Braid Group Representations Research Articles

FERMIONS AND LINK INVARIANTS

This paper deals with various aspects of knot theory when fermionic degrees of freedom are taken into account in the braid group representations and in the state models. We discuss how the Ř matrix for the Alexander polynomial arises from the Fox differential calculus, and how it is related to the quantum group Uqgl(1,1). We investigate new families of solutions of the Yang Baxter equation obtained from "linear" representations of the braid group and exterior algebra. We study state models associated with Uqsl(n,m), and in the case n=m=1 a state model for the multivariable Alexander polynomial. We consider invariants of links in solid handlebodies and show how the non trivial topology lifts the boson fermion degeneracy that is present in S3. We use "gauge like" changes of basis to obtain invariants in thickened surfaces Σ×[0,1].

Representations of braid group and Birman-Wenzl algebra

The necessary and sufficient condition for a braid group representation with a cubic reduction relation to give rise to a representation of the Birman-Wenzl algebra is given. For the case with an arbitrary order reduction relation, a wider sufficient condition is proposed.

NEW SOLUTIONS OF YANG-BAXTER EQUATION AND NEW LINK POLYNOMIALS

A new family of braid group representations associated with Lie algebras An, Bn, Cn and Dn is proved to be Birman-Wenzl algebra, then Yang-Baxterize them to trigonometric and rational solutions of YBE. New link polynomials are set up.

PERMUTATION REPRESENTATIONS OF BRAID GROUPS OF SURFACES

In this paper all primitive representations of braid groups on strings of surfaces of finite type in groups of permutations of symbols are found. As an application it is proved that the groups of pure braids of surfaces are characteristic subgroups of the braid groups. These results generalize Artin's classical results on Artin's braid groups.

New constant and trigonometric 4*4 solutions to the Yang-Baxter equations

A classification of triangular constant solutions of the Yang-Baxter equations is performed. The formulae for the Baxterization of braid group representation are used for finding trigonometric solutions of the Yang-Baxter equations from the constant ones. Some of the obtained solutions correspond to known solutions but several new ones are obtained as well. The calculations confirm that the Baxterization formulae for more than two eigenvalues of the R-matrix are not universal.

The Braid Group Representation of IRF Model

The braid group representations (BGRs) corresponding to IRF model are discussed by solving the spectral-independent Yang-Baxter Equation in the cases and 3. The BGRs obtained here are constant matrices.

Link polynomials related to the new braid group representations

By introducing a modified diagonal matrix h, the properties of Markov moves are examined for the non-standard representations of the braid group associated with the fundamental representations of An, Bn, Cn, and Dn. It is shown that link polynomials can be constructed from those braid group representations. The skein relations of link polynomials are obtained explicitly.

EXPLICIT TRIGONOMETRIC YANG-BAXTERIZATION

We present an explicit prescription for trigonometric Yang-Baxterization. Given a braid group representation (BGR) in appropriate form, our prescription generates explicit solutions to the quantum Yang-Baxter equations (QYBE). We have proved the correctness of this prescription, together with a variation of it, for BGR’s having two or three unequal eigenvalues. All the known explicit results obtained by Jimbo and Bazhanov are reproduced. More YB solutions are obtained from “standard” BGR’s associated with fundamental and higher dimensional representations of simple Lie algebras. Our prescription also applies to the new “nonstandard” BGR’s, which are beyond the reach of present quantum group techniques but obtainable by our previous direct method. New explicit examples include the standard ones with 6 of SU(3), 10 of SU(5) and exotic ones with C2 and D2, etc. The classical limit of our YB solutions is also studied. It turns out that some of our exotic QYB solutions have an unusual classical limit. Our results suggest that the QYBE embodies a much richer structure than we thought.

New hierarchy of colored-braid-group representations.

Novel solutions of the Yang-Baxter relation for a series of multistate vertex models are presented in the braid-group limit. The obtained family of solutions gives a sequence of ``colored''-braid-group representations. Possible generalizations of the multivariable Alexander polynomial are discussed.

Weight conservation condition and structure of the braid group representation

A weight conservation condition for the S-matrix of the braid group representation, from which non-vanishing elements of the S-matrix can be determined, is introduced. A method of weight lattice analysis of representations of Lie algebra is suggested and used to give structures of braid group representations in various cases. This makes it feasible for braid group representations to be obtained by solving the spectral parameter independent Yang-Baxter equation directly, which leads to several new braid group representations.

