A formula is provided to explicitly describe global dimensions of all kinds of tree type finite dimensional $k$-algebras for $k$ an algebraic closed field. In particular, it is pointed out that if the underlying tree type quiver has $n$ vertices, then the maximum global dimension is $n\,-\,1$.

We prove that two quiver operator algebras can be isometrically isomorphic only if the quivers (=directed graphs) are isomorphic. We also show how the graph can be recovered from certain representations of the algebra.

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPick(A) is the group of triangle auto-equivalences of Db(mod A) induced by two-sided tilting complexes. We study the group DPick(A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPick, as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPick(A) on a certain infinite quiver Γirr. This representation is faithful when the quiver Δ of A is a tree, and then DPick(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPick(A). When A is hereditary, DPick(A) coincides with the full group of k-linear triangle auto-equivalences of Db(mod A). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to Db(mod A. These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.