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118
Constructing the extended Haagerup planar algebra
, 2009
"... We construct a subfactor planar algebra, and as a corollary a subfactor, with the ‘extended Haagerup´ principal graph pair. This is the last open case from Haagerup’s 1993 list of potential principal graphs of subfactors with index in the range (4, 3 + √ 3). We prove that the subfactor planar algeb ..."
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Cited by 43 (16 self)
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We construct a subfactor planar algebra, and as a corollary a subfactor, with the ‘extended Haagerup´ principal graph pair. This is the last open case from Haagerup’s 1993 list of potential principal graphs of subfactors with index in the range (4, 3 + √ 3). We prove that the subfactor planar algebra with these principal graphs is unique. We give a skein theoretic description, and a description as a subalgebra generated by a certain element in the graph planar algebra of its principal graph. We give an explicit algorithm for evaluating closed diagrams using the skein theoretic description. This evaluation algorithm is unusual because intermediate steps may increase the number of generators in a diagram.
Classification of twodimensional local conformal nets with c < 1 and 2cohomology vanishing for tensor categories
 Commun. Math. Phys
, 2004
"... We classify twodimensional local conformal nets with parity symmetry and central charge less than 1, up to isomorphism. The maximal ones are in a bijective correspondence with the pairs of ADE Dynkin diagrams with the difference of their Coxeter numbers equal to 1. In our previous classification ..."
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Cited by 37 (16 self)
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We classify twodimensional local conformal nets with parity symmetry and central charge less than 1, up to isomorphism. The maximal ones are in a bijective correspondence with the pairs of ADE Dynkin diagrams with the difference of their Coxeter numbers equal to 1. In our previous classification of onedimensional local conformal nets, Dynkin diagrams D2n+1 and E7 do not appear, but now they do appear in this classification of twodimensional local conformal nets. Such nets are also characterized as twodimensional local conformal nets with µindex equal to 1 and central charge less than 1. Our main tool, in addition to our previous classification results for onedimensional nets, is 2cohomology vanishing for certain tensor categories related to the Virasoro tensor categories with central charge less than 1.
Quantum automorphism groups of homogeneous graphs
 J. Funct. Anal
"... Let X be a finite graph, with edges colored and possibly oriented, such that an oriented edge and a nonoriented one cannot have same color. The universal Hopf algebra H(X) coacting on X is in general non commutative, infinite dimensional, bigger than the algebra of functions on the usual symmetry g ..."
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Cited by 35 (14 self)
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Let X be a finite graph, with edges colored and possibly oriented, such that an oriented edge and a nonoriented one cannot have same color. The universal Hopf algebra H(X) coacting on X is in general non commutative, infinite dimensional, bigger than the algebra of functions on the usual symmetry group G(X). For a graph with no edges Tannakian duality makes H(X) correspond to a TemperleyLieb algebra. We study some versions of this correspondence.
A theory of dimension
 KTHEORY
, 1997
"... In which a theory of dimension related to the Jones index and based on the notion of conjugation is developed. An elementary proof of the additivity and multiplicativity of the dimension is given and there is an associated trace. Applications are given to a class of endomorphisms of factors and to ..."
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Cited by 32 (1 self)
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In which a theory of dimension related to the Jones index and based on the notion of conjugation is developed. An elementary proof of the additivity and multiplicativity of the dimension is given and there is an associated trace. Applications are given to a class of endomorphisms of factors and to the theory of subfactors. An important role is played by a notion of amenability inspired by the work of Popa.
Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The comp ..."
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Cited by 30 (9 self)
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We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
A primer of Hopf algebras
 Insitut des Hautes Études Scientifiques, IHES/M/06/04
, 2006
"... Summary. In this paper, we review a number of basic results about socalled Hopf algebras. We begin by giving a historical account of the results obtained in the 1930’s and 1940’s about the topology of Lie groups and compact symmetric spaces. The climax is provided by the structure theorems due to H ..."
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Cited by 30 (0 self)
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Summary. In this paper, we review a number of basic results about socalled Hopf algebras. We begin by giving a historical account of the results obtained in the 1930’s and 1940’s about the topology of Lie groups and compact symmetric spaces. The climax is provided by the structure theorems due to Hopf, Samelson, Leray and Borel. The main part of this paper is a thorough analysis of the relations between Hopf algebras and Lie groups (or algebraic groups). We emphasize especially the category of unipotent (and prounipotent) algebraic groups, in connection with MilnorMoore’s theorem. These methods are a powerful tool to show that some algebras are free polynomial rings. The last part is an introduction to the combinatorial aspects of polylogarithm functions and the corresponding multiple zeta values. 1 Introduction.............................................
Canonical tensor product subfactors
"... Canonical tensor product subfactors (CTPS’s) describe, among other things, the embedding of chiral observables in twodimensional conformal quantum field theories. A new class of CTPS’s is constructed some of which are associated with certain modular invariants, thereby establishing the expected exi ..."
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Cited by 27 (10 self)
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Canonical tensor product subfactors (CTPS’s) describe, among other things, the embedding of chiral observables in twodimensional conformal quantum field theories. A new class of CTPS’s is constructed some of which are associated with certain modular invariants, thereby establishing the expected existence of the corresponding twodimensional theories. 1 Introduction and
On Galois extensions of braided tensor categories, mapping class group representations and simple current extensions
 In preparation
"... We show that the author’s notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed Gcategories, recently introduced for the purposes of 3manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G non ..."
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Cited by 26 (5 self)
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We show that the author’s notion of Galois extensions of braided tensor categories [22], see also [3], gives rise to braided crossed Gcategories, recently introduced for the purposes of 3manifold topology [31]. The Galois extensions C ⋊ S are studied in detail, and we determine for which g ∈ G nontrivial objects of grade g exist in C ⋊ S. 1
The monoidal EilenbergMoore construction and bialgebroids
"... Abstract. Monoidal functors U: C → M with left adjoints determine, in a universal way, monoids T in the category of oplax monoidal endofunctors on ”quantum groupoids ” we derive Tannaka duality between left adjointable monoidal functors and bimonads. Bialgebroids, i.e., Takeuchi’s ×Rbialgebras, app ..."
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Cited by 22 (3 self)
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Abstract. Monoidal functors U: C → M with left adjoints determine, in a universal way, monoids T in the category of oplax monoidal endofunctors on ”quantum groupoids ” we derive Tannaka duality between left adjointable monoidal functors and bimonads. Bialgebroids, i.e., Takeuchi’s ×Rbialgebras, appear as the special case when T has also a right adjoint. Street’s 2category of monads then leads to a natural definition of the 2category of bialgebroids. Contents