Quadratic forms and iterated tilted algebras

Abstract

Given a finite quiver A without oriented cycles, an algebra A is called an iterated tilted algebra of type A [2] if there exists a sequence of algebras A = A,, A 1) . ..) A,,,, where A, is the path algebra of A, and a sequence of tilting modules T>, (0 , and every indecomposable A ,-module satisfies either Hom,J T’, M) = 0 or Exta,( T’, M) = 0. If m < 1, A is called a tilted algebra of type A [25]. The representation theory of iterated tilted algebras was proved to be closely related to that of a class of symmetric algebras, namely, the trivial extension algebras; see [3,4, 18, 27, 291. Recently, they were also shown to arise naturally in the study of the derived category of a finite dimensional algebra; see [23, 24, 71. Iterated tilted algebras of type A, where the underlying graph of A is a Dynkin diagram, were studied in [ 1, 2, 8, 223, and the iterated tilted algebras of Euclidean type A,,, (m 3 1) were classified in [S]. 55 0021-8693/90 $3.00