Abstract
Assume that a basic algebra A over an algebraically closed field with a basic set A0 of primitive idempotents has the property that
for all
Let n be a nonzero integer, and
and
two automorphisms of the repetitive category
of A with jump n (namely, they send
to
where
is the i-th copy of A in
for all
). If
and
coincide on the objects and if there exists a map
such that
for all morphisms
then the orbit categories
and
are isomorphic as
-graded categories.
Acknowledgments
We would like to thank Junichi Miyachi for asking us about the existence of examples that gives a negative solution to the conjecture above, Manuel Saorín for sending us his unpublished paper [Citation11] (the published version of which is [Citation10]) and Steffen Koenig for informing us the proof of Lemmas 5.1 and 5.2. Finally we also would like to thank the referee for helpful suggestions about relationships between this paper and the work of [Citation7], which relates the main result with the inner automorphisms of algebras (see Lemma 4.3).
Notes
1 When B is regarded as a category by fixing a basic set of primitive idempotents of B as the set of objects, these automorphisms are not assumed to send objects of B to objects of B.
2 Note that if then
can be given by
because
for all
and