Universal tangle invariant and commutants of quantum algebras

Lee, H C

doi:10.1088/0305-4470/29/2/019

Cited by 3 publications

(6 citation statements)

References 21 publications

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“…Reshetikhin [72,Section 4] and Lee [48] considered universal invariants of .1; 1/-tangles (ie, tangles with one endpoint on the top and one on the bottom) with values in the center of a quantum group, which can be thought of as the .1; 1/-tangle version of the universal link invariant. Universal invariants are further generalized to more general oriented tangles by Lee [50;49;51] and Ohtsuki [65]. Kauffman [35] and Kauffman and Radford [38] defined functorial versions of universal tangle invariant for a generalization of ribbon Hopf algebra which is called "(oriented) quantum algebra".…”

Section: Remark 11mentioning

confidence: 99%

Bottom tangles and universal invariants

Habiro

2006

Algebr. Geom. Topol.

128

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A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of B, which provides an especially convenient way to generate all the bottom tangles, and hence all the framed, oriented links, via closure. We also define a kind of "braided Hopf algebra action" on the set of bottom tangles.Using the universal invariant of bottom tangles associated to each ribbon Hopf algebra H , we define a braided functor J from B to the category Mod H of left H -modules. The functor J, together with the set of generators of B, provides an algebraic method to study the range of quantum invariants of links. The braided Hopf algebra action on bottom tangles is mapped by J to the standard braided Hopf algebra structure for H in Mod H .Several notions in knot theory, such as genus, unknotting number, ribbon knots, boundary links, local moves, etc are given algebraic interpretations in the setting involving the category B. The functor J provides a convenient way to study the relationships between these notions and quantum invariants. 57M27; 57M25, 18D10 IntroductionThe notion of category of tangles (see Yetter [84] and Turaev [80]) plays a crucial role in the study of the quantum link invariants. One can define most quantum link invariants as braided functors from the category of (possibly colored) framed, oriented tangles to other braided categories defined algebraically. An important class of such functorial tangle invariants is introduced by Reshetikhin and Turaev [74]: Given a ribbon Hopf algebra H over a field k , there is a canonically defined functor F W T H ! Mod

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Section: Remark 11mentioning

confidence: 99%

Bottom tangles and universal invariants

Habiro

2006

Algebr. Geom. Topol.

128

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“…The same multiparameter universal R matrix can also be obtained in a different way. Denote R in (17) as R(H 1 , H 2 ). Let u = 1 and define q H ′ = q H t −J , we obtain a single-parameter U q gl(2) and universal R matrix denoted by R(H ′ , H ′ ):…”

Section: Quantum Doublementioning

confidence: 99%

“…As is well known, the Yang-Baxter equation (YBE) [1,8] plays an essential role in the study of quantum groups (QG) and quantum algebras (QA) [2,3,4,5,6,7,8], integrable models [9,10,11,12], as well as in the construction of knot or link invariants [13,14,15,16,17,18,19]. For instance, in the Faddeev-Reshetikhin-Takhtajan (FRT) approach [4,5,6] to construct quantum groups or quantum algebras, one has to find an R matrix, which is a matrix solution of YBE [8], then using this R matrix as the input, substituting it into the RT T or RLL relations to get the quantum group or quantum algebra as the output.…”

Section: Introductionmentioning

confidence: 99%

Colored solutions of the Yang–Baxter equation from representations of Uqgl(2)

Luan

Lee

Zhang

2000

Journal of Mathematical Physics

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Representations of the quantum doubles of finite group algebras and spectral parameter dependent solutions of the Yang-Baxter equation J. Math. Phys. 47, 103511 (2006) We study the Hopf algebra structure and the highest weight representation of a multiparameter version of U q gl(2). The Hopf algebra maps of this algebra are explicitly given. We show that the multiparameter universal R matrix can be constructed directly as a quantum double intertwiner without using Reshetikhin's twisting transformation. We find there are two types highest weight representations for this algebra: type a corresponds to the qeneric q and type b corresponds to the case that q is a root of unity. When applying the representation theory to the multiparameter universal R matrix, both standard and nonstandard colored solutions of the Yang-Baxter equation are obtained.

