Gendo-symmetric algebras, comultiplications and homological properties

点击翻译按钮获取中文摘要

Gendo-symmetric algebras are the Morita-Tachikawa correspondents of symmetric algebras. Examples include classical and quantised Schur algebras (with parameters n at least r), blocks of the Bernstein-Gelfand-Gelfand category O of a semisimple complex Lie algebra and Auslander algebras of blocks of cyclic defect of group algebras.In recent joint work with Ming Fang, a comultiplication on gendo-symmetric algebras has been discovered. This comultiplication has been shown to characterise gendo-symmetric algebras and their dominant dimension.The PhD project that is the main part of this proposal aims at investigating this comultiplication and comparing it with known comultiplications on Frobenius algebras and on weak bialgebras. A major aim is to use a comultiplicative analogue of the bar complex to prove Nakayama's conjecture for gendo-symmetric algebras. Another major property of gendo-symmetric algebras is their behaviour under derived equivalences, which is to be taken as a starting point for general results about derived equivalences: Under assumptions to be determined, derived equivalences between gendo-symmetric or more general algebras induce derived equivalences between symmetric centraliser subalgebras, and derived equivalences preserve global or dominant dimension, another unexpected phenomenon. The assumptions needed are expected to be in terms of dominant dimension, which is the key homological ingredient of the whole project.