Koszul and quasi-Koszul algebras obtained by tilting
Given a finite-dimensional algebra, we present sufficient conditions on the projective presentation of the algebra modulo its radical for a tilted algebra to be a Koszul algebra and for the endomorphism ring of a tilting module to be a quasi-Koszul algebra. One condition we impose is that the algebra has global dimension no greater than $2$. One of the main techniques is studying maps between the direct summands of the tilting module. Some applications are given. We also show that a Brenner–Butler tilted algebra is simply connected if and only if the original algebra is simply connected.