Mirror-reflective algebras and Tachikawa's second conjecture

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.