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On exotic modular tensor categories
 Commun. Contemp. Math
"... Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a tot ..."
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Cited by 14 (7 self)
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Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 nontrivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 nontrivial prime UMTCs are given in Section 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Section 6. 1.
Orthogonal and Symplectic Quantum Matrix Algebras and CayleyHamilton Theorem for them
, 2005
"... For families of orthogonal and symplectic types quantum matrix (QM) algebras, we derive corresponding versions of the CayleyHamilton theorem. For a wider family of BirmanMurakamiWenzl type QMalgebras, we investigate a structure of its characteristic subalgebra (the subalgebra in which the coeff ..."
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Cited by 8 (3 self)
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For families of orthogonal and symplectic types quantum matrix (QM) algebras, we derive corresponding versions of the CayleyHamilton theorem. For a wider family of BirmanMurakamiWenzl type QMalgebras, we investigate a structure of its characteristic subalgebra (the subalgebra in which the coefficients of characteristic polynomials take values). We define 3 sets of generating elements of the characteristic subalgebra and derive recursive Newton and Wronski relations between them. For the family of the orthogonal type QMalgebras, additional reciprocal relations for the generators of the characteristic subalgebra are obtained.
On a family of nonunitarizable ribbon categories
 Math Z
, 2005
"... Abstract. We consider several families of categories. The first are quotients of H. Andersen’s tilting module categories for quantum groups of Lie type B at odd roots of unity. The second consists of categories of type BC constructed from idempotents in BMWalgebras. Our main result is to show that ..."
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Cited by 7 (5 self)
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Abstract. We consider several families of categories. The first are quotients of H. Andersen’s tilting module categories for quantum groups of Lie type B at odd roots of unity. The second consists of categories of type BC constructed from idempotents in BMWalgebras. Our main result is to show that these families coincide as braided tensor categories using a recent theorem of Tuba and Wenzl. By appealing to similar results of Blanchet and Beliakova we obtain another interesting equivalence with these two families of categories and the quantum group categories of Lie type C at odd roots of unity. The morphism spaces in these categories can be equipped with a Hermitian form, and we are able to show that these categories are never unitary, and no braided tensor category sharing the Grothendieck semiring common to these families is unitarizable. 1.
Knot polynomial identities and quantum group coincidences
"... Abstract We construct link invariants using the D2n subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between smal ..."
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Cited by 4 (2 self)
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Abstract We construct link invariants using the D2n subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between small modular categories involving the even parts of the D2n planar algebras. We discuss the origins of these coincidences, explaining the role of SO levelrank duality, KirbyMelvin symmetry, and properties of small Dynkin diagrams. One of these coincidences involves G2 and does not appear to be related to levelrank duality. AMS Classification 18D10; 57M27 17B10 81R05 57R56
Some nonbraided fusion categories of rank 3, arXiv: 0704.0208
"... Abstract. We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion we describe a convenient, c ..."
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Cited by 2 (1 self)
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Abstract. We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion we describe a convenient, concrete and useful variation of graphical calculus for fusion categories, discuss pivotality and sphericity in this framework, and give a short and elementary reproof of the fact that the quadruple dual functor is naturally isomorphic to the identity. 1.
THE NEIGENVALUE PROBLEM AND TWO APPLICATIONS
"... Abstract. We consider the classification problem for compact Lie groups G ⊂ U(n) which are generated by a single conjugacy class with a fixed number N of distinct eigenvalues. We give an explicit classification when N = 3, and apply this to extract information about Galois representations and braid ..."
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Abstract. We consider the classification problem for compact Lie groups G ⊂ U(n) which are generated by a single conjugacy class with a fixed number N of distinct eigenvalues. We give an explicit classification when N = 3, and apply this to extract information about Galois representations and braid group representations. 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.
Quantum automorphism groups and SO(3)deformations
"... We show that any compact quantum group having the same fusion rules as the ones of SO(3) is the quantum automorphism group of a pair (A,ϕ), where A is a finite dimensional C∗algebra endowed with a homogeneous faithful state. We also study the representation category of the quantum automorphism grou ..."
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We show that any compact quantum group having the same fusion rules as the ones of SO(3) is the quantum automorphism group of a pair (A,ϕ), where A is a finite dimensional C∗algebra endowed with a homogeneous faithful state. We also study the representation category of the quantum automorphism group of (A,ϕ) when ϕ is not necessarily positive, generalizing some known results, and we discuss the possibility of classifying the cosemisimple (not necessarily compact) Hopf algebras whose corepresentation semiring is isomorphic to that of SO(3). 1 Introduction and main results The quantum automorphism group of a measured finite dimensional C∗algebra (A,ϕ) (i.e. a finitedimensional C∗algebra A endowed with a faithful state ϕ) has been defined by Wang in [24] as the universal object in the category of compact quantum groups acting on (A,ϕ). The corresponding compact Hopf algebra is denoted by Aaut(A,ϕ).