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119
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 114 (8 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
Morava Ktheories and localisation
 MEM. AMER. MATH. SOC
, 1999
"... We study the structure of the categories of K(n)local and E(n)local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcategories, and give characterisations of small, dualisable, and K(n)nilpotent spectra ..."
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Cited by 105 (19 self)
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We study the structure of the categories of K(n)local and E(n)local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcategories, and give characterisations of small, dualisable, and K(n)nilpotent spectra. We give a number of useful extensions to the theory of vn self maps of finite spectra, and to the theory of Landweber exactness. We show that certain rings of cohomology operations are left Noetherian, and deduce some powerful finiteness results. We study the Picard group of invertible K(n)local spectra, and the problem of grading homotopy groups over it. We prove (as announced by Hopkins and Gross) that the BrownComenetz dual of MnS lies in the Picard group. We give a detailed analysis of some examples when n =1 or 2, and a list of open problems.
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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Invariants of piecewiselinear 3manifolds
 Trans. Amer. Math. Soc
, 1996
"... Abstract. This paper presents an algebraic framework for constructing invariants of closed oriented 3manifolds by taking a state sum model on a triangulation. This algebraic framework consists of a tensor category with a condition on the duals which we have called a spherical category. A signican ..."
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Cited by 74 (5 self)
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Abstract. This paper presents an algebraic framework for constructing invariants of closed oriented 3manifolds by taking a state sum model on a triangulation. This algebraic framework consists of a tensor category with a condition on the duals which we have called a spherical category. A signicant feature is that the tensor category is not required to be braided. The main examples are constructed from the categories of representations of involutive Hopf algebras and of quantised enveloping algebras at a root of unity. The purpose of this paper is to present an algebraic framework for constructing invariants of closed oriented 3manifolds. The construction is in the spirit of topological quantum eld theory and the invariant is calculated from a triangulation of the 3manifold. The data for the construction of the invariant is a tensor category
Duality for modules over finite rings and applications to coding theory
 AMER. J. MATH
, 1999
"... This paper sets a foundation for the study of linear codes over finite rings. The finite Frobenius rings are singled out as the most appropriate for coding theoretic purposes because two classical theorems of MacWilliams, the extension theorem and the MacWilliams identities, generalize from finite ..."
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Cited by 56 (4 self)
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This paper sets a foundation for the study of linear codes over finite rings. The finite Frobenius rings are singled out as the most appropriate for coding theoretic purposes because two classical theorems of MacWilliams, the extension theorem and the MacWilliams identities, generalize from finite fields to finite Frobenius rings. It is over Frobenius rings that certain key identifications can be made between the ring and its complex characters.
Nondegenerate invariant bilinear forms on non–associative algebras, Preprint Freiburg THEP 92/3, to appear
 Acta Math. Univ. Comenianae
"... Abstract. A bilinear form f on a nonassociative algebra A is said to be invariant iff f(ab, c) = f(a, bc) for all a, b, c ∈ A. Finitedimensional complex semisimple Lie algebras (with their Killing form) and certain associative algebras (with a trace) carry such a structure. We discuss the ideal st ..."
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Cited by 44 (0 self)
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Abstract. A bilinear form f on a nonassociative algebra A is said to be invariant iff f(ab, c) = f(a, bc) for all a, b, c ∈ A. Finitedimensional complex semisimple Lie algebras (with their Killing form) and certain associative algebras (with a trace) carry such a structure. We discuss the ideal structure of A if f is nondegenerate and introduce the notion of T ∗extension of an arbitrary algebra B (i.e. by its dual space B ∗ ) where the natural pairing gives rise to a nondegenerate invariant symmetric bilinear form on A: = B ⊕ B ∗. The T ∗extension involves the third scalar cohomology H3 (B, K) if B is Lie and the second cyclic cohomology HC 2 (B) if B is associative in a natural way. Moreover, we show that every nilpotent finitedimensional algebra A over an algebraically closed field carrying a nondegenerate invariant symmetric bilinear form is a suitable T ∗extension. As a Corollary, we prove that every complex Lie algebra carrying a nondegenerate invariant symmetric bilinear form is always a special type of Manin pair in the sense of Drinfel’d but not always isomorphic to a Manin triple. Examples involving the Heisenberg and filiform Lie algebras (whose third scalar cohomology is computed) are discussed. 1.
