A bocs theoretic characterization of gendo-symmetric algebras

Gendo-symmetric algebras were recently introduced by Fang and König in [7]. An algebra is called gendo-symmetric in case it is isomorphic to the endomorphism ring of a generator over a finite dimensional symmetric algebra. We show that a finite dimensional algebra A over a field K   is gendo-symmetric if and only if there is a bocs-structure on (A,D(A))(A,D(A)), where D=HomK(−,K)D=HomK(−,K) is the natural duality. Assuming that A   is gendo-symmetric, we show that the module category of the bocs (A,D(A))(A,D(A)) is equivalent to the module category of the algebra eAe, when e is an idempotent such that eA is the unique minimal faithful projective-injective right A-module. We also prove some new results about gendo-symmetric algebras using the theory of bocses.