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MUTATIONS AND POINTING FOR BRAUER TREE ALGEBRAS

Schaps, Mary 大阪大学 DOI:10.18910/76681

2020.07

概要

Brauer tree algebras are important and fundamental blocks in the representation theory of finite dimensional algebras. In this research, we present a combination of two main approaches to the tilting theory of Brauer tree algebras. The first approach is the theory initiated by Rickard, providing a direct link between an ordinary Brauer tree algebra and the Brauer star algebra. This approach was continued by Schaps-Zakay with their theory of pointing the tree. The second approach is the theory developed by Aihara, relating to the sequence of mutations from the ordinary Brauer tree algebra to the Brauer star algebra. Our main purpose in this research is to combine these two approaches. We first find an algorithm based on centers which are all terminal edges, for which we are able to obtain a tilting complex constructed from irreducible complexes of length two [13], which is obtained from a sequence of mutations. In [1], Aihara gave an algorithm for reducing from tree to star by mutations and showed that it gave a two-term tree-to-star complex. We prove that Aihara’s complex is obtained from the corresponding completely folded Rickard tree-to-star complex by a permutation of projectives.

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参考文献

[1] T. Aihara: Mutating Brauer trees, Math. J. Okayama Univ. 56 (2014), 1–16.

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[10] J. Rickard: Derived categories and stable equivalence, J. Pure and Appl. Algebra. 61 (1989), 303–317.

[11] J. Rickard and M. Schaps: Folded tilting complexes for Brauer tree algebras, Adv. Math. 171 (2002), 169–182.

[12] M. Schaps: Deformations, tiltings, and decomposition matrices; in Representations of Algebras and Re- lated Topics, Fields Institute Publication 45, Amer. Math. Soc., Providence, RI, 2005, 345–356.

[13] M. Schaps and E. Zakay-Illouz: Combinatorial partial tilting complexes for the Brauer star algebras; in Representations of Algebras (Sa˜o Paulo, 1999). Lecture Notes in Pure and Appl. Math. 224, Dekker, New York 2002, 187–207.

[14] M. Schaps and E. Zakay-Illouz: Pointed Brauer trees, J. Algebra 246 (2001), 647–672.

[15] C.A. Weibel: An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, Cambridge, 1994.

[16] Z. Zvi: Combining mutations and pointing for Brauer trees, M.Sc. thesis, Bar-Ilan University, 2016.

[17] A. Zvonareva: On the derived Picard group of the Brauer star algebra, arXiv:1401.6952v3, 2015.

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