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A FINITENESS PROPERTY FOR BRAIDED FUSION CATEGORIES
"... Abstract. We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations ..."
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Abstract. We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has property F if the associated braid group representations factor over a finite group, and suggest that categories of integral FrobeniusPerron dimension are precisely those with property F. 1.
BRAID REPRESENTATIONS FROM QUANTUM GROUPS OF EXCEPTIONAL LIE TYPE
"... Abstract. We study the problem of determining if the braid group representations obtained from quantum groups of types E, F and G at roots of unity have infinite image or not. In particular we show that when the fusion categories associated with these quantum groups are not weakly integral, the brai ..."
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Abstract. We study the problem of determining if the braid group representations obtained from quantum groups of types E, F and G at roots of unity have infinite image or not. In particular we show that when the fusion categories associated with these quantum groups are not weakly integral, the braid group images are infinite. This provides further evidence for a recent conjecture that weak integrality is necessary and sufficient for the braid group representations associated with any braided fusion category to have finite image. 1.
Formal proof, computation, and the construction problem in algebraic geometry
"... It has become a classical technique to turn to theoretical computer science to provide computational tools for algebraic geometry. A more recent transformation is that now we also get logical tools, and these too should be useful in the study of algebraic varieties. The purpose of this note is to co ..."
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It has become a classical technique to turn to theoretical computer science to provide computational tools for algebraic geometry. A more recent transformation is that now we also get logical tools, and these too should be useful in the study of algebraic varieties. The purpose of this note is to consider a very small part of this picture, and try to motivate the study of computer theoremproving techniques by looking at how they might be relevant to a particular class of problems in algebraic geometry. This is only an informal discussion, based more on questions and possible research directions than on actual results. This note amplifies the themes discussed in my talk at the “Arithmetic and Differential Galois Groups ” conference (March 2004, Luminy), although many specific points in the discussion were only finished more recently. I would like to thank: André Hirschowitz and Marco Maggesi, for their invaluable insights about computerformalized mathematics as it relates
ON THE CLASSIFICATION OF THE GROTHENDIECK RINGS OF NONSELFDUAL MODULAR CATEGORIES
"... Abstract. We develop a symbolic computational approach to classifying lowrank modular fusion categories, up to finite ambiguity. By a generalized form of Ocneanu rigidity due to Etingof, Ostrik and Nikshych, it is enough to classify modular fusion algebras of a given rank–that is, to determine the p ..."
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Abstract. We develop a symbolic computational approach to classifying lowrank modular fusion categories, up to finite ambiguity. By a generalized form of Ocneanu rigidity due to Etingof, Ostrik and Nikshych, it is enough to classify modular fusion algebras of a given rank–that is, to determine the possible Grothendieck rings with modular realizations. We use this technique to classify modular categories of rank at most 5 that are nonselfdual, i.e. those for which some object is not isomorphic to its dual object. 1.
A QUATERNIONIC BRAID REPRESENTATION (AFTER GOLDSCHMIDT AND JONES)
"... Abstract. We show that the braid group representations associated with the (3, 6)quotients of the Hecke algebras factor over a finite group. This was known to experts going back to the 1980s, but a proof has never appeared in print. Our proof uses an unpublished quaternionic representation of the b ..."
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Abstract. We show that the braid group representations associated with the (3, 6)quotients of the Hecke algebras factor over a finite group. This was known to experts going back to the 1980s, but a proof has never appeared in print. Our proof uses an unpublished quaternionic representation of the braid group due to Goldschmidt and Jones. Possible topological and categorical generalizations are discussed. 1.
LOCALIZATION OF UNITARY BRAID GROUP REPRESENTATIONS
"... Abstract. Governed by locality, we explore a connection between unitary braid group representations associated to a unitary Rmatrix and to a simple object in a unitary braided fusion category. Unitary Rmatrices, namely unitary solutions to the YangBaxter equation, afford explicitly local unitary ..."
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Abstract. Governed by locality, we explore a connection between unitary braid group representations associated to a unitary Rmatrix and to a simple object in a unitary braided fusion category. Unitary Rmatrices, namely unitary solutions to the YangBaxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary Rmatrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q = e πi/6 specialization of the unitary Jones representation of the braid groups can be localized by a unitary 9 × 9 Rmatrix. Actually this Jones representation is the first one in a family of theories (SO(N), 2) for an odd prime N> 1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science. 1.
GENERALIZED AND QUASILOCALIZATIONS OF BRAID GROUP REPRESENTATIONS
"... Abstract. We develop a theory of localization for braid group representations associatedwithobjectsinbraidedfusioncategoriesand,moregenerally,toYangBaxter operators in monoidal categories. The essential problem is to determine when a family of braid representations can be uniformly modelled upon a ..."
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Abstract. We develop a theory of localization for braid group representations associatedwithobjectsinbraidedfusioncategoriesand,moregenerally,toYangBaxter operators in monoidal categories. The essential problem is to determine when a family of braid representations can be uniformly modelled upon a tensor power of a fixed vector space in such a way that the braid group generators act “locally”. Although related to the notion of (quasi)fiber functors for fusion categories,remarkably,suchlocalizationscanexistforrepresentationsassociated with objects of nonintegral dimension. We conjecture that such localizations exist precisely when the object in question has dimension the squareroot of an integer and prove several key special cases of the conjecture. 1.
Tensor categories: A selective guided tour ∗
, 2008
"... These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively nontechnical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something int ..."
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These are the – only lightly edited – lecture notes for a short course on tensor categories. The coverage in these notes is relatively nontechnical, focussing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something interesting in them. Once the basic definitions are given, the focus is mainly on klinear categories with finite dimensional homspaces. Connections with quantum groups and low dimensional topology are pointed out, but these notes have no pretension to cover the latter subjects at any depth. Essentially, these notes should be considered as annotations to the extensive bibliography. We also recommend the recent review [33], which covers less ground in a deeper way. 1 Tensor categories 1.1 Strict tensor categories
ON THE CLASSIFICATION OF NONSELFDUAL MODULAR CATEGORIES
, 907
"... Abstract. We classify pseudounitary modular categories of rank at most 5 under the assumption that some simple object is not isomorphic to its dual. Our approach uses Gröbner basis computations, and suggests a general computational procedure for classifying lowrank modular categories. 1. ..."
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Abstract. We classify pseudounitary modular categories of rank at most 5 under the assumption that some simple object is not isomorphic to its dual. Our approach uses Gröbner basis computations, and suggests a general computational procedure for classifying lowrank modular categories. 1.
Microscopic description of 2d topological phases, duality and 3d state sums
, 907
"... Doubled topological phases introduced by Kitaev, Levin and Wen supported on two dimensional lattices are Hamiltonian versions of three dimensional topological quantum field theories described by the TuraevViro state sum models. We introduce the latter with an emphasis on obtaining them from theorie ..."
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Doubled topological phases introduced by Kitaev, Levin and Wen supported on two dimensional lattices are Hamiltonian versions of three dimensional topological quantum field theories described by the TuraevViro state sum models. We introduce the latter with an emphasis on obtaining them from theories in the continuum. Equivalence of the previous models in the ground state are shown in case of the honeycomb lattice and the gauge group being a finite group by means of the wellknown duality transformation between the group algebra and the spin network basis of lattice gauge theory. An analysis of the ribbon operators describing excitations in both types of models and the three dimensional geometrical interpretation are given. 1