Tiago Cruz "Relative Auslander pairs and a new higher Auslander correspondence"

A famous result in representation theory is the Auslander’s correspondence which connects finite-dimensional algebras of finite representation-type with Auslander algebras. Over the years, many generalisations of Auslander algebras have been proposed: for instance n-Auslander algebras (by Iyama), n-minimal Auslander–Gorenstein algebras (by Iyama and Solberg), among others. All of the concepts above require the existence of a faithful projective-injective module and use classical dominant dimension. Now replace the faithful projective-injective module with a self-orthogonal module and classical dominant dimension with relative dominant dimension with respect to a module and you get a relative Auslander-Gorenstein pair.

In this talk, we introduce Auslander pairs and how they fit in a higher dimensional generalisation of the Auslander correspondence. Further, we illustrate that some q-Schur algebras are precisely the homological counterparts of infinite global dimension Temperley-Lieb algebras under a new higher Auslander correspondence. This talk is based on recent joint works with C. Psaroudakis and K. Erdmann.