Derivation Algebra in Noncommutative Group Algebras

For a generally infinite noncommutative discrete group G, we study derivation algebras in the group algebra of G in terms of characters on a groupoid associated with the group. We obtain necessary conditions for a character to define a derivation. Using these conditions, we analyze some examples. In particular, we describe a derivation algebra in the case when the group is a nilpotent group of rank 2.