Christine Riedtmann

Skew group algebras in the representation theory of artin algebras

Then the corresponding crossed product algebra /i * ?G, or /i x G for short, has an elements CgiEG ,gi A.-; li ~/i. Addition is componentwise, and multiplication is given by & = g(A)g and g, g2 = y(g, , g2) m. In this paper we assume that the values of y lie in the center Z(A) of/i. Hence the action of G on /i is given by a group homomorphism G-P Aut(/i), and (3) can be left out. In the special case that y is the trivial map we write /1G instead of/i * G, and the elements as C,,,c li gi. AG is then called a skew group ring. There is a lot of literature on skew group algebras and crossed product algebras, and on the relationship with the ring AC whose elements are those elements of A left fixed by G. Much work has been done on which properties of li are inherited by n * G or AC. Some of the work on the relationship between these rings has its roots in trying to develop a Galois theory for noncommutative rings. We refer to [3, 7, 13-15, 19, 21, 23-25, 27, 281 and their references. In this paper we study these constructions when /i is an artin algebra and G a finite group of order n such that n is invertible in /1. Under these assumptions the construction preserves central properties of interest in the

Orbit closures and rank schemes

Let A be a finitely generated associative algebra over an algebraically closed field k, and consider the variety mod A.k/ of A-module structures on k d . In case A is of finite representation type, equations defining the closure x OM are known for M 2 mod A.k/; they are given by rank conditions on suitable matrices associated with M . We study the schemes CM defined by such rank conditions for modules over arbitraryA, comparing them with similar schemes defined for representations of quivers and obtaining results on singularities. One of our main theorems is a description of the ideal of x OM for a representation M of a quiver of type An, a result Lakshmibai and Magyar established for the equioriented quiver of type An in [12]. Mathematics Subject Classification (2010). Primary 14L30; Secondary 14B05, 16G20, 16G70.

On stable blocks of Auslander-algebras

The Auslander-algebra EA of an algebra A of finite representation type is the endomorphism algebra of the direct sum M = E? M, of one copy of each indecomposable A-module. A stable block of EA is a connected direct factor of the residue algebra of EA modulo the two-sided ideal generated by the projections of M to the M,s that are not stable under DTr. This paper describes the stable blocks whose quiver is a stable translation-quiver of class A,, or D/,.