Braid Group Representations Research Articles

A New Representation of Birman-Wenzl Algebra and Solution of Yang-Baxter Equation11The project partially supported by National Natural Science Foundation of China and the Local Natural Science Foundation of Xinjiang.

Corresponding to a new braid group representation, a representation of Birman-Wenzl algebra is provided. The trigonometric type solutions of spectral-dependent Yang-Baxter equation are obtained in the framework of the Yang-Baxterization, and the related vertex models are also mentioned.

Quantum harmonic oscillator algebra and link invariants

Associated with the q-deformation of the harmonic oscillator algebra we define an infinite dimensional braid group representation on the Hilbert space of the harmonic oscillator, and an extended Yang–Baxter system in the sense of Turaev. The corresponding link invariant is computed in some particular cases and coincides with the inverse of the Alexander–Conway polynomial. The R matrix of Uq(h4) can be interpreted as defining a baxterization of the intertwiners for semicyclic representations of SU(2)q at q=e2πi/N in the N→∞ limit. Finally we define new multicolored braid group representations and study their relation to the multivariable Alexander–Conway polynomial.

Spinning braid-group representation and the fractional quantum Hall effect

The path-integral approach to representing braid group is generalized for particles with spin. Introducing the notion of charged winding number in the super-plane, we represent the braid-group generators as homotopically constrained Feynman kernels. In this framework, super Knizhnik-Zamolodchikov operators appear naturally in the hamiltonian, suggesting the possibility of spinning nonabelian anyons. We then apply our formulation to the study of fractional quantum Hall effect (FQHE). A systematic discussion of the ground states and their quasi-hole excitations is given. We obtain Laughlin, Halperin and Moore-Read states as exact ground-state solutions to the respective hamiltonians associated to the braid-group representations. The energy gap of the quasi-excitation is also obtainable from this approach.

Extