## LOCALIZATION OF UNITARY BRAID GROUP REPRESENTATIONS

Citations: | 1 - 0 self |

### BibTeX

@MISC{Rowell_localizationof,

author = {Eric C. Rowell and Zhenghan Wang},

title = {LOCALIZATION OF UNITARY BRAID GROUP REPRESENTATIONS},

year = {}

}

### OpenURL

### Abstract

Abstract. Governed by locality, we explore a connection between unitary braid group representations associated to a unitary R-matrix and to a simple object in a unitary braided fusion category. Unitary R-matrices, namely unitary solutions to the Yang-Baxter 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 R-matrix. 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 R-matrix. 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.

### Citations

577 |
The Theory of Matrices
- Gantmacher
- 1960
(Show Context)
Citation Context ... the (block permutation) form: ⎛ ⎞ 0 C0 · · · 0 . 0 0 .. 0 AΓ := ⎜ . . ⎝ . . .. ⎟ Cp−2⎠ Cp−1 0 · · · 0 notice that the Ci are not necessarily square matrices, but CiCi+1 is well-defined. Theorem 4.2 (=-=[Gt]-=-). Suppose that A is irreducible of period p. Then: (a) A p is block diagonal with primitive blocks B0, · · · , Bp−1. (b) A has a real eigenvalue λ such that (i) |α| ≤ λ for all eigenvalues α of A and... |

274 |
Hecke algebra representations of braid groups and link polynomials
- Jones
(Show Context)
Citation Context ...quences of braid group representations is finite dimensional quotients of the braid group algebras KBn where K is some field. A typical example are the Temperley-Lieb algebras TLn(q) for q a variable =-=[J2]-=-. These are the quotients of C(q)Bn by ideals containing the element (σi − q)(σi + 1) ∈ C(q)Bn. The ensuing sequence of finite dimensional algebras · · · ⊂ TLn(q) ⊂ TLn+1(q) ⊂ · · · are semisimple. As... |

132 |
Coxeter graphs and towers of algebras
- Goodman, Harpe, et al.
- 1989
(Show Context)
Citation Context ...e centralizer algebras End(X ⊗n ). To describe the results we will need some graph-theoretic notions associated with sequences of semisimple finite-dimensional C-algebras ordered under inclusion. See =-=[GHJ]-=- and graph-theoretical references therein for relevant (standard) definitions. The main tool we employ is the Perron-Frobenius Theorem. Our approach is fairly standard, see [W1] for arguments of a sim... |

115 |
Hecke algebras of type An and subfactors
- Wenzl
- 1988
(Show Context)
Citation Context ...tions. Moreover, it is clear that the Bn generators σi have finite order in the image of these representation. Other examples of this type of construction are the Hecke-algebras and BMW-algebras (see =-=[W1]-=- and [W2]). Braided fusion categories (see eg. [ENO]) are a modern marriage of these two sources of braid group representations. These are fusion categories equipped with a family of natural transform... |

109 | Topological quantum computation - Freedman, Kitaev, et al. - 2000 |

94 |
Quantum Groups. Graduate Texts in Mathematics, vol. 155
- Kassel
- 1995
(Show Context)
Citation Context ...oups Bn on W ⊗n via σi → RW i where R W i := I ⊗i−1 W ⊗ R ⊗ I⊗n−i−1 W . The most ubiquitous source of R-matrices is finite dimensional semisimple quasitriangular (or braided) Hopf algebras H (see eg. =-=[Ks]-=-). A pair (W, R) as in Definition 2.1 is sometimes called a braided vector space, which play a key role in the Andruskiewitch-Schneider program [AS] for classifying pointed Hopf algebras.4 ERIC C. RO... |

87 | A modular functor which is universal for quantum computation - Freedman, Larsen, et al. |

77 | Nilpotent fusion categories
- Gelaki, Nikshych
(Show Context)
Citation Context ... σi have finite order in the image of these representation. Other examples of this type of construction are the Hecke-algebras and BMW-algebras (see [W1] and [W2]). Braided fusion categories (see eg. =-=[ENO]-=-) are a modern marriage of these two sources of braid group representations. These are fusion categories equipped with a family of natural transformations: for each pair of objects X, Y one has an inv... |

75 | Simulation of topological field theories by quantum computers
- Freedman, Kitaev, et al.
- 2002
(Show Context)
Citation Context ...mula of TQFTs can be considered as hidden locality of the Jones representations (see Remark 2.4). The gluing formula in TQFTs as hidden locality underlies the simulation of TQFTs by quantum computers =-=[FKW]-=-. Interesting unitary R-matrices are very difficult to construct. This observation reflects a conflict of explicit locality with unitarity in braid group representations. The only systematic unitary s... |

