Lie derivation and Hochschild cohomology of an extension of path algebras

Let k be a field and Q a finite simply laced quiver without oriented cycles. Firstly, we prove that each Lie derivation of the generalized one-point extension of path algebra kQ is of the standard form, and the standard decomposition is unique. Secondly, we calculated the k-dimensions of all the Hochschild cohomology of the generalized one-point extension of A type path algebra, and gave a description of the cup product in the Hochschild cohomology ring.