| 概要 | As is well-known, derived category and derived equivalence are
widely used in many different branches in mathematics. In the modern
representation theory of algebras and groups, they are also of
particular interest. The following are three of the fundamental
questions in studying derived equivalences:
(1) How to characterize derived equivalences ?
(2) How to decide two given derived categories (or algebras) being
derived-equivalent ?
(3) How to construct new derived equivalences from given ones?
The first question was answered by Rickard's beautiful Morita
theory for derived categories. The second question is hard, and
still open (just think of Broue's abelian defect group conjecture).
As to the last question, not much is known.
In this talk, we shall mainly consider the last question and provide
partial answers from algebraic point of view. Roughly speaking, we
shall give two methods to construct derived equivalences: One is to
form the so-called $\Phi$-Auslander-Yoneda algebras of generators,
or cogenerators, and the other is to form endomorphism algebras of
modules appearing in an almost split sequence. We apply these
methods to Frobenius algebras.
We shall start with basic definitions and examples, and present some
recent results in this direction. |