**Lleonard Rubio y Degrassi****, Universita' di** **Padova**

**Title****: **
**On the Lie algebra structure of the first Hochschild cohomology **

**Abstract:** Let A
be a finite dimensional algebra over an algebraically closed
field. Hochschild cohomology records crucial information about A:
its first degree component, denoted by HH^{^1}(A),
is a Lie algebra and it is invariant under Morita and derived
equivalences. For symmetric algebras, it is also invariant under
stable equivalences of Morita type. Although HH^{^1}
is a powerful invariant, its Lie structure has been calculated
only for few families of algebras.

In this talk I will show how the Lie structure of HH^{^1}
is strongly related with the Ext-quiver of A. More precisely, if
we assume that the Ext-quiver of A is a simple directed graph,
then the Lie algebra of HH^{^1}(A)
is solvable. For quivers containing loops, I will determine
sufficient conditions for the solvability of HH^{^1}.
Finally, I will apply these criteria to show the solvability of
the first Hochschild cohomology of blocks with cyclic defect, all
tame blocks of finite groups and some wild algebras.

This is part of two joint works with Markus Linckelmann, and with
Andrea Solotar and Sibylle Schroll.