Infinite index subalgebras of depth two
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- by Lars Kadison PDF
- Proc. Amer. Math. Soc. 136 (2008), 1523-1532 Request permission
Abstract:
An algebra extension $A | B$ 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 $A | B$ is right depth two and right balanced if and only if $A | \ B$ is $T$-Galois with respect to left projective, right $R$-bialgebroid $T$.References
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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
- Received by editor(s): December 1, 2006
- Published electronically: January 17, 2008
- Communicated by: Martin Lorenz
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 1523-1532
- MSC (2000): Primary 16W30; Secondary 46L37, 81R50
- DOI: https://doi.org/10.1090/S0002-9939-08-09077-1
- MathSciNet review: 2373579