2001/08/09 by Lars Kadison, L. Kadison, Kadison, L. +3
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA) #Quantum Algebra (math.QA) #Rings and Algebras (math.RA) #math.OA #math.QA #math.RA
paper · pdf · doi:10.48550/arxiv.math/0108067
2 new sections added, 37 pages
openalex publication_date 2001/08/09 · arxiv created 2001/11/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a general notion of depth two for ring homomorphism N --> M, and derive Morita equivalence of the step one and three centralizers, R = CM(N) and C = EndN-M(M øN M), via dual bimodules and step two centralizers A = EndNMN and B = (M øN M)N, in a Jones tower above N --> M. Lu's bialgebroids Endk A' and A' øk A'op over a k-algebra A' are generalized to left and right bialgebroids A and B with B the R-dual bialgebroid of A. We introduce Galois-type actions of A on M and B on EndNM when MN is a balanced module. In the case of Frobenius extensions M | N, we prove an endomorphism ring theorem for depth two. Further in the case of irreducible extensions, we extend previous results on Hopf algebra and weak Hopf algebra actions in subfactor theory [Szymanski, Nikshych-Vainerman] and its generalizations [Kadison-Nikshych: RA/0107064, RA/0102010] by methods other than nondegenerate pairing. As a result, we have concrete expressions for the Hopf or weak Hopf algebra structures on the step two centralizers. Semisimplicity of B is equivalent to separability of the extension M | N. In the presence of depth two, we show that biseparable extensions are QF.