Galois actions on rings and finite Galois coverings.
Galois coverings and splitting properties of the ideal generated by halflines
Given a locally bounded k-category R and a group acting freely on R we study the properties of the ideal generated by a class of indecomposable locally finite-dimensional modules called halflines (Theorem 3.3). They are applied to prove that under certain circumstances the Galois covering reduction to stabilizers, for the Galois covering F: R → R/G, is strictly full (Theorems 1.5 and 4.2).
Galois coverings and the Clebsch-Gordan problem for quiver representations
We study the Clebsch-Gordan problem for quiver representations, i.e. the problem of decomposing the point-wise tensor product of any two representations of a quiver into its indecomposable direct summands. For this purpose we develop results describing the behaviour of the point-wise tensor product under Galois coverings. These are applied to solve the Clebsch-Gordan problem for the double loop quivers with relations αβ = βα = αⁿ = βⁿ = 0. These quivers were originally studied by I. M. Gelfand and...
Galois coverings and the problem of axiomatization of the representation type of algebras.
Galois coverings of representation-infinite algebras.
General sheaves over weighted projective lines
We develop a theory of general sheaves over weighted projective lines. We define and study a canonical decomposition, analogous to Kac's canonical decomposition for representations of quivers, study subsheaves of a general sheaf, general ranks of morphisms, and prove analogues of Schofield's results on general representations of quivers. Using these, we give a recursive algorithm for computing properties of general sheaves. Many of our results are proved in a more abstract setting, involving a hereditary...
Generalized Brauer tree orders
Generalized canonical algebras and standard stable tubes
We introduce a new wide class of finite-dimensional algebras which admit families of standard stable tubes (in the sense of Ringel [17]). In particular, we prove that there are many algebras of arbitrary nonzero (finite or infinite) global dimension whose Auslander-Reiten quivers admit faithful standard stable tubes.
Generalized coil enlargements of algebras
Generalized injectivity
Generalized one-point extensions.
Generalized projectivity
Generalized quivers associated to reductive groups
We generalize the definition of quiver representation to arbitrary reductive groups. The classical definition corresponds to the general linear group. We also show that for classical groups our definition gives symplectic and orthogonal representations of quivers with involution inverting the direction of arrows.
Generalized support varieties for finite group schemes.
Generic extensions of nilpotent k[T]-modules, monoids of partitions and constant terms of Hall polynomials
We prove that the monoid of generic extensions of finite-dimensional nilpotent k[T]-modules is isomorphic to the monoid of partitions (with addition of partitions). This gives us a simple method for computing generic extensions, by addition of partitions. Moreover we give a combinatorial algorithm that calculates the constant terms of classical Hall polynomials.
Geometry of modules over tame quasi-tilted algebras
Geometry of noncommutative algebras
There has been several attempts to generalize commutative algebraic geometry to the noncommutative situation. Localizations with good properties rarely exist for noncommutative algebras, and this makes a direct generalization difficult. Our point of view, following Laudal, is that the points of the noncommutative geometry should be represented as simple modules, and that noncommutative deformations should be used to obtain a suitable localization in the noncommutative situation. Let A be an algebra...
Global Dimension two orders are quasi-hereditary.
Global dimensions of dual extension algebras.
Graded blocks of group algebras with dihedral defect groups
We investigate gradings on tame blocks of group algebras whose defect groups are dihedral. For this subfamily of tame blocks we classify gradings up to graded Morita equivalence, we transfer gradings via derived equivalences, and we check the existence, positivity and tightness of gradings. We classify gradings by computing the group of outer automorphisms that fix the isomorphism classes of simple modules.