## From Quantum Groups to Unitary Modular Tensor Categories

Venue: | CONTEMPORARY MATHEMATICS |

Citations: | 9 - 6 self |

### BibTeX

@MISC{Rowell_fromquantum,

author = {Eric C. Rowell},

title = { From Quantum Groups to Unitary Modular Tensor Categories},

year = {}

}

### OpenURL

### Abstract

Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are “physically feasible.” We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasizing explicit computations.

### Citations

1006 |
Introduction to Lie algebras and representation theory
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(Show Context)
Citation Context ...iption of the MTCs derived from quantum groups can be understood with little more than a firm grasp on the theory of representations of simple finite-dimensional Lie algebras found in Humphrey’s book =-=[Hum]-=- or any other introductory text. 1.2. Motivation. Recently, an application of unitary MTCs to quantum computing has been proposed by Freedman and Kitaev and advanced in the series of papers ([FKW], [F... |

912 |
Enumerative Combinatorics
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(Show Context)
Citation Context ... [2, 2, 3, 3, 4, 4, 5, 6] 30 F4, ℓ even [2, 4, 4, 6] 18 F4, ℓ odd [2, 2, 3, 4] 13 G2, 3 | ℓ [3, 6] 12 G2, 3 ∤ ℓ [2, 3] 7 from the multiset T . Any standard reference on generating functions (see e.g. =-=[Sn]-=-) will provide enough details about generating functions to prove the following: Lemma 4.9. The number PT (n) of partitions of n into parts from the multiset T has generating function: ∏ ∞∑ 1 = PT (n)... |

855 |
Quantum groups
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(Show Context)
Citation Context ...ion theory of quantum groups has proven to be a useful tool and a fruitful source of examples in many areas of mathematics. The general definition of a quantum group was given around 1985 by Drinfeld =-=[D]-=- and independently Jimbo [Ji] as a general method for finding solutions to the quantum Yang-Baxter equation. These solutions led to new representations of Artin’s braid group Bn and connections to lin... |

767 |
A guide to quantum groups
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Citation Context ...nd F, and m = 3 for Lie type G. It can be shown that T is a (non-semisimple) ribbon Ab-category (see [A] and [TW1]). The ribbon structure on T comes from the (ribbon) Hopf algebra structure on U (see =-=[ChP]-=-), i.e. the antipode, comultiplication, R-matrix, quantum Casimir etc. The set of indecomposable tilting modules with dim(X) = 0 (categorical dimension) generates a tensor ideal I ⊂ T , and semisimpli... |

526 |
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Citation Context ...lds from so-called modular Hopf algebras, examples of which can be found among quantum groups at roots of unity (see [RT] and [TW1] for examples, much simplified by constructions in [A]). When Witten =-=[Wi]-=- introduced the notion of a topological quantum field theory (TQFT) relating ideas from quantum field theory to manifold invariants, 2000 Mathematics Subject Classification. Primary 20G42; Secondary 2... |

504 |
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Citation Context ... of view. In this article, we take quantum group to mean the “classical” q-deformation of the universal enveloping algebra of a simple complex finite dimensional Lie algebra as in the book by Lusztig =-=[L]-=-, rather than the broader class of Hopf algebras the term sometimes describes. 1.1. Background. The representation theory of quantum groups has proven to be a useful tool and a fruitful source of exam... |

291 |
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Citation Context ...to link invariants. In fact, specializations of the famous polynomial invariants of Jones [J], the six-authored paper [HOMFLY] and Kauffman [Kf] have been obtained in this way. Reshetikhin and Turaev =-=[RT]-=- used this connection to derive invariants of 3-manifolds from so-called modular Hopf algebras, examples of which can be found among quantum groups at roots of unity (see [RT] and [TW1] for examples, ... |

243 |
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Citation Context ...edure in [RT] to construct 3-manifold invariants and TQFTs. The concept of an MTC is an algebraic version of 3-dimensional TQFT, and the concepts are thought to be equivalent (see the introduction to =-=[T2]-=-). Besides the quantum group approach to finding MTCs, there are several other general constructions. A geometric construction using link invariants and tangle categories was introduced in [T2] and ad... |

