The Ingalls-Thomas Bijections

Given a finite acyclic quiver Q with path algebra kQ, Ingalls and Thomas have exhibited a bijection between the set of Morita equivalence classes of support-tilting modules and the set of thick subcategories of mod kQ and they have collected a large number of further bijections with these sets. We add some additional bijections and show that all these bijections hold for arbitrary hereditary artin algebras. The proofs presented here seem to be of interest also in the special case of the path algebra of a quiver.Comment: This is a modified version of an appendix which was written for the paper "The numbers of support-tilting modules for a Dynkin algebra" (see arXiv:1403.5827v1

Fakieh, “Strongly principal ideals of rings with involution

Abstract The notion of strongly principal ideal groups for associative rings was introduced in Mathematics Subject Classification: 16W1

The number of support-tilting modules for a Dynkin algebra

A. A. Obaid M, Khalid Nauman S, M. Fakieh W, Ringel CM. The number of support-tilting modules for a Dynkin algebra. Journal of Integer Sequences. 2015;18(10): 15.10.6

The number of complete exceptional sequences for a Dynkin algebra

The Dynkin algebras are the hereditary artin algebras of finite representation type. The paper determines the number of complete exceptional sequences for any Dynkin algebra. Since the complete exceptional sequences for a Dynkin algebra of Dynkin type ∆ correspond bijectively to the maximal chains in the lattice of non-crossing partitions of type ∆, the calculations presented here may also be considered as a categorification of the corresponding result for non-crossing partitions

The number of complete exceptional sequences for a Dynkin algebra.

Obaid M, Nauman K, S. M. Al-Shammakh W, Fakieh W, Ringel CM. The number of complete exceptional sequences for a Dynkin algebra. Colloquium Mathematicum. 2013;133(2):197-210