Dual Bialgebroids for Depth Two Ring Extensions
| dc.creator | Kadison, L. | |
| dc.creator | Szlachanyi, K. | |
| dc.date | 2001-08-09 | |
| dc.date | 2001-11-20 | |
| dc.date.accessioned | 2026-07-07T04:42:56Z | |
| dc.date.available | 2026-07-07T04:42:56Z | |
| dc.description | 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 = C_M(N) and C = End_{N-M}(M ø_N M), via dual bimodules and step two centralizers A = End_NM_N and B = (M ø_N M)^N, in a Jones tower above N --> M. Lu's bialgebroids End_k A' and A' ø_k {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 End_NM when M_N 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. | |
| dc.description | 2 new sections added, 37 pages | |
| dc.identifier | https://arxiv.org/abs/math/0108067 | |
| dc.identifier | http://arxiv.org/abs/math/0108067 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/61998 | |
| dc.subject | Rings and Algebras | |
| dc.subject | Operator Algebras | |
| dc.subject | Quantum Algebra | |
| dc.title | Dual Bialgebroids for Depth Two Ring Extensions | |
| dc.type | text |