Frobenius bimodules and flat-dominant dimensions  

作　　者：Changchang Xi 

机构地区：[1]School of Mathematical Sciences,Capital Normal University,Beijing,100048,China

出　　处：《Science China Mathematics》2021年第1期33-44,共12页中国科学：数学（英文版）

基　　金：supported by the Beijing Natural Science Foundation(Grant No.1192004);National Natural Science Foundation of China(Grant No.11331006)。

摘　　要：We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules.This is motivated by the Nakayama conjecture and an approach of MartinezVilla to the Auslander-Reiten conjecture on stable equivalences.We show that the Frobenius parts of Frobenius extensions are again Frobenius extensions.Furthermore,let A and B be finite-dimensional algebras over a field k,and let domdim(_AX)stand for the dominant dimension of an A-module X.If_BM_A is a Frobenius bimodule,then domdim(A)domdim(_BM)and domdim(B)domdim(_AHom_B(M,B)).In particular,if B■A is a left-split(or right-split)Frobenius extension,then domdim(A)=domdim(B).These results are applied to calculate flat-dominant dimensions of a number of algebras:shew group algebras,stably equivalent algebras,trivial extensions and Markov extensions.We also prove that the universal(quantised)enveloping algebras of semisimple Lie algebras are QF-3 rings in the sense of Morita.

关 键 词：dominant dimension extension of algebras Frobenius bimodule Frobenius part universal enveloping algebra 

分 类 号：O15[理学—数学]