2019/03/19 by Changchang Xi, Xi, Changchang · 2 citations
Mathematics · #16E10 #16S50 #17B35 #18G20 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Primary 16D20 #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Secondary 17B37
paper · pdf · doi:10.48550/arxiv.1903.07921
openalex publication_date 2019/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Martinez-Villa to the Auslander-Reiten conjecture on stable equivalences. We show that the Frobenius parts of Frobenius extensions are again Frobenius extensions. Further, let A and B be finite-dimensional algebras over a field k, and let \dm(AX) stand for the dominant dimension of an A-module X. If BMA is a Frobenius bimodule, then \dm(A)≤ \dm(BM) and \dm(B)≤ \dm(A\HomB(M, B)). In particular, if B⊆ A is a left-split (or right-split) Frobenius extension, then \dm(A)=\dm(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. Finally, we prove that the universal (quantised) enveloping algebras of semisimple Lie algebras are QF-3 rings in the sense of Morita.