Frobenius monads and pseudomonoids
 2CATEGORIES COMPANION 73
, 2004
"... Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalenc ..."
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Cited by 39 (4 self)
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Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to &quot;strongly separable &quot; Frobenius algebras and &quot;weak monoidal Morita equivalence&quot;. Wreath products of Frobenius algebras are discussed.
Mapping class group action on quantum doubles
 Commun. Math. Phys
, 1995
"... Abstract: We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly nonsemisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT’ ..."
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Cited by 36 (3 self)
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Abstract: We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly nonsemisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT’s is circumvented. We find compact formulae for the S ±1matrices using the canonical, non degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford’s relations between the canonical forms and the moduli of integrals. We analyze the projective SL(2,Z)action on the center of Uq(sl2) for q an l = 2m + 1st root of unity. It appears that the 3m + 1dimensional representation decomposes into an m + 1dimensional finite representation and a 2mdimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation of SL(2,Z) and the finite, mdimensional representation, obtained from the truncated TQFT Since the seminal paper of Atiyah [A] on the abstract definition of a topological quantum field theory (TQFT) much progress has been made in finding non trivial examples and
Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry
, 1995
"... Given a finite dimensional C Hopf algebra H and its dual H we construct the infinite crossed product A = : : : ? / H ? / H ? / H ? / : : : and study its superselection sectors. A is the observable algebra of a generalized quantum spin chain with Horder and Hdisorder symmetries, where ..."
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Cited by 19 (5 self)
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Given a finite dimensional C Hopf algebra H and its dual H we construct the infinite crossed product A = : : : ? / H ? / H ? / H ? / : : : and study its superselection sectors. A is the observable algebra of a generalized quantum spin chain with Horder and Hdisorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If H = j CG is a group algebra then A becomes an ordinary Gspin model. We classify all DHRsectors of A  relative to some Haag dual vacuum representation  and prove that their symmetry is described by the Drinfeld double D(H). To achieve this we construct localized coactions ae : A ! A\Omega D(H) and use a certain compressibility property to prove that they are universal amplimorphisms on A. In this way the double D(H) can be recovered from the observable algebra A as a universal cosymmetry. 1 Email: NILL@omega.physik.fuberlin.de Supported by the DFG, SFB 288 "Differentialgeometrie und Quant...
Weak Hopf algebras I. Integral theory and C*structure
 WEAK HOPF ALGEBRAS II. REPRESENTATION THEORY, DIMENSIONS AND THE MARKOV TRACE. PREPRINT MATH.QA/9906045
, 1998
"... We give an introduction to the theory of weak Hopf algebras proposed recently as a coassociative alternative of weak quasiHopf algebras. We follow an axiomatic approach keeping as close as possible to the ”classical ” theory of Hopf algebras. The emphasis is put on the new structure related to the ..."
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Cited by 19 (0 self)
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We give an introduction to the theory of weak Hopf algebras proposed recently as a coassociative alternative of weak quasiHopf algebras. We follow an axiomatic approach keeping as close as possible to the ”classical ” theory of Hopf algebras. The emphasis is put on the new structure related to the presence of canonical subalgebras AL and AR in any weak Hopf algebra A that play the role of noncommutative numbers in many respects. A theory of integrals is developed in which we show how the algebraic properties of A, such as the Frobenius property, or semisimplicity, or innerness of the square of the antipode, are related to the existence of nondegenerate, normalized, or Haar integrals. In case of C∗weak Hopf algebras we prove the existence of a unique Haar measure h ∈ A and of a canonical grouplike element g ∈ A implementing the square of the antipode and factorizing into left and right elements g = gL g −1 R, gL ∈ AL, gR ∈ AR. Further discussion of the C∗case will be presented in Part II.