242 |
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(Show Context)
Citation Context ... has proven to be a useful tool and a fruitful source of examples in many areas of mathematics. The general definition of a quantum group was given around 1985 by Drinfeld [D] and independently Jimbo =-=[Ji]-=- as a general method for finding solutions to the quantum Yang-Baxter equation. These solutions led to new representations of Artin’s braid group Bn and connections to link invariants. In fact, specia... |

221 |
A polynomial invariant of knots via von Neumann algebras
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Citation Context ...um Yang-Baxter equation. These solutions led to new representations of Artin’s braid group Bn and connections to link invariants. In fact, specializations of the famous polynomial invariants of Jones =-=[J]-=-, the six-authored paper [HOMFLY] and Kauffman [Kf] have been obtained in this way. Reshetikhin and Turaev [RT] used this connection to derive invariants of 3-manifolds from so-called modular Hopf alg... |

175 |
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Citation Context ...ach Lie algebra g and integer ℓ > 〈ρ, ϑj〉 and to apply standard combinatorics to count the number of partitions into parts in S(g, ℓm). The first task is easily accomplished with the help of the book =-=[Bo]-=-. Table 2 lists the results of these computations, where ℓ0 := min{ℓ : ℓ ≥ 〈ρ, ϑj〉 + 1} is the minimal non-degenerate value of ℓ. Let PT (n) denote the number of partitions of n into parts in a multis... |

173 |
Quantum Groups, Graduate Text
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(Show Context)
Citation Context ...on of unitarity. 2.1. Axioms. In this subsection we outline the axioms for the categories we are interested in. We follow the paper [T1], and refer to that paper or the books by Turaev [T2] or Kassel =-=[K]-=- for a complete treatment. Let O be a category defined over a subfield k ⊂ C. A modular tensor category is a semisimple ribbon Ab-category O with finitely many isomorphism classes of simple objects sa... |

166 |
Categories for the Working Mathematician, Graduate Texts
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(Show Context)
Citation Context ...he constructions in [RT] (after reconciling notation). Modular Hopf algebras were replaced by the more general framework of modular tensor categories (MTCs) by Turaev [T1] (building on definitions in =-=[Mac]-=- and [JS]), axiomatizing the conditions necessary for the procedure in [RT] to construct 3-manifold invariants and TQFTs. The concept of an MTC is an algebraic version of 3-dimensional TQFT, and the c... |

157 |
The geometry of tensor calculus
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- 1991
(Show Context)
Citation Context ...ctions in [RT] (after reconciling notation). Modular Hopf algebras were replaced by the more general framework of modular tensor categories (MTCs) by Turaev [T1] (building on definitions in [Mac] and =-=[JS]-=-), axiomatizing the conditions necessary for the procedure in [RT] to construct 3-manifold invariants and TQFTs. The concept of an MTC is an algebraic version of 3-dimensional TQFT, and the concepts a... |

125 |
Lectures on tensor categories and modular functors
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- 2001
(Show Context)
Citation Context ...imple. We denote this scalar by θi. Standard arguments show that the entries of the S-matrix are determined by the categorical dimensions, the fusion rules and the twists on these simple classes (see =-=[BK]-=-): Si,j = 1 ∑ N k i∗ (2.1) ,jdkθk. θiθj k Provided O is modular the S-matrix determines the fusion rules via the Verlinde formula (see [BK]). To express the formula we must introduce the quantity D2 ∑... |

104 |
Quantum groups, Graduate Texts in Mathematics 155
- Kassel
- 1995
(Show Context)
Citation Context ...on of unitarity. 2.1. Axioms. In this subsection we outline the axioms for the categories we are interested in. We follow the paper [T1], and refer to that paper or the books by Turaev [T2] or Kassel =-=[K]-=- for a complete treatment. Let O be a category defined over a subfield k ⊂ C. A modular tensor category is a semisimple ribbon Ab-category O with finitely many isomorphism classes of simple objects sa... |

82 | Simulation of topological Field theories by quantum computers”, Comm.Math.Phys
- Freedman, Kitaev, et al.
(Show Context)
Citation Context ...ook [Hum] or any other introductory text. 1.2. Motivation. Recently, an application of unitary MTCs to quantum computing has been proposed by Freedman and Kitaev and advanced in the series of papers (=-=[FKW]-=-, [FKLW], [FLW] and [FNSWW]). Their topological model for quantum computing has a major advantage over the “classical” qubit model in that errors are corrected on the physical level and so has a more ... |

