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Aslak Bakke Buan 《中国科学 数学(英文版)》2019,(7)
We survey some recent results generalizing classical tilting theory to a theory of two-term silting objects. In particular, this includes a generalized Brenner-Butler theorem, and a homological characterization of algebras obtained by two-term silting from hereditary algebras. 相似文献
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Aslak Bakke Buan Robert J. Marsh Idun Reiten 《Transactions of the American Mathematical Society》2007,359(1):323-332
We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.
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Aslak Bakke Buan Robert J. Marsh Idun Reiten Gordana Todorov with an Appendix coauthored in addition by P. Caldero B. Keller 《Proceedings of the American Mathematical Society》2007,135(10):3049-3060
Cluster algebras are commutative algebras that were introduced by Fomin and Zelevinsky in order to model the dual canonical basis of a quantum group and total positivity in algebraic groups. Cluster categories were introduced as a representation-theoretic model for cluster algebras. In this article we use this representation-theoretic approach to prove a conjecture of Fomin and Zelevinsky, that for cluster algebras with no coefficients associated to quivers with no oriented cycles, a seed is determined by its cluster. We also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.
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