Gendo-symmetric algebras, comultiplications and homological properties

gendo对称代数是对称代数的森田-立川对应。例子包括经典和量子化Schur代数（参数n至少为r），半简单复李代数的Bernstein-Gelfand-Gelfand范畴O的块，群代数的循环缺陷块的Auslander代数。在最近与方明的合作中，发现了一个关于性对称代数的乘法。这种乘法已被证明是性别对称代数及其主导维数的特征。该博士项目是该提案的主要部分，旨在研究该乘法，并将其与已知的Frobenius代数和弱双代数上的乘法进行比较。一个主要的目的是利用棒状复合体的乘法模拟来证明Nakayama猜想对于geno对称代数。属对称代数的另一个主要性质是它们在派生等价下的行为，这是关于派生等价的一般结果的起点：在待确定的假设下，属对称或更一般代数之间的派生等价导致对称中心化子代数之间的派生等价，派生等价保持全局或主导维数，这是另一个意想不到的现象。所需的假设预计是在主导维度方面，这是整个项目的关键同源成分。

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.