63 |
Representations of quantum algebras
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(Show Context)
Citation Context ...e highly advanced state of the theory of representations of quantum groups at roots of unity provided by the pioneering work of many including Lusztig ([L]) and Andersen and his co-authors ([A], [AP] =-=[APW]-=-). The description of the MTCs derived from quantum groups can be understood with little more than a firm grasp on the theory of representations of simple finite-dimensional Lie algebras found in Hump... |

57 |
Quantum invariants of 3-manifolds associated with classical simple Lie algebras
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- 1993
(Show Context)
Citation Context ...khin and Turaev [RT] used this connection to derive invariants of 3-manifolds from so-called modular Hopf algebras, examples of which can be found among quantum groups at roots of unity (see [RT] and =-=[TW1]-=- for examples, much simplified by constructions in [A]). When Witten [Wi] introduced the notion of a topological quantum field theory (TQFT) relating ideas from quantum field theory to manifold invari... |

51 | Vertex operator algebras, the Verlinde conjecture and modular tensor categories
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(Show Context)
Citation Context ...⋉ F(G) of the group algebra of a finite group with its (Hopf algebra) dual and can be found in the book [BK]. MTCs have also been constructed from other sources, such as vertex operator algebras (see =-=[Hu]-=-). There are two indirect constructions that should be mentioned. One is the quantum double technique of Müger [Mg] (inspired by the Drinfeld double of a Hopf algebra) by which an MTC is constructed b... |

47 |
Infinite dimensional Lie algebras, theta functions and modular forms, Adv
- Kac, Petersen
- 1984
(Show Context)
Citation Context ..., q) the cases where ℓ is divisible by m have been mainly studied in the literature. The invertibility of S for Lie types A and C with q = e πi/ℓ was shown in [TW1] using the work of Kac and Peterson =-=[KP]-=-, and a complete treatment (for all Lie types with q = e πi/ℓ ) is found in [Ki]. The invertibility can be extended to other values of q by the following Galois argument, which is found in [TW2] in a ... |

46 |
A.: A new polynomial invariant of knots and links
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- 1985
(Show Context)
Citation Context ...se solutions led to new representations of Artin’s braid group Bn and connections to link invariants. In fact, specializations of the famous polynomial invariants of Jones [J], the six-authored paper =-=[HOMFLY]-=- and Kauffman [Kf] have been obtained in this way. Reshetikhin and Turaev [RT] used this connection to derive invariants of 3-manifolds from so-called modular Hopf algebras, examples of which can be f... |

45 |
Introduction to quantum groups, Birkhäuser
- Lusztig
- 1993
(Show Context)
Citation Context ...ication that motivates this research. In this article, we take quantum group to mean the “classical” q-deformation of the universal enveloping algebra of a Kac-Moody algebra as in the book by Lusztig =-=[L]-=-, rather than the broader class of Hopf algebras the term sometimes describes. 1.1. Background. The representation theory of quantum groups has proven to be a useful tool and a fruitful source of exam... |

44 |
Tensor products of quantized tilting modules
- Andersen
- 1992
(Show Context)
Citation Context ...riants of 3-manifolds from so-called modular Hopf algebras, examples of which can be found among quantum groups at roots of unity (see [RT] and [TW1] for examples, much simplified by constructions in =-=[A]-=-). When Witten [Wi] introduced the notion of a topological quantum field theory (TQFT) relating ideas from quantum field theory to manifold invariants, 2000 Mathematics Subject Classification. Primary... |

44 | The two-eigenvalue problem and density of Jones representation of braid groups
- Freedman, Larsen, et al.
(Show Context)
Citation Context ...y other introductory text. 1.2. Motivation. Recently, an application of unitary MTCs to quantum computing has been proposed by Freedman and Kitaev and advanced in the series of papers ([FKW], [FKLW], =-=[FLW]-=- and [FNSWW]). Their topological model for quantum computing has a major advantage over the “classical” qubit model in that errors are corrected on the physical level and so has a more feasible thresh... |

42 |
Fusion categories arising from semisimple Lie algebras
- Andersen, Paradowski
- 1995
(Show Context)
Citation Context ...en more useful than Formula (3.1) for computing the entries of the S-matrix, as computing the twists θλ, q-dimensions dλ (see below) and fusion coefficients Nν λ,µ (via the quantum Racah formula, see =-=[AP]-=- and [S2]) is more straightforward than summing over the Weyl group. The twist coefficients for simple objects are also well known: θλ = q 〈λ,λ+2ρ〉 , as are the categorical q-dimensions: where [n] = q... |

