Mirror-reflective algebras and Tachikawa's second conjecture

TL;DR

This paper introduces mirror-reflective algebras and their reduced forms, establishing their relationships and applications in constructing higher-dimensional algebras and reformulating Tachikawa's second conjecture for symmetric algebras.

Contribution

It develops new procedures for constructing mirror-reflective algebras, links them via recollements, and applies these to systematically build higher-dimensional algebras and reformulate Tachikawa's conjecture.

Findings

Mirror-reflective algebras are constructed from given algebras with idempotents.

These algebras are connected through recollements of derived categories.

The approach provides new methods for higher-dimensional algebra construction and conjecture reformulation.

Abstract

Given an algebra with an idempotent, we introduce two procedures to construct families of new algebras, termed mirror-reflective algebras and reduced mirror-reflective algebras. We then establish connections among these algebras by recollements of derived module categories. In case of given algebras being gendo-symmetric, we show that the (reduced) mirror-reflective algebras are symmetric and provide new methods to construct systematically both higher dimensional (minimal) Auslander-Gorenstein algebras and gendo-symmetric algebras of higher dominant dimensions. This leads to a new formulation of Tachikawa's second conjecture for symmetric algebras in terms of idempotent stratifications.