Title: The derived Picard group and representations of quivers

Publication status: "Proceedings of the 32nd Symposium on Ring Theory and Representation Theory," Ed. J. Miyachi, Tokyo Gakugei Univ., Japan, 2000.

The derived Picard group DPicA of a ring A is the group of auto-equivalences of the derived category D^b(Mod A) induced by tilting complexes. When A is either local or commutative, DPic(A) is a product of the usual (noncommutative) Picard group Pic(A) and a cyclic group; in this sense DPic(A) has a noncommutative geometry content.

In this talk we consider the case where A is the path algebra of a finite quiver \Delta over a field k (i.e. A is a finite dimensional hereditary k-algebra). There is a natural action of DPic(A) on a certain infinite quiver. This action is faithful when \Delta is a tree, and otherwise a connected linear algebraic group may occur as a factor of DPic(A). At any rate we get an effective description of DPic(A); examples will be shown.

In the hereditary case DPic(A) coincides with the group of triangle auto-equivalences of the derived category of A-modules. This means that we can calculate the groups of auto-equivalences of various derived categories occurring, for example, in noncommutative geometry. (Joint work with J. Miyachi.)

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