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45
From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
 J. Pure Appl. Alg
, 2003
"... We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground f ..."
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Cited by 52 (6 self)
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We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = FVect, where F is a field. An object X ∈ A with twosided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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Hopf algebra actions on strongly separable extensions of depth two
 Adv. in Math. 163
, 2001
"... Abstract. We bring together ideas in analysis of Hopf ∗algebra actions on II1 subfactors of finite Jones index [9, 24] and algebraic characterizations of Frobenius, Galois and cleft Hopf extensions [14, 13, 3] to prove a noncommutative algebraic analogue of the classical theorem: a finite field ex ..."
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Cited by 21 (20 self)
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Abstract. We bring together ideas in analysis of Hopf ∗algebra actions on II1 subfactors of finite Jones index [9, 24] and algebraic characterizations of Frobenius, Galois and cleft Hopf extensions [14, 13, 3] to prove a noncommutative algebraic analogue of the classical theorem: a finite field extension is Galois iff it is separable and normal. Suppose N ֒ → M is a separable Frobenius extension of kalgebras split as Nbimodules with a trivial centralizer CM(N). Let M1: = End(MN) and M2: = End(M1)M be the endomorphism algebras in the Jones tower N ֒ → M ֒ → M1 ֒ → M2. We show that under depth 2 conditions on the second centralizers A: = CM1 (N) and B: = CM2 (M) the algebras A and B are semisimple Hopf algebras dual to one another and such that M1 is a smash product of M and A, and that M is a BGalois extension of N. 1.
On comatrix corings and bimodules
, 2008
"... To any bimodule which is finitely generated and projective on one side one can associate a coring, known as a comatrix coring. A new description of comatrix corings in terms of data reminiscent of a Morita context is given. It is also studied how properties of bimodules are reflected in the associat ..."
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Cited by 10 (2 self)
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To any bimodule which is finitely generated and projective on one side one can associate a coring, known as a comatrix coring. A new description of comatrix corings in terms of data reminiscent of a Morita context is given. It is also studied how properties of bimodules are reflected in the associated comatrix corings. In particular it is shown that separable bimodules give rise to coseparable comatrix corings, while Frobenius bimodules induce Frobenius comatrix corings.
An Approach to Hopf Algebras via Frobenius Coordinates
, 1999
"... In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in Section 3. In Section 4 we study the Frobenius structure of a ..."
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Cited by 9 (4 self)
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In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in Section 3. In Section 4 we study the Frobenius structure of an FHalgebra H [25] and extend two recent theorems in [8]. We obtain two Radford formulas for the antipode in H and generalize in Section 7 the results on its order in [10]. We study the Frobenius structure on an FHsubalgebra pair in Sections 5 and 6. In Section 8 we show that the quantum double of H is symmetric and unimodular.
Galois theory for bialgebroids, depth two and normal Hopf subalgebras
"... Abstract. We reduce certain proofs in [16, 11, 12] to depth two quasibases from one side only, a minimalistic approach which leads to a characterization of Galois extensions for finite projective bialgebroids without the Frobenius extension property. We prove that a proper algebra extension is a lef ..."
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Cited by 9 (6 self)
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Abstract. We reduce certain proofs in [16, 11, 12] to depth two quasibases from one side only, a minimalistic approach which leads to a characterization of Galois extensions for finite projective bialgebroids without the Frobenius extension property. We prove that a proper algebra extension is a left TGalois extension for some right finite projective left bialgebroid over some algebra R if and only if it is a left depth two and left balanced extension. Exchanging left and right in this statement, we have a characterization of right Galois extensions for left finite projective right bialgebroids. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a HopfGalois extension. We characterize finite weak HopfGalois extensions using an alternate Galois canonical mapping with several corollaries: that these are depth two and that surjectivity of the Galois mapping implies its bijectivity.
Frobenius extensions and weak Hopf algebras
 J. Algebra
"... Abstract. We study a symmetric Markov extension of kalgebras N ֒ → M, a certain kind of Frobenius extension with conditional expectation that is tracial on the centralizer and dual bases with a separability property. We place a depth two condition on this extension, which is essentially the require ..."
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Cited by 8 (7 self)
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Abstract. We study a symmetric Markov extension of kalgebras N ֒ → M, a certain kind of Frobenius extension with conditional expectation that is tracial on the centralizer and dual bases with a separability property. We place a depth two condition on this extension, which is essentially the requirement that the Jones tower N ֒ → M ֒ → M1 ֒ → M2 can be obtained by taking relative tensor products with centralizers A = CM1 (N) and B = CM2 (M). Under this condition, we prove that N ֒ → M is the invariant subalgebra pair of a weak Hopf algebra action by A, i.e., that N = MA. The endomorphism algebra M1 = EndNM is shown to be isomorphic to the smash product algebra M#A. We also extend results of Szymański [26], Vainerman and the second author [18], and the authors [11].
LarsonSweedler theorem and the role of grouplike elements in weak Hopf algebras
 QA/0111045. Matematiska Institutionen, Göteborg University, S412 96 Göteborg, Sweden Email address: lkadison@c2i.net
"... We extend the Larson–Sweedler theorem [10] to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a nondegenerate left integral. We show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and i ..."
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Cited by 6 (0 self)
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We extend the Larson–Sweedler theorem [10] to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a nondegenerate left integral. We show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and invertible modules. We also reveal the connection of invertible modules to left and right grouplike elements in the dual weak Hopf algebra. Defining distinguished left and right grouplike elements we derive the Radford formula [15] for the fourth power of the antipode in a weak Hopf algebra and prove that the order of the antipode is finite up to an inner automorphism by a grouplike element in the trivial subalgebra A T of the underlying weak Hopf algebra A.
TOWERS OF CORINGS
, 2002
"... The notion of a Frobenius coring is introduced, and it is shown that any such coring produces a tower of Frobenius corings and Frobenius extensions. ..."
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Cited by 5 (1 self)
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The notion of a Frobenius coring is introduced, and it is shown that any such coring produces a tower of Frobenius corings and Frobenius extensions.
ANCHOR MAPS AND STABLE MODULES IN DEPTH TWO
, 2006
"... Abstract. An algebra extension A  B is right depth two if its tensorsquare A ⊗B A is in the Dress category Add BAA. We consider necessary conditions for right, similarly left, D2 extensions in terms of partial Ainvariance of twosided ideals in A contracted to the centralizer. Finite dimensional a ..."
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Cited by 4 (4 self)
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Abstract. An algebra extension A  B is right depth two if its tensorsquare A ⊗B A is in the Dress category Add BAA. We consider necessary conditions for right, similarly left, D2 extensions in terms of partial Ainvariance of twosided ideals in A contracted to the centralizer. Finite dimensional algebras extending central simple algebras are shown to be depth two. Following P. Xu, left and right bialgebroids over a base algebra R may be defined in terms of anchor maps, or representations on R. The anchor maps for the bialgebroids S = End BAB and T = End AA ⊗B AA over the centralizer R = CA(B) are the modules SR and RT studied in [11, 15, 8], which provide information about the bialgebroids and the extension [9]. The anchor maps for the Hopf algebroids in [17, 10] reverse the order of right multiplication and action by a Hopf algebra element, and lift to the isomorphism in [20]. We sketch a theory of stable Amodules and their endomorphism rings and generalize the smash product decomposition in [7, Prop. 1.1] to any Amodule. We observe that Schneider’s coGalois theory in [21] provides examples of codepth two, such as the quotient epimorphism of a finite dimensional normal Hopf subalgebra. 1. Introduction and