A bocs theoretic characterization of gendo-symmetric algebras

Abstract

Gendo-symmetric algebras were recently introduced by Fang and König in [FanKoe]. 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)), where D = Hom K (−, K) is the natural duality. Assuming that A is gendosymmetric, we show that the module category of the bocs (A, D(A)) is isomorphic 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.2010 Mathematics Subject Classification. Primary 16G10, 16E10.

A bimodule approach to dominant dimension

“…For more on gendo-symmetric algebras we refer to [FanKoe] and [Mar2]. M is called an n-th syzygy module in case M ∼ = Ω n (N ) for some other A-module N .…”

Upper bounds for the dominant dimension of Nakayama and related algebras

“…A finite dimensional algebra B is called a Morita algebra, if it is isomorphic to the endomorphism ring of a module M , which is a generator of a selfinjective algebra A (see [KerYam]). If A is even symmetric, then B is called a gendo-symmetric algebra (see [FanKoe] and [Mar2] for other characterisations).…”

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“…In the meantime more attention is paid to coalgebraic aspects of finite dimensional algebras. For this we refer to the recent paper [13] by R. Marczinzik and the references given there. 7.4.…”