Abstract
In an earlier paper, Raphaël Rouquier and the author introduced the group of self-equivalences of a derived category. In the case of a Brauer tree algebra, we determined a nontrivial homomorphism of the Artin braid group to this group of self-equivalences. The class of Brauer tree algebras include blocks of finite group rings over a large enough field with cyclic defect groups. In the present paper we give an integral version of this homomorphism. Moreover, we identify some interesting arithmetic subgroups with natural groups of self-equivalences of the derived category.
Similar content being viewed by others
References
Drozd, Y.: Finitely generated quadratic modules, Manuscripta Math. 104 (2001), 239–256.
Keller, B.: A remark on tilting theory and DG-algebras, Manuscripta Math. 79 (1993), 247–253.
König, S. and Zimmermann, A.: Derived Equivalences for Group Rings (with contributions by Bernhard Keller, Markus Linckelmann, Jeremy Rickard and Raphaël Rouquier), Lecture Notes in Math. 1685, Springer-Verlag, Berlin, 1998.
Newman, M.: Integral Matrices, Academic Press, New York, 1972.
Reiner, I.: Maximal Orders, Academic Press, New York, 1975.
Rickard, J.: Derived categories and stable equivalence, J. Pure Appl. Algebra 61 (1989), 303–317.
Rickard, J.: Lifting theorems for tilting complexes, J. Algebra 142 (1989), 383–393.
Rickard, J.: Derived equivalences as derived functors, J. London Math. Soc. 43 (1991), 37–48.
Roggenkamp, K. W.: Blocks of cyclic defect and Green orders, Comm. Algebra 20 (1992), 1715–1734.
Roggenkamp, K. W. and Scott, L. L.: Units in metabelian group rings: nonsplitting examples for normalized units, J. Pure Appl. Algebra 27 (1983), 299–314.
Rouquier, R. and Zimmermann, A.: A Picard group for derived module categories, Proc. London Math. Soc. 87 (2003) 197–225.
Serre, J. P.: Corps Locaux, Hermann, Paris, 1962.
Zimmermann, A.: Derived equivalences of orders, In: Bautista, de la Peña and Martinez (eds), Proc. ICRA VII, Mexico, Can. Math. Soc. Conf. Proc. 18 (1996), 721–749.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zimmermann, A. Automorphisms of Green Orders and Their Derived Categories. Algebras and Representation Theory 7, 19–34 (2004). https://doi.org/10.1023/B:ALGE.0000019388.23407.e2
Issue Date:
DOI: https://doi.org/10.1023/B:ALGE.0000019388.23407.e2