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15
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.
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.
TopologicalLike Features in Diagrammatical Quantum Circuits
, 2007
"... In this paper, we revisit topologicallike features in the extended Temperley– Lieb diagrammatical representation for quantum circuits including the teleportation, dense coding and entanglement swapping. We perform these quantum circuits and derive characteristic equations for them with the help of ..."
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In this paper, we revisit topologicallike features in the extended Temperley– Lieb diagrammatical representation for quantum circuits including the teleportation, dense coding and entanglement swapping. We perform these quantum circuits and derive characteristic equations for them with the help of topologicallike operations. Furthermore, we comment on known diagrammatical approaches to quantum information phenomena from the perspectives of both tensor categories and topological quantum field theories. Moreover, we remark on the proposal for categorical quantum physics and information to be described by dagger ribbon categories.
Unitary braid representations with finite image”, arXiv: math.GR/0805.4222v1
"... Abstract. We characterize unitary representations of braid groups Bn of degree linear in n and finite images of such representations of degree exponential in n. 1. ..."
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Abstract. We characterize unitary representations of braid groups Bn of degree linear in n and finite images of such representations of degree exponential in n. 1.
Finite linear quotients of B3 of low dimension
"... Abstract. We study the problem of deciding whether or not the image of an irreducible representation of the braid group B3 of degree ≤ 5 has finite image if we are only given the eigenvalues of a generator. We provide a partial algorithm that determines when the images are finite or infinite in all ..."
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Abstract. We study the problem of deciding whether or not the image of an irreducible representation of the braid group B3 of degree ≤ 5 has finite image if we are only given the eigenvalues of a generator. We provide a partial algorithm that determines when the images are finite or infinite in all but finitely many cases, and use these results to study examples coming from quantum groups. Our technique uses two classification theorems and the computational group theory package GAP. 1.
MODULAR CATEGORIES, INTEGRALITY AND EGYPTIAN FRACTIONS
"... Abstract. It is a wellknown result of Etingof, Nikshych and Ostrik that there are finitely many inequivalent integral modular categories of any fixed rank n. This follows from a doubleexponential bound on the maximal denominator in an Egyptian fraction representation of 1. A naïve computer search ..."
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Abstract. It is a wellknown result of Etingof, Nikshych and Ostrik that there are finitely many inequivalent integral modular categories of any fixed rank n. This follows from a doubleexponential bound on the maximal denominator in an Egyptian fraction representation of 1. A naïve computer search approach to the classification of rank n integral modular categories using this bound quickly overwhelms the computer’s memory (for n ≥ 7). We use a modified strategy: find general conditions on modular categories that imply integrality and study the classification problem in these limited settings. The first such condition is that the order of the twist matrix is 2, 3, 4 or 6 and we obtain a fairly complete description of these classes of modular categories. The second condition is that the unit object is the only simple nonselfdual object, which is equivalent to odddimensionality. In this case we obtain a (linear) improvement on the bounds and employ numbertheoretic techniques to obtain a classification for rank at most 11 for odddimensional modular categories. 1.