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16
Noncommutative symmetric functions VI. Free quasisymmetric functions and related algebras
 Internat. J. Algebra Comput
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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 54 (7 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
Adjointable monoidal functors and quantum groupoids, Hopf algebras in noncommutative geometry and physics
 Lecture Notes in Pure and
"... Abstract. Every monoidal functor G: C → M has a canonical factorization through the category RMR of bimodules in M over some monoid R in M in which the factor U: C → RMR is strongly unital. Using this result and the characterization of the forgetful functors MA → RMR of bialgebroids A over R given b ..."
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Cited by 13 (2 self)
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Abstract. Every monoidal functor G: C → M has a canonical factorization through the category RMR of bimodules in M over some monoid R in M in which the factor U: C → RMR is strongly unital. Using this result and the characterization of the forgetful functors MA → RMR of bialgebroids A over R given by Schauenburg [15] together with their bimonad description given by the author in [18] here we characterize the ”long ” forgetful functors MA → RMR → M of both bialgebroids and weak bialgebras. 1.
COTENSOR PRODUCTS OF MODULES
, 2002
"... Abstract. Let C beacoalgebraoverafieldkand A its dual algebra. The category of Ccomodules is equivalent to a category of Amodules. We use this to interpret the cotensor product M□N of two comodules in terms of the appropriate Hochschild cohomology of the Abimodule M ⊗ N, whenAis finitedimensiona ..."
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Cited by 5 (0 self)
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Abstract. Let C beacoalgebraoverafieldkand A its dual algebra. The category of Ccomodules is equivalent to a category of Amodules. We use this to interpret the cotensor product M□N of two comodules in terms of the appropriate Hochschild cohomology of the Abimodule M ⊗ N, whenAis finitedimensional, profinite, graded or differentialgraded. The main applications are to Galois cohomology, comodules over the Steenrod algebra, and the homology of induced fibrations. 1.
Frobenius Rational Loop Algebra
, 2005
"... Recently R. Cohen and V. Godin have proved that the loop homology IH∗(LM; Ik) of a closed oriented manifold M with coefficients in a field Ik has the structure of a unital Frobenius algebra without counit. In this paper we describe explicitly the dual of the coproduct H∗(LM; Q) → (H∗(LM; Q) ⊗ H∗(L ..."
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Cited by 2 (1 self)
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Recently R. Cohen and V. Godin have proved that the loop homology IH∗(LM; Ik) of a closed oriented manifold M with coefficients in a field Ik has the structure of a unital Frobenius algebra without counit. In this paper we describe explicitly the dual of the coproduct H∗(LM; Q) → (H∗(LM; Q) ⊗ H∗(LM; Q)) ∗−m and prove it respects the Hodge decomposition.
The double algebraic view of finite quantum groupoids
 Journal of Algebra
"... Abstract. Double algebra is the structure modelled by the properties of the ordinary and the convolution product in Hopf algebras, weak Hopf algebras and Hopf algebroids if a Frobenius integral is given. The Hopf algebroids possessing a Frobenius integral are precisely the Frobenius double algebras ..."
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Cited by 2 (1 self)
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Abstract. Double algebra is the structure modelled by the properties of the ordinary and the convolution product in Hopf algebras, weak Hopf algebras and Hopf algebroids if a Frobenius integral is given. The Hopf algebroids possessing a Frobenius integral are precisely the Frobenius double algebras in which the two multiplications satisfy distributivity. The double algebra approach makes it manifest that all comultiplications in such measured Hopf algebroids are of the AbramsKadison type, i.e., they come from a Frobenius algebra structure in some bimodule category. Antipodes for double algebras correspond to the ConnesMoscovici ‘deformed ’ antipode as we show by discussing Hopf and weak Hopf algebras from the double algebraic point of view. Frobenius algebra extensions provide further examples that need not be distributive. 1.
STRING TOPOLOGY OF CLASSIFYING SPACES
, 2007
"... Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG: = map(S 1, BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H∗(LBG) of this loop space. We ..."
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Cited by 2 (0 self)
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Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG: = map(S 1, BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H∗(LBG) of this loop space. We prove that when taken with coefficients in a field the homology of LBG is a homological conformal field theory. As a byproduct of our main theorem, we get on the cohomology H ∗ (LBG) a BValgebra structure.
A Note on \Modules, Comodules, and Cotensor Products over Frobenius Algebras "
"... coordinates developed recently by Kadison, one has a direct proof of Abrams ’ characterization for Frobenius algebras in terms of comultiplication (see L. Kadison (1999)). For any Frobenius algebra, by using the explicit comultiplication, the explicit correspondence ..."
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coordinates developed recently by Kadison, one has a direct proof of Abrams ’ characterization for Frobenius algebras in terms of comultiplication (see L. Kadison (1999)). For any Frobenius algebra, by using the explicit comultiplication, the explicit correspondence
QuasiFrobenius Rational Loop Algebra
, 2004
"... Recently R. Cohen and V. Godin have proved that the loop homology IH∗(LM) of a closed oriented manifold M with coefficients in a field Ik has the structure of a Frobenius algebra without counit. In this short note we prove that when the characteristic of Ik is zero and if M is 1connected then this ..."
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Recently R. Cohen and V. Godin have proved that the loop homology IH∗(LM) of a closed oriented manifold M with coefficients in a field Ik has the structure of a Frobenius algebra without counit. In this short note we prove that when the characteristic of Ik is zero and if M is 1connected then this algebraic structure depends only on the rational homotopty type of M. We describe explicitly the dual coproduct