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).