39 |
Catégories prémodulaires, modularisations et invariants des variétés de dimension 3
- Bruguières
- 2000
(Show Context)
Citation Context ... which an MTC is constructed by “doubling” a monoidal category with some further technical properties. An example of this approach is the finite group algebra construction mentioned above. Bruguières =-=[Br]-=- describes conditions under which one may modularize a category that satisfies all of the axioms of an MTC except the modularity condition (called a pre-modular category. This corresponds essentially ... |

38 |
C*-Tensor categories from quantum groups
- Wenzl
- 1998
(Show Context)
Citation Context ...rix is a scalar multiple of the first column (or row). For the categories C(g, ℓ, q) Kirillov Jr. conjectured [Ki] the existence of a positive-definite Hermitian structure, and it was proved by Wenzl =-=[W]-=- in some cases and independently by Xu [Xu]. We describe the modularity and unitarity of the categories C(g, ℓ, q) first for the cases can be handled uniformly, and then consider those that must be co... |

20 |
Integral modular categories and integrality of quantum invariants at roots of unity of prime order
- Masbaum, Wenzl
(Show Context)
Citation Context ...A. For Lie type Ar corresponding to g = slr+1 we have m = 1 and d = r+1. Bruguières [Br] shows that one has modularity for q = e zπi/ℓ if and only if gcd(z, (r + 1)ℓ) = 1. Moreover, Masbaum and Wenzl =-=[MW]-=- show that when gcd(ℓ, r + 1), the subcategory of C(slr+1, ℓ, q) generated by the objects labelled by integer weights forms a modular subcategory of rank 1/(r+1) times the rank of the full category. T... |

20 | Fusion categories of rank 2
- Ostrik
(Show Context)
Citation Context ...g, and the author in [LRW]. Another problem is to prove the conjecture of Z. Wang: There are finitely many MTCs of a fixed rank (see Subsection 2.2). This has been verified for ranks 1,2,3 and 4: see =-=[O1]-=- and [O2] for ranks 2 and 3 respectively, and [BRSW] for both ranks 3 and 4. It is with this conjecture in mind that we provide generating functions for ranks of categories in Subsection 4.7. Acknowle... |

16 |
Semisimple and modular categories from link invariants
- Turaev, Wenzl
- 1997
(Show Context)
Citation Context ... the quantum group approach to finding MTCs, there are several other general constructions. A geometric construction using link invariants and tangle categories was introduced in [T2] and advanced in =-=[TW2]-=-, but all examples that have been carried out lead to MTCs also obtained from quantum groups. Although it is expected that there are non-trivial examples of MTCs that cannot be derived from quantum gr... |

12 |
Modular categories and 3-manifold invariants
- Turaev
- 1992
(Show Context)
Citation Context ...were immediately available from the constructions in [RT] (after reconciling notation). Modular Hopf algebras were replaced by the more general framework of modular tensor categories (MTCs) by Turaev =-=[T1]-=- (building on definitions in [Mac] and [JS]), axiomatizing the conditions necessary for the procedure in [RT] to construct 3-manifold invariants and TQFTs. The concept of an MTC is an algebraic versio... |

11 |
Modular categories of types
- Beliakova, Blanchet
(Show Context)
Citation Context ... constructions. A geometric construction using link invariants and tangle categories was introduced in [T2], advanced by Turaev and Wenzl in [TW2] and somewhat simplified by Blanchet and Beliakova in =-=[BB]-=-. However, all examples that have been carried out lead to MTCs also obtainable from quantum groups. Yet another construction of MTCs from representations of vertex operator algebras has recently appe... |

11 |
Topological quantum computation. Mathematical challenges of the 21st century (Los
- Freedman, Larsen, et al.
- 2000
(Show Context)
Citation Context ...m] or any other introductory text. 1.2. Motivation. Recently, an application of unitary MTCs to quantum computing has been proposed by Freedman and Kitaev and advanced in the series of papers ([FKW], =-=[FKLW]-=-, [FLW] and [FNSWW]). Their topological model for quantum computing has a major advantage over the “classical” qubit model in that errors are corrected on the physical level and so has a more feasible... |

