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Two-term silting complexes over algebras with small Loewy length and complete special biserial algebras

青木, 利隆 名古屋大学

2021.06.16

概要

Tilting theory plays an important role in the study of many areas of mathematics. Central notions of tilting theory are tilting complexes and silting complexes, which are generalizations of a progenerator in Morita theory. In fact, the endomorphism algebra of a tilting complex is derived equivalent to the original algebra. In this thesis, we mainly study two-term silting complexes. In representation theory of finite dimensional algebras, the g-vectors of two-term silting complexes are important numerical invariant. In the first part of this thesis, we classify all two-term silting complexes and their g-vectors over algebras with radical square zero. Using this result, we also study symmetric algebras with radical cube zero and determine the number of two-term tilting complexes over them. In the second part of this thesis, we study two-term silting theory for (complete) gentle algebras. A central role is played by their geometric realization on marked surfaces. Using our surface model, we show that a union of g-vector cones is dense in the real Grothendieck group. A main ingredient of our proof is the asymptotic behavior of g-vectors under Dehn twists. We also study a special class of special biserial algebras, called Brauer tree algebras, and show that any Brauer tree algebra has 2n n two-term tilting complexes, where n is the number of edges of the corresponding Brauer tree.

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