(joint with Ming Fang)

Defining gendo-symmetric algebras is an attempt to give one answer to three seemingly rather different questions:

(a) Algebraic Lie theory starts with Hopf algebras - group algebras, coordinate rings, enveloping algebras - and then often moves to rather different algebras (finite dimensional of finite global dimension) - Schur algebras, blocks of the BGG category O, etc. What kind of structure do the latter algebras have in common with the former?

(b) Group algebras of finite groups, or more generally symmetric algebras, do have a comultiplication. Is there a larger class of finite dimensional algebras with a comultiplication?

(c) Dominant dimension of an algebra conjecturally characterises self-injective algebras. It also characterises the strength of double centraliser properties such as Schur-Weyl duality. How does it relate to other homological data, which can be read off, for instance, a Hochschild (co)complex?