10 |
The n-eigenvalue problem and two
- Larsen, Rowell, et al.
- 2005
(Show Context)
Citation Context ...up. This is related to a sine qua non of quantum computation known as universality. Progress towards answering this question has been made in [FLW] and was extended by Larsen, Wang, and the author in =-=[LRW]-=-. Another problem is to prove the conjecture of Z. Wang: There are finitely many MTCs of a fixed rank (see Subsection 2.2). This has been verified for ranks 1,2,3 and 4: see [O1] and [O2] for ranks 2 ... |

7 | On a family of non-unitarizable ribbon categories
- Rowell
- 2005
(Show Context)
Citation Context ...-Turaev [LT]. It is shown in [TW2] that the subcategory of C(so2r+1, ℓ, q) generated by objects labelled by integer weights is modular. In Table 1 this is denoted Z(Br). Combining the computations in =-=[R1]-=- and the modularity criterion of [Br] one has: Theorem 4.3. The category C(so2r+1, ℓ, q) with ℓ odd is modular if and only if q ℓ = −1 and r is odd. Proof. By the modularity criterion we wish to show ... |

7 |
Jones-Witten invariants for non-simply connected Lie groups and the geometry of the Weyl alcove. arXiv: math.QA/9905010
- Sawin
- 1999
(Show Context)
Citation Context ... 5. In Table 1 this subcategory is denoted by Z(Ar). 4.3. Type B, ℓ odd. The categories C(so2r+1, ℓ, q) with C(so2r+1, ℓ, q)ℓ odd has been considered to some extent by several authors including Sawin =-=[S1]-=-, [S2] and Le-Turaev [LT]. It is shown in [TW2] that the subcategory of C(so2r+1, ℓ, q) generated by objects labelled by integer weights is modular. In Table 1 this is denoted Z(Br). Combining the com... |

7 |
A class of P, Tinvariant topological phases of interacting electrons
- Freedman, Nayak, et al.
(Show Context)
Citation Context ...troductory text. 1.2. Motivation. Recently, an application of unitary MTCs to quantum computing has been proposed by Freedman and Kitaev and advanced in the series of papers ([FKW], [FKLW], [FLW] and =-=[FNSWW]-=-). Their topological model for quantum computing has a major advantage over the “classical” qubit model in that errors are corrected on the physical level and so has a more feasible threshold. For a v... |

6 |
On braided tensor categories of type
- Tuba, Wenzl
(Show Context)
Citation Context ...the following definition: Definition 4.4. A pre-modular category O is called unitarizable if O is tensor equivalent to a unitary pre-modular category O ′ . Using a structure theorem of Tuba and Wenzl =-=[TbW]-=- it is shown in [R2] that: Theorem 4.5. Fix q with q 2 a primitive ℓth root of unity. Then no braided tensor category sharing the Grothendieck semiring of the category C(so2r+1, ℓ, q) with ℓ odd is un... |

5 | A class of P, T -invariant topological phases of interacting electrons - Freedman, Nayak, et al. - 2004 |

5 |
On an inner product in modular categories
- Jr
- 1996
(Show Context)
Citation Context ...ieck semiring and denoted Gr(O). If {X0 = 1, X1, . . .,Xn−1} is a set of representatives of these isomorphism classes, the rank of O is n. The axioms guarantee that we have (using Kirillov’s notation =-=[Ki]-=-): Xi ⊗ Xj ∼ = ∑ k N k i,jXk for some Nk i,j ∈ N. These structure coefficients of Gr(O) are called the fusion rules of O. Having fixed an ordering of simple objects as above, the structure coefficient... |

5 |
λ-lattices from quantum groups. preprint Teodor Banica Institut de Mathématiques de Luminy
- Xu, Standard
(Show Context)
Citation Context ...h 3|ℓ or E7 with ℓ even. Following a conjecture of Kirillov Jr. [Ki], Wenzl [W] showed that the Hermitian form on C(g, ℓ, q) is positive definite for the uniform cases for certain values of q, and Xu =-=[X]-=- independently showed some of the cases covered by Wenzl. Their results are summarized in: Theorem 4.2 (Wenzl/Xu). The categories C(g, ℓ, q) are unitary when m|ℓ and q = e πi/ℓ . 4.2. Type A. For Lie ... |

