On the first relative Hochschild cohomology and contracted fundamental group

In this paper we investigate the Lie algebra structure of the first relative Hochschild cohomology and its relation with the relative notion of fundamental group. Let $A,B$ be finite-dimensional basic $k$-algebras over an algebraically closed field of characteristic zero, such that $Q_B$ is a subquiver of $Q_A$. We show that if the complement of $Q_A$ by the arrows of $Q_B$ is a simple directed graph, then the first relative Hochschild cohomology ${\text{HH}}^1(A\mathrm{\mid }B)$ is a solvable Lie algebra. We also compute the Lie algebra structure of the first relative Hochschild cohomology for radical square zero algebras and for dual extension algebras of directed monomial algebras. Finally, we introduce the notion of fundamental group for a pair of an algebra $A$ and a subalgebra $B$ and we construct the relative version of the map from the dual fundamental group into the first Hochschild cohomology.