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“…One might think that F (L) would be labeled by ρ k , but it has been shown [19] that Eqs. Reidemeister [21], begins by making a planar projection of a link, called a link diagram, in such a way that the projection is a network whose only nodes are either one of two kinds of crossings -overcrossing or undercrossing -each being a four-valent planar diagram.…”

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confidence: 99%

“…Topologically equivalent classes of links are classified according to isotopic classes of link diagrams. The connection to a gauge group either through a skein relation [3], the braid group [22] or directly [19] is made by mapping the overcrossing (undercrossing, resp.) to (the inverse of, resp.)…”

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confidence: 99%

Quantum holonomy in three-dimensional general covariant field theory and the link invariant

1997

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We consider quantum holonomy of some three-dimensional general covariant non-Abelian field theory in the Landau gauge and confirm a previous result partially proven. We show that quantum holonomy retains metric independence after explicit gauge fixing and hence possesses the topological property of a link invariant. We examine the generalized quantum holonomy defined on a multicomponent link and discuss its relation to a polynomial for the link. ͓S0556-2821͑97͒02314-X͔ PACS number͑s͒: 11.15.Tk, 02.40.Pc, 03.65.Fd Some time ago it was shown that quantum holonomy in a three-dimensional general covariant non-Abelian gauge field theory possesses topological information of the link on which the holonomy operator is defined ͓1͔. The quantum holonomy operator was shown to be a central element of the gauge group so that, in a given representation of the gauge group, it is a matrix that commutes with the matrix representations of all other operators in the group. In an irreducible representation, it is proportional to the identity matrix. Quantum holonomy should therefore in general have more information on the link invariant than the quantum Wilson loop which, for the SU͑2͒ Chern-Simons quantum field theory, was shown by Witten ͓2͔ to yield the Jones polynomial ͓3͔. Horne ͓4͔ extended Witten's result to some other Lie groups. The difference between quantum holonomy and the Wilson loop becomes apparent in the SU(N͉N) Chern-Simons theory, where the quantum Wilson loop vanishes identically for any link owing to the property of supertrace, but the quantum holonomy ͓1͔ yields the important AlexanderConway polynomial ͓5-8͔.However, the argument used in Ref.͓1͔ was based only on the formal properties of the functional integral and complications that may arise from the necessity for gauge fixing in any actual computation were not taken into consideration. In addition, in a case when a metric is needed for gauge fixing, the metric independence of quantum holonomy may be violated. Furthermore, in the standard Faddeev-Popov technique used for gauge fixing, ghost fields and auxiliary fields that are introduced reduce the original local gauge symmetry to Becchi-Rouet-Stora-Tyutin ͑BRST͒ symmetry, and it is no longer certain that the formal arguments and manipulations used in Ref. ͓1͔ to derive its results are still valid. As well, the case of the quantum holonomy defined on multicomponent links was not explicitly considered.The main aim of the present work is to clarify these problems. We explicitly work in the Landau gauge 1 and confirm the results obtained in Ref. ͓1͔ for the case of a onecomponent contour. We then show that a quantum holonomy operator defined on a n-component link, which by construction is a tensor product of those operators defined on the components, is a central element of the universal enveloping algebra of the Lie algebra of the gauge group and, when evaluated on a set of n irreducible representations of the gauge group, has a uniquely defined eigenvalue that is a polynomial invariant of the link.The quantum h...

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Universal tangle invariant and commutants of quantum algebras

Cited by 3 publications

References 21 publications

Bottom tangles and universal invariants

Bottom tangles and universal invariants

Colored solutions of the Yang–Baxter equation from representations of Uqgl(2)

Quantum holonomy in three-dimensional general covariant field theory and the link invariant

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