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Given an algebra with an idempotent, we introduce two procedures to constructfamilies of new algebras, termed mirror-reflective algebras and reducedmirror-reflective algebras. We then establish connections among these algebrasby recollements of derived module categories. In case of given algebras beinggendo-symmetric, we show that the (reduced) mirror-reflective algebras aresymmetric and provide new methods to construct systematically both higherdimensional (minimal) Auslander-Gorenstein algebras and gendo-symmetricalgebras of higher dominant dimensions. This leads to a new formulation ofTachikawa's second conjecture for symmetric algebras in terms of idempotentstratifications.

Mirror-reflective algebras and Tachikawa's second conjecture