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Classifying quasi-hereditary structures of some quiver algebras

Banff International Research Station for Mathematical Innovation and Discovery

Quasi-hereditary algebras were introduced by L. Scott in the context of the work of E. Cline, B. Parshall and L. Scott on highest weight categories arising in the representation theory of Lie algebras and algebraic groups. The notion of a quasi-hereditary algebra depends on a partial order given to the set of simple modules. In particular an algebra may be quasi-hereditary for one partial order but not for another one, even in the hereditary case. After recalling basic definitions, we will introduce an equivalence relation on the set of all partial orders giving a quasi-hereditary algebra, calling the equivalence classes quasi-hereditary structures. In the case of the path algebra of an equioriented quiver of type A, we will classify all its quasi-hereditary structures in terms of tilting modules, highlighting its nice combinatorial properties. Then we will generalise this classification to any orientation. As a complementary example we will discuss a class of quiver algebras with a unique quasi-hereditary structure. Time permitting, we will introduce a partial order on the set of all quasi-hereditary structures and give some examples. This is joint work in progress with Yuta Kimura and Baptiste Rognerud.

Mathematics; Associative rings and algebras; Algebraic geometry; Representation theory

video/mp4

Author affiliation: Bielefeld University

2020-03-02

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/73642

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/