60 |
Quantum groups and subfactors of type
- Wenzl
- 1990
(Show Context)
Citation Context ...reover, it is clear that the Bn generators σi have finite order in the image of these representation. Other examples of this type of construction are the Hecke-algebras and BMW-algebras (see [W1] and =-=[W2]-=-). Braided fusion categories (see eg. [ENO]) are a modern marriage of these two sources of braid group representations. These are fusion categories equipped with a family of natural transformations: f... |

57 | Set-theoretical solutions to the quantum Yang-Baxter equation
- Etingof, Schedler, et al.
- 1999
(Show Context)
Citation Context ...difficult to construct. This observation reflects a conflict of explicit locality with unitarity in braid group representations. The only systematic unitary solutions that we know of are permutations =-=[ESS]-=-. Smooth deformations of such solutions seem to be all trivial. We believe that any smooth deformation R(h) of the identity R(0) = id on a Hilbert space V with an orthonormal basis {ei}, i = 1, . . .,... |

54 |
Pointed Hopf algebras, New directions in Hopf algebras
- Andruskiewitsch, Schneider
- 2002
(Show Context)
Citation Context ...asitriangular (or braided) Hopf algebras H (see eg. [Ks]). A pair (W, R) as in Definition 2.1 is sometimes called a braided vector space, which play a key role in the Andruskiewitch-Schneider program =-=[AS]-=- for classifying pointed Hopf algebras.4 ERIC C. ROWELL AND ZHENGHAN WANG We would like to point out that these representations ρR are explicitly local, in the sense that the representation space is ... |

43 | The two-eigenvalue problem and density of jones representation of braid groups - Freedman, Larsen, et al. - 2002 |

40 |
Braid groups, Hecke algebras and type II1 factors, in Geometric methods in operator algebras (Kyoto
- Jones
- 1983
(Show Context)
Citation Context ...ence. 12 ERIC C. ROWELL AND ZHENGHAN WANG Jones evaluations of links can be obtained from two representations of the braid groups: the original unitary Jones representation from von Neumman algebras =-=[J1]-=-, and the non-unitary representation from Drinfeld’s SU(2) R-matrix [Tu2]. While it is obvious that any R-matrix braid group representation is local, the unitary Jones representation cannot be local a... |

28 | Anyons in an exactly solved model and beyond
- Kitaev
- 2006
(Show Context)
Citation Context ...only 8 distinguishable categories in this infinite family. These 16 different unitary TQFTs realize nicely the 16-fold way as algebraic models of anyonic quantum systems which encode anyon statistics =-=[K1]-=-. We make the following: Definition 6.1. If the braid representations from any simple object in a unitary modular tensor category can be localized, we will say the corresponding TQFT can be localized.... |

28 |
Quantum invariants of knots and
- Turaev
- 1994
(Show Context)
Citation Context ...n be obtained from two representations of the braid groups: the original unitary Jones representation from von Neumman algebras [J1], and the non-unitary representation from Drinfeld’s SU(2) R-matrix =-=[Tu2]-=-. While it is obvious that any R-matrix braid group representation is local, the unitary Jones representation cannot be local as in the case of R-matrix because the dimensions of the representation sp... |

26 | A magnetic model with a possible Chern-Simons phase - Freedman |

24 |
Markov traces and II1 factors in conformal field theory
- Boer, Goeree
- 1991
(Show Context)
Citation Context ...r N even for ℓ = N −1 and N, respectively. These level 2 categories (SO(N), 2) have been studied elsewhere, see [Gn] and [NR]. The corresponding fusion rules also appeared in older work, see [J3] and =-=[dBG]-=-. In light of Conjectures 3.1, 4.1 and [RSW, Conjecture 6.6], we expect that that the (unitary) braid group representations associated with (SO(N), 1) and (SO(N), 2) (a) have finite image and (b) can ... |

24 |
the quantum field computer
- Freedman, PNP
- 1998
(Show Context)
Citation Context ...eral conjectures and discuss possible connections to physics and computer science. 1. Introduction One of the questions when a topological quantum field theory (TQFT) computational model was proposed =-=[F1]-=- is whether or not there are non-trivial unitary R-matrices, i.e., unitary solutions to the constant Yang-Baxter equation. In particular, are there unitary R-matrices (with algebraic entries) whose re... |