3 |
Classification of Modular Tensor Categories I: Low-rank cases, in preparation
- Belinschi, Rowell, et al.
(Show Context)
Citation Context ...prove the conjecture of Z. Wang: There are finitely many MTCs of a fixed rank (see Subsection 2.2). This has been verified for ranks 1,2,3 and 4: see [O1] and [O2] for ranks 2 and 3 respectively, and =-=[BRSW]-=- for both ranks 3 and 4. It is with this conjecture in mind that we provide generating functions for ranks of categories in Subsection 4.7. Acknowledgements. The author wishes to thank the referees fo... |

3 |
Quantum groups and ribbon G-categories
- Turaev
(Show Context)
Citation Context ...tegory is denoted by Z(Ar). 4.3. Type B, ℓ odd. The categories C(so2r+1, ℓ, q) with C(so2r+1, ℓ, q)ℓ odd has been considered to some extent by several authors including Sawin [S1], [S2] and Le-Turaev =-=[LT]-=-. It is shown in [TW2] that the subcategory of C(so2r+1, ℓ, q) generated by objects labelled by integer weights is modular. In Table 1 this is denoted Z(Br). Combining the computations in [R1] and the... |

2 |
Quantum groups at roots of unity and modularity, preprint, arXiv: math.QA/0308281
- Sawin
(Show Context)
Citation Context ...icted attention to the root lattice. While in most cases one can appeal to the results in [A] to fix this problem, the only general proof this author is aware of is in a more recent preprint by Sawin =-=[S2]-=-. 3.2. Other Constructions. As was mentioned in the introduction, the quantum group constructions exhausts the body of explicitly known non-trivial examples. The geometric construction alluded to in t... |

2 |
Tensor categories arising from quantum groups and BMW-algebras at odd roots of unity, thesis
- Rowell
- 2003
(Show Context)
Citation Context .... Using the formula 2.1 with µ = γ, the explicit computations of dλ and θλ and the obstruction equation Sγ,λ = dγdλ the result follows from Scholium 4.11 of [R1]. □ The subject of the author’s thesis =-=[R2]-=- (the results of which can be found in [R1]) is the question of unitarizability of the family of categories C(so2r+1, ℓ, q) with ℓ odd. Using an analysis of the characters of the Grothendieck semiring... |

2 |
λ-lattices from quantum groups, Invent
- Xu, Standard
- 1998
(Show Context)
Citation Context ...mn (or row). For the categories C(g, ℓ, q) Kirillov Jr. conjectured [Ki] the existence of a positive-definite Hermitian structure, and it was proved by Wenzl [W] in some cases and independently by Xu =-=[Xu]-=-. We describe the modularity and unitarity of the categories C(g, ℓ, q) first for the cases can be handled uniformly, and then consider those that must be considered individually as well as a few subc... |

1 |
An invariant of regular isotopy
- Kaufmann
- 1990
(Show Context)
Citation Context ...w representations of Artin’s braid group Bn and connections to link invariants. In fact, specializations of the famous polynomial invariants of Jones [J], the six-authored paper [HOMFLY] and Kauffman =-=[Kf]-=- have been obtained in this way. Reshetikhin and Turaev [RT] used this connection to derive invariants of 3-manifolds from so-called modular Hopf algebras, examples of which can be found among quantum... |

1 |
From subfactors to topology II. The quantum double of tensor categories and subfactors
- Müger
(Show Context)
Citation Context ...have also been constructed from other sources, such as vertex operator algebras (see [Hu]). There are two indirect constructions that should be mentioned. One is the quantum double technique of Müger =-=[Mg]-=- (inspired by the Drinfeld double of a Hopf algebra) by which an MTC is constructed by “doubling” a monoidal category with some further technical properties. An example of this approach is the finite ... |

1 |
Pre-modular categories of rank 3, preprint arXiv
- Ostrik
(Show Context)
Citation Context ...e author in [LRW]. Another problem is to prove the conjecture of Z. Wang: There are finitely many MTCs of a fixed rank (see Subsection 2.2). This has been verified for ranks 1,2,3 and 4: see [O1] and =-=[O2]-=- for ranks 2 and 3 respectively, and [BRSW] for both ranks 3 and 4. It is with this conjecture in mind that we provide generating functions for ranks of categories in Subsection 4.7. Acknowledgements.... |