DERIVED TUBULARITY: A COMPUTATIONAL APPROACH

Let A be a finite dimensional algebra over an algebraically closed field. We denote by modA the category of nite dimensional left A-modules and by D b(A) the derived category of modA (see for example [12] for definitions). We say that two algebras A and B are derived equivalent if their derived categories Db(A) and Db(B) are equivalent as triangulated categories. An important problem in the Representation Theory of Algebras has been to characterize those algebras A which are derived equivalent to well understood classes of algebras, for instance to representation-finite hereditary algebras [11,12], to tame hereditary algebras [11] or to tubular algebras [13,1]. The aim of this work is to discuss the above mentioned characterizations from a computational point of view. In particular, we present an algorithm to decide whether or not certain classes of algebras are derived equivalent to a tubular algebra. This algorithm has been imple-mented in a C++ program which will be available as part of the CREP package, see [8]. Let A be an algebra as above with nite global dimension. Let K0(A) ! Zn be its Grothendieck group which we will consider equipped with a (non-symmetric) bilinear form h;iA such that for two modules X; Y 2 modA we have, h[X]; [Y]iA