17 | Unitary solutions to the Yang-Baxter equation in dimension four
- Dye
- 2003
(Show Context)
Citation Context ...d. 3. Unitary Yang-Baxter operators In this section we discuss the braid group images associated with unitary solutions to the YBE. In case dim(W) = 2, all unitary solutions R have been classified in =-=[D]-=- (following the complete classification in [H]), up to conjugation by an operator of the form Q ⊗ Q, Q ∈ End(W). All but one of the conjugacy classes are have a monomial representative: a matrix that ... |

17 |
On a certain value of the Kauffman polynomial
- Jones
- 1989
(Show Context)
Citation Context ...omplication for such a localization is that no description of the centralizer algebras End((Xε) ⊗n ) as quotients of CBn is known, except for N = 3 (Temperley-Lieb algebras), N = 5 (BMW-algebras, see =-=[J4]-=-) and N = 7 (see [Wb]). In [dBG] a similar situation is studied corresponding to the rational conformal field theory associated with the WZW theory corresponding to (SO(N), 2) for N odd. The principal... |

16 | Self-dual solutions of the startriangle relations in ZN models - Fateev, Zamolodchikov - 1982 |

14 |
Representations of the braid group B3
- Tuba, Wenzl
(Show Context)
Citation Context ...) at Q = e 7πi/10 . The details are not particularly enlightening, but the approach is the following: first determine the B3-reprentation associated with the simple object Xε1 in C(so10, Q, 10) using =-=[TbW]-=-. Check the defining relations for Cm(−i, e πi/10 ) are satisfied by the images of the B3 generators and that the values of the trace coincide with those of the uniquely defined trace on Cm(−i, e πi/1... |

13 | Extraspecial 2-groups and images of braid group representations
- Franko, Rowell, et al.
(Show Context)
Citation Context ...a localization of (ρX, End(X ⊗n )) then one expects the invariant InvX(L) to be directly related to TR(L, µi, α, β). For example, the localization of the Jones representations at level 2 described in =-=[FRW]-=- has an enhancement and the relationship between the two invariants is explicitly described. 3. Unitary Yang-Baxter operators In this section we discuss the braid group images associated with unitary ... |

13 |
All solutions to the constant quantum Yang-Baxter equation in two dimensions, Phys. Lett. A 165
- Hietarinta
- 1992
(Show Context)
Citation Context ...ction we discuss the braid group images associated with unitary solutions to the YBE. In case dim(W) = 2, all unitary solutions R have been classified in [D] (following the complete classification in =-=[H]-=-), up to conjugation by an operator of the form Q ⊗ Q, Q ∈ End(W). All but one of the conjugacy classes are have a monomial representative: a matrix that is the product of a diagonal matrix D and a pe... |

13 | On classification of modular tensor categories
- Rowell, Stong, et al.
(Show Context)
Citation Context ... 2 ∈ N for all simple Xi (see [ENO]). A braided fusion category C is said to have property F if for each X ∈ C the Bn-representations on End(X ⊗n ) have finite image for all n. Then Conjecture 6.6 of =-=[RSW]-=- states: a braided fusion category C has property F if, and only if, FPdim(X) 2 ∈ N for each simple object X. Notice that this conjecture has no formulation for general braided tensor categories, for ... |

10 | The n-eigenvalue problem and two - Larsen, Rowell, et al. - 2005 |

9 |
Metaplectic link invariants
- Goldschmidt, Jones
- 1989
(Show Context)
Citation Context ..., n − 1) is defined on generators of Bn by p−1 ∑ γn(σi) = ζ j=0 ω j2 u j i where ζ is a normalization factor ensuring that γn(σi) is unitary (regarded as an operator on ES(ω, n − 1)). It was shown in =-=[GJ]-=- that the image of the braid group under this representation is a finite group. In fact, for n odd the analysis in [GJ] shows that, projectively, γn(Bn) is isomorphic to the finite simple group PSp(n ... |

8 | From quantum groups to unitary modular tensor categories, from: “Representations of algebraic groups, quantum groups
- Rowell
- 2006
(Show Context)
Citation Context ...ups at roots of unity. For each simple Lie algebra g and a 2ℓth root of unity q one may construct such a category which we denote by C(g, q, ℓ). For details we refer the reader to [BK] and the survey =-=[R1]-=-. In the physics literature these categories are often denoted by (G, k) where G is the compact Lie group with Lie algebra g and k is the level, which is a linear function of the dual Coxeter number o... |

