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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Infinite index subalgebras of depth two


Author: Lars Kadison
Journal: Proc. Amer. Math. Soc. 136 (2008), 1523-1532
MSC (2000): Primary 16W30; Secondary 46L37, 81R50
Published electronically: January 17, 2008
MathSciNet review: 2373579
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Abstract: An algebra extension $ A\, \vert\, B$ is right depth two in this paper if its tensor-square is $ A$-$ B$-isomorphic to a direct summand of any (not necessarily finite) direct sum of $ A$ with itself. For example, normal subgroups of infinite groups, infinitely generated Hopf-Galois extensions and infinite-dimensional algebras are depth two in this extended sense. The added generality loses some duality results obtained in the finite theory (Kadison and Szlachányi, 2003) but extends the main theorem of depth two theory, as for example in (Kadison and Nikshych, 2001). That is, a right depth two extension has right bialgebroid $ T = (A \otimes_B A)^B$ over its centralizer $ R = C_A(B)$. The main theorem: An extension $ A\, \vert\, B$ is right depth two and right balanced if and only if $ A\, \vert\, B$ is $ T$-Galois with respect to left projective, right $ R$-bialgebroid $ T$.


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Additional Information

Lars Kadison
Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104-6395
Email: lkadison@math.upenn.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09077-1
PII: S 0002-9939(08)09077-1
Received by editor(s): December 1, 2006
Published electronically: January 17, 2008
Communicated by: Martin Lorenz
Article copyright: © Copyright 2008 American Mathematical Society