Lleonard Rubio y
Degrassi
First Hochschild cohomology and stable equivalence obtained by
gluing idempotents
Abstract: There is a class of stable equivalences which is
given by using bimodules that are projective on one side, but not
on the other. More precisely, let A be a finite dimensional
algebra with a simple projective module and a simple injective
module. Assume that B is a subalgebra of A having the same
Jacobson radical. Then B is constructed by gluing the
corresponding idempotents of A, that is, by identifying the two
idempotents belonging to the simple projective module and to the
simple injective module, respectively.
In this case HH^1(A), the first Hochschild cohomology of A, is not
isomorphic to HH^1(B). However, in joint work with Yuming Liu and
Can Wen we have shown that for monomial algebras there is still a
relation between these two Lie algebras: HH^1(A) is isomorphic to a
quotient of HH^1(B).