7 |
Braid group representations from quantum doubles of finite groups
- Etingof, Rowell, et al.
(Show Context)
Citation Context ...entations as follows: let ˇ R ∈ End(H ⊗H) be the solution to the YBE for the vector space H so that (H, ˇ R) is a braided vector space. With respect to the usual (tensor product) basis for H ⊗ H (see =-=[ERW]-=-) ˇ R is a permutation matrix and hence unitary. Moreover, standard algebraic considerations show that (H, ˇ R) is a localization of (ρH, EndH(H ⊗n )). We hasten to point out that the main result of [... |

7 |
The level 2 and 3 modular invariants for the orthogonal algebras
- Gannon
(Show Context)
Citation Context ...d to quantum group categories C(soN, q, ℓ) for ℓ = 2N−2 and ℓ = 2N, respectively, and for N even for ℓ = N −1 and N, respectively. These level 2 categories (SO(N), 2) have been studied elsewhere, see =-=[Gn]-=- and [NR]. The corresponding fusion rules also appeared in older work, see [J3] and [dBG]. In light of Conjectures 3.1, 4.1 and [RSW, Conjecture 6.6], we expect that that the (unitary) braid group rep... |

6 |
Strongly multiplicity free modules for Lie algebras and quantum groups
- Lehrer, Zhang
- 2006
(Show Context)
Citation Context ...e well-known “Fibonacci theory.” Here we have just 4 simple objects 1, X, Y and Z with Bratteli diagram in Figure 1. Moreover, the image of the braid group generates the algebras End(X ⊗n ) (see, eg. =-=[LZ]-=-). Observe that the Bratteli diagram is cyclically 1-partite in this case. The�� �� �� � �� �� �� 14 ERIC C. ROWELL AND ZHENGHAN WANG Figure 1. Bratteli diagram for C(sl2, q, 5) X ��� ��� �� 1 Y ����... |

6 |
Invariant tensors for spin representations of so(7
- Westbury
(Show Context)
Citation Context ...a localization is that no description of the centralizer algebras End((Xε) ⊗n ) as quotients of CBn is known, except for N = 3 (Temperley-Lieb algebras), N = 5 (BMW-algebras, see [J4]) and N = 7 (see =-=[Wb]-=-). In [dBG] a similar situation is studied corresponding to the rational conformal field theory associated with the WZW theory corresponding to (SO(N), 2) for N odd. The principal graph and Bratteli d... |

5 | An algebra-level version of a link-polynomial identity of Lickorish - Larsen, Rowell |

5 | A finiteness property for braided fusion categories
- Naidu, Rowell
(Show Context)
Citation Context ...tum group categories C(soN, q, ℓ) for ℓ = 2N−2 and ℓ = 2N, respectively, and for N even for ℓ = N −1 and N, respectively. These level 2 categories (SO(N), 2) have been studied elsewhere, see [Gn] and =-=[NR]-=-. The corresponding fusion rules also appeared in older work, see [J3] and [dBG]. In light of Conjectures 3.1, 4.1 and [RSW, Conjecture 6.6], we expect that that the (unitary) braid group representati... |

5 |
Semisimple and modular tensor categories from link invariants
- Turaev, Wenzl
- 1997
(Show Context)
Citation Context ...b)object of X ⊗n . Thus ⊕ i W n n , i is a faithful End(X ⊗n )-module as well as for ρX(CBn) (although the W n i might be reducible as Bn-representations). If the category in question is unitary (see =-=[TW]-=-) then the Bn representations are also unitary. The representations obtained from quotients of Temperley-Lieb, Hecke and BMW algebras at roots of unity can be obtained in this setting by choosing an a... |

4 |
Two paradigms for topological quantum computation
- Rowell
- 2009
(Show Context)
Citation Context ... if and only if ℓ ̸∈ {3, 4, 6}. Also it is worth observing that the Jones polynomial evaluations at these roots of unity are “classical” (see [J2]) and can be exactly computed in polynomial time (see =-=[R3]-=- for a discussion of this phenomonon). 6. Generalizations The categories C(sl2, q, 4) and C(sl2, q, 6) are members of two families of weakly integral, unitary braided fusion categories. In this sectio... |

3 | Braid representations from quantum groups of exceptional Lie type - Rowell |

2 |
Braid group representations via the Yang Baxter Equation
- Franko
- 2007
(Show Context)
Citation Context ...he conjugacy classes are have a monomial representative: a matrix that is the product of a diagonal matrix D and a permutation matrix P. The corresponding braid group representations were analyzed in =-=[Fr]-=-, and found to have finite image under the assumption that R = DP has finite order. The only unitary solution not of monomial form is: ⎛ ⎞ 1 √ 2 1 0 0 1 0 1 −1 0 ⎜ ⎟ ⎝ 0 1 1 0⎠ −1 0 0 1 . The correspo... |

2 |
Notes on subfactors and statistical mechanics. Braid group, knot theory and statistical mechanics
- Jones
- 1989
(Show Context)
Citation Context ...y, and for N even for ℓ = N −1 and N, respectively. These level 2 categories (SO(N), 2) have been studied elsewhere, see [Gn] and [NR]. The corresponding fusion rules also appeared in older work, see =-=[J3]-=- and [dBG]. In light of Conjectures 3.1, 4.1 and [RSW, Conjecture 6.6], we expect that that the (unitary) braid group representations associated with (SO(N), 1) and (SO(N), 2) (a) have finite image an... |

2 |
Extraspecial two-groups, generalized Yang-Baxter equations and braiding quantum gates, Quantum Inf
- Rowell, Zhang, et al.
(Show Context)
Citation Context ...e general quasi-localization that incorporates this approach, but this would take us too far afield so we ignore this possibility for the present. • A generalized version of the YBE was introduced in =-=[RZWG]-=-. The idea is that we can consider R ∈ End(W ⊗k ) for k > 2 and some vector space W satisfying: (R ⊗ Idm)(Idm ⊗ R)(R ⊗ Idm) = (Idm ⊗ R)(R ⊗ Idm)(Idm ⊗ R) where Idm := I ⊗m W with m ≥ 1. Such generaliz... |

2 |
The Yang-Baxter equation and invaraints of links
- Turaev
- 1988
(Show Context)
Citation Context ...-matrix.Associated with any object X in a modular category is a link invariant InvX(L). Given an R-matrix one may also define link invariants TR(L, µi, α, β) for any enhancement (µi, α, β) of R (see =-=[Tu1]-=-). If (W, R) is a localization of (ρX, End(X ⊗n )) then one expects the invariant InvX(L) to be directly related to TR(L, µi, α, β). For example, the localization of the Jones representations at level... |

2 |
Topological Quantum Computation
- Wang
- 2010
(Show Context)
Citation Context ... space V with an orthonormal basis {ei}, i = 1, . . .,m would be of the form R(h)(ei ⊗ ej) = e ihHij(h) (ei ⊗ ej) for some smooth family of matrices Hij(h). Topological quantum computation (TQC) (see =-=[Wa]-=- for references) stirred great interests in unitary representations of the braid groups, which describe the statistics of quasi-particles in condensed matter physics [NSSFD]. In TQC, unitary matrices ... |

1 |
Anyon Condensation and Continuous Topological Phase Transitions
- Barkeshli, Wen
(Show Context)
Citation Context ... by Turing machines [J4, KMM]. These results provide evidence to our conjectures. The theory (SO(3), 2) ∼ = (SU(2), 4) has been proposed as a description of the FQH liquid at filling fraction ν = 8/3 =-=[BW]-=-. We believe a similar proposal can be made for (SO(p), 2) for filling fraction ν = 2 + 2/p when p is prime. Of particular interests is the p = 5 case due to the existence of the FQH liquid at ν = 12/... |

1 | On braided fusion categories I arXiv:0906.0620 - Drinfeld, Gelaki, et al. |

1 |
Wang Projective Ribbon Permutation Statistics: a Remnant of non-Abelian Braiding in Higher Dimensions
- Freedman, Hastings, et al.
(Show Context)
Citation Context ...y computable by Turing machines [FZ, J4, KMM, LRW, R3]. Property F TQFTs might also be easier to find in real systems and relevant to the non-abelian statistics of extended objects in dimension three =-=[FHNQWW]-=-. We also conjecture that topological quantum computing models from Property F TQFTs can be simulated efficiently by Turing machines. 2. Localization of braid group representations The (n-strand) brai... |

1 | Cyclotomic invariants for links - Kobayashi, Murakami, et al. - 1988 |

1 |
The representations of TemperleyLieb algebra and entanglement in a Yang-Baxter system
- Sun, Wang, et al.
(Show Context)
Citation Context ...re: (1, 2, 1) for (1, Y, Z) and ( √ 3, √ 3) for (X, X ′ ). We remark that we obtained the localization matrix R in Theorem 5.1 by modifying a solution to the two-parameter version of the YBE found in =-=[Setal]-=-. With a bit more work we could have obtained these from Jones’ [J1] description in terms of automorphisms of 3-groups. This approach will be described below in a more general setting. The following v... |