Communicated by I. Reiten
MSC:16S35 18E30 a b s t r a c tWe show that the main results of Happel- Rickard-Schofield (1988) and Happel-ReitenSmalø (1996) on piecewise hereditary algebras are coherent with the notion of group action on an algebra. Then, we take advantage of this compatibility and show that if G is a finite group acting on a piecewise hereditary algebra A over an algebraically closed field whose characteristic does not divide the order of G, then the resulting skew group algebra A[G] is also piecewise hereditary.Let k be an algebraically closed field. For a finite dimensional k-algebra A, we denote by mod A the category of finite dimensional left A-modules, and by D b (A) the (triangulated) derived category of bounded complexes over mod A. Let H be a connected hereditary abelian k-category. Following [7] (compare [5,9]), we say that A is piecewise hereditary of type H if it is derived equivalent to H , that is D b (A) is triangle-equivalent to the derived category D b (H) of bounded complexes over H. Over the years, piecewise hereditary algebras have been widely investigated and proved to be related with many other topics, such as the simply connected algebras and the trivial extensions, the self-injective algebras of polynomial growth and the strong global dimension.Hereditary categories H having tilting objects are of special interest in representation theory of algebras. The endomorphism algebras End H T of tilting objects T in H, called quasitilted algebras, were introduced and studied in [8]. It is well-known that H and End H T are derived equivalent. When k is algebraically closed, it was shown by Happel [6] that H is either derived equivalent to a finite dimensional hereditary k-algebra H or derived equivalent to a category of coherent sheaves cohX on a weighted projective line X (in the sense of [3]).
Let C be a finite dimensional algebra of global dimension at most two. A partial relation extension is any trivial extension of C by a direct summand of its relation C − C-bimodule. When C is a tilted algebra, this construction provides an intermediate class of algebras between tilted and cluster tilted algebras. The text investigates the representation theory of partial relation extensions. When C is tilted, any complete slice in the Auslander-Reiten quiver of C embeds as a local slice in the Auslander-Reiten quiver of the partial relation extension; Moreover, a systematic way of producing partial relation extensions is introduced by considering direct sum decompositions of the potential arising from a minimal system of relations of C.
We give a construction of algebras, called articulation, which is a specific gluing of two non-simple algebras. Then, we describe the Auslander–Reiten theory of an algebra obtained in this way. This allows us to characterize the articulated algebras which are laura, left (or right) glued, weakly shod, shod, quasi-tilted or tilted.
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