Derived Picard Groups of Finite-Dimensional Hereditary Algebras |
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Authors: | Jun-Ichi Miyachi and Amnon Yekutieli |
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Institution: | (1) Department of Mathematics, Tokyo Gakugei University, Koganei-shi, Tokyo, 184, Japan;(2) Department Mathematics and Computer Science, Ben Gurion University, Be'er Sheva, 84105, Israel |
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Abstract: | Let A be a finite-dimensional algebra over a field k. The derived Picard group DPic
k
(A) is the group of triangle auto-equivalences of D>
b( mod A) induced by two-sided tilting complexes. We study the group DPic
k
(A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPic
k
, as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic
k
(A) on a certain infinite quiver irr. This representation is faithful when the quiver of A is a tree, and then DPic
k
(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic
k
(A). When A is hereditary, DPic
k
(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. |
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Keywords: | derived category Picard group finite dimensional algebra quiver |
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