Stable equivalences occur frequently in the representation theory of finite-dimensional algebras; however, these equivalences are poorly understood. An interesting class of stable equivalences is obtained by ‘gluing’ two idempotents. 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 identifying the two idempotents belonging to the simple projective module and to the simple injective module, respectively. In this talk we will compare the first Hochschild cohomology groups of finite-dimensional monomial algebras under gluing two arbitrary idempotents (hence not necessarily inducing a stable equivalence). As a corollary, we will show that stable equivalences obtained by gluing two idempotents provide ‘some functoriality’ to the first Hochschild cohomology, that is, HH1(A) is isomorphic to a quotient of HH1(B).
This video is part of the European Non-Associative Algebra Seminar series.
