λ has a directed Auslander–Reiten quiver. Received: 11 November 1999     

There are enough Azumaya algebras on surfaces

Using Maruyama's theory of elementary transformations, I show that the Brauer group surjects onto the cohomological Brauer group for separated geometrically normal algebraic surfaces. As an application, I infer the existence of nonfree vector bundles on proper normal algebraic surfaces. Received: 14 April 2000 / Accepted: 26 February 2001 / Published online: 23 July 2001     

On holomorphic principal bundles over a compact Riemann surface admitting a flat connection

Let G be a connected reductive linear algebraic group over , and X a compact connected Riemann surface. Let be a Levi factor of some parabolic subgroup of G, with its maximal abelian quotient. We prove that a holomorphic G-bundle over X admits a flat connection if and only if for every such L and every reduction of the structure group of to L, the -bundle obtained by extending the structure group of is topologically trivial. For , this is a well-known result of A. Weil. Received: 1 December 2000 / Revised version: 2 April 2001 / Published online: 24 September 2001     

On minimal disjoint degenerations of modules over tame path algebras

For representations of tame quivers the degenerations are controlled by the dimensions of various homomorphism spaces. Furthermore, there is no proper degeneration to an indecomposable. Therefore, up to common direct summands, any minimal degeneration from M to N is induced by a short exact sequence 0→U→M→V→0 with indecomposable ends that add up to N. We study these ‘building blocs’ of degenerations and we prove that the codimensions are bounded by two. Therefore, a quiver is Dynkin resp. Euclidean resp. wild iff the codimension of the building blocs is one resp. bounded by two resp. unbounded. We explain also that for tame quivers the complete classification of all the building blocs is a finite problem that can be solved with the help of a computer.     

Finiteness properties for subgroups of GL (n,\mathbb Z)(n,\mathbb Z)

Abstract. We construct finitely presented subgroups of GL that have infinitely many conjugacy classes of finite subgroups. This answers a question of Grunewald and Platonov. We suggest a variation on their question. Received: 26 August 1999 / Revised: 28 September 1999 / Published online: 8 May 2000     

Axiomatization of abelian-by-G groups for a finite group G

We show that, for each finite group G, there exists an axiomatization of the class of abelian-by-G groups with a single sentence. In the proof, we use the definability of the subgroups M ⁿ in an abelian-by-finite group M, and the Auslander-Reiten sequences for modules over an Artin algebra. Received: 15 March 1996 / Published online: 18 July 2001     

Unique tensor factorization of algebras

Tensor product decomposition of algebras is known to be non-unique in many cases. But, as will be shown here, a -indecomposable, finite-dimensional -algebra A has an essentially unique tensor factorization into non-trivial, -indecomposable factors . Thus the semiring of isomorphism classes of finite-dimensional -algebras is a polynomial semiring . Moreover, the field of complex numbers can be replaced by an arbitrary field of characteristic zero if one restricts oneself to schurian algebras. Received: 5 October 1998