Index of subfactors and statistics of quantum fields

The endomorphism semigroup End(M) of an infinite factor M is endowed with a natural conjugation (modulo inner automorphisms) {Mathematical expression}, where I³ is the canonical endomorphism of M into I(M). In Quantum Field Theory conjugate endomorphisms are shown to correspond to conjugate superselection sectors in the description of Doplicher, Haag and Roberts. On the other hand one easily sees that conjugate endomorphisms correspond to conjugate correspondences in the setting of A. Connes. In particular we identify the canonical tower associated with the inclusion[Figure not available: see fulltext.] relative to a sector I. As a corollary, making use of our previously established index-statistics correspondence, we completely describe, in low dimensional theories, the statistics of a selfconjugate superselection sector I with 3 or less channels, in particular of sectors with statistical dimension d(I)<2, by obtaining the braid group representations of V. Jones and Birman, Wenzl and Murakami. The statistics is thus described in these cases by the polynomial invariants for knots and links of Jones and Kauffman. Selconjugate sectors are subdivided into real and pseudoreal ones and the effect of this distinction on the statistics is analyzed. The FYHLMO polynomial describes arbitrary 2-channels sectors. © 1990 Springer-Verlag.

Quantum groups constructed from the non-standard braid group representations in the Faddeev-Reshetikhin-Takhtajan approach

Quantum groups constructed from the non-standard braid group representations in the Faddeev-Reshetikhin-Takhtajan approach

New solutions of Yang-Baxter equations: Birman-Wenzl algebra and quantum group structures

New braid group representations associated with the fundamental representations of Bn, Cn and Dn are derived. They are shown to satisfy Birman-Wenzl algebra. By Yang-Baxterization the authors obtain new solutions of Yang-Baxter equations, including the twisted extensions. Quantum group structures related to these new solutions are explicitly given.

From representations of the braid group to solutions of the Yang-Baxter equation

A systematic method is developed for constructing solutions of the Yang-Baxter equation from given braid group representations, arising from such finite dimensional irreps of quantum groups that any irrep can be affinized and the tensor product of the irrep with itself is multiplicity-free. The main tool used in the construction is a tensor product graph, whose circuits give rise to consistency conditions. A maximal tree of this graph leads to an explicit formula for the quantum R-matrix when the consistency conditions are satisfied. As examples, new solutions of the Yang-Baxter equation are found, corresponding to braid group generators associated with the symmetric and antisymmetric tensor irreps of U q [ gl( m)], a spinor irrep of U q [ so(2 n)]. and the minimal irreps of U q [E 6] and U q [E 7].

New solutions of Braid group representations associated with Yang–Baxter equation

By directly solving quantum Yang–Baxter equations (QYBE) in terms of the extended expansion scheme, besides the known solutions we find new family of solutions of braid group representations associated with the fundamental representations of An, Bn, Cn, and Dn.

Anyons on a torus: Braid group, Aharonov-Bohm period, and numerical study.

We present a careful construction of anyons on a torus starting with the braid-group analysis. The rules of Wen, Dagotto, and Fradkin for putting anyons on a torus are reproduced with some minor improvements. The existence of noncontractible loops leads to braid-group representations characterized not only by anyon statistics \ensuremath{\theta} but also by the magnetic fluxes ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ threading through the holes of the torus. The three parameters are tangled with each other. We explore the symmetries of the torus to separate the effects of ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ from those of \ensuremath{\theta}. It is shown that the anyon system always has a smaller period \ensuremath{\theta}/\ensuremath{\pi} in ${\mathrm{\ensuremath{\Phi}}}_{\mathit{x}}$ and ${\mathrm{\ensuremath{\Phi}}}_{\mathit{y}}$ than the natural period 1. We perform several numerical calculations to investigate the spectral flow and consistency of the method and find interesting features in the spectral flow, which are relevant in understanding the fractional quantum Hall effect.

QUANTUM LIE SUPERALGEBRAS AND “NON-STANDARD” BRAID GROUP REPRESENTATIONS

The formulae of the Faddeev-Reshetikhin-Takhtajan (FRT) method in supersymmetric case are presented transparently and consistently. With the help of these formulae, the simplest “non-standard” solution of braid group representation (BGR) is re-examined. The result shows that the hidden symmetry associated with this “non-standard” BGR is indeed the q-deformed Lie superalgebra U q ( gl (1|1)).

A new quantum group associated with a ?nonstandard? braid group representation

A new quantum group is derived from a ‘nonstandard’ braid group representation by employing the Faddeev-Reshetikhin-Takhtajan constructive method. The classical limit is not a Lie superalgebra, despite relations like x 2−y 2=0. We classify all finite-dimensional irreducible representations of the new Hopf algebra and find only one- and two-dimensional ones.

NEW SOLUTIONS OF THE YANG-BAXTER EQUATION AND THEIR YANG-BAXTERIZATION

Through the examples associated with simple Lie algebras B2, C2 and D2, an approach of constructing new solutions of the spectral-independent Yang-Baxter equation (i.e. the braid group representations) was shown. These new solutions possess some particular features, such as that they do not have the classical limit in the usual sense of Refs. 2, 5, etc., and give rise to the new solutions of the x-dependent Yang-Baxter equation by using the Yang-Baxterization prescription.

Yang-Baxterization of braid group representation associated with the seven-dimensional representation of G2

The explicit expression of a solution that relates to the seven-dimensional representation of the Lie algebra G2 of the quantum Yang-Baxter equation (QYBE) is obtained by applying the Yang-Baxterization procedure to the braid group representation (BGR). The result is consistent with an earlier one derived by a different method.