Pure Maths Colloquium: A Gentle Introduction to Auslander-Retien Theory
27 November 2017 16:00 in CM221
Adachi-Iyama-Reiten recently introduced the notion of tau-tilting theory, a generalization of the classical tilting theory, which heavily relies on the Auslander-Reiten theory. This generalization, as its precursor, soon found many applications and established new connections between different areas, including representation theory of algebras, geometry, combinatorics, lattice theory, etc. String and gentle algebras, however, have received a lot of attention in the past few decades. In particular, the work of Butler-Ringel (in 1987) bolstered their phenomenal growth. Due to their fruitful combinatorics, providing a tractable framework and including a vast family of algebras, gentle algebras have prominent applications in diverse areas, including representation theory, cluster algebra, symplectic geometry, etc.
In this talk, after a short explanation of the above-mentioned topics, we will look at some interactions between them to show how one sheds light on the study of the other. Recapping some preliminaries in the first half, in the remainder I will introduce a canonical method to embed every arbitrary gentle algebra into a well-behaved gentle algebra, where we can find an explicit combinatorial description of Hom(M,\tau(N)) and Ext(N,M), for each pair of string modules M and N. If time permits, we will see some aspects of \tau-tilting theory and further use our results to describe the lattice of (functorially finite) torsion classes of gentle algebras by means of kisses between strings and also analyze the notion of mutation in terms of concrete combinatorics.
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