Hopf Algebra Actions on Strongly Separable Extensions of Depth Two

By Lars Kadison, D. Nikshych. 2001

52 citations recorded by the source corpus.

Abstract We bring together ideas in analysis on Hopf *-algebra actions on II 1 subfactors of finite Jones index [9, 24] and algebraic characterizations of Frobenius, Galois and cleft Hopf extensions [3, 13, 14] to prove a non-commutative algebraic analogue of the classical theorem: a finite degree field extension is Galois iff it is separable and normal. Suppose N ↪ M is a separable Frobenius extension of k -algebras with trivial centralizer C M ( N ) and split as N -bimodules. Let M 1 ≔End( M N ) and M 2 ≔End( M 1 ) M be the endomorphism algebras in the Jones tower N ↪ M ↪ M 1 ↪ M 2 . We place depth 2 conditions on its second centralizers A ≔ C M 1 ( N ) and B ≔ C M 2 ( M ). We prove that A and B are semisimple Hopf algebras dual to one another, that M 1 is a smash product of M and A , and that M is a B -Galois extension of N .

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d636066ed97c0f478db25a72d3c3a2c3095cfbc7

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In Algebra and Number Theory

1. Korn'el Szlach'anyi
2. Lars Kadison
3. G. Bòhm
4. Ross Street
5. V. Ostrik
6. Brian Day

In Geometry and Topology

1. Korn'el Szlach'anyi
2. Lars Kadison
3. G. Bòhm
4. Ross Street
5. V. Ostrik
6. Brian Day

In Mathematical Physics

1. Korn'el Szlach'anyi
2. Lars Kadison
3. G. Bòhm
4. Ross Street
5. V. Ostrik
6. Brian Day

Citations in the corpus

• Bialgebroid actions on depth two extensions and duality (2003, methodology): “Nikshych and the first author [21] extend the depth two notion and results of [43] to an algebraic analog without trace: certain semisimple Hopf algebras are shown to have a Galois action on split separable Frobenius extensions with trivial centralizer.”