Resumen

Joint work with Lleonard Rubio y Degrassi. Relative Hochschild cohomology was first defined by Hochschild in 1956. It has been used by Gerstenhaber-Schack in the context of deformation, and recently by Cibils, Lanzilotta, Marcos, Schroll and Solotar to describe how Hochschild cohomology of bound quiver changes when adding or deleting an arrow of a bound quiver. Let A be a bound quiver algebra and let B be a subalgebra with the same semisimple part. We give a sufficient condition for the first relative Hochschild cohomology to be solvable as a Lie algebra. We also defined the contracted fundamental group and show, similar to the normal case, that the dual maps into the first relative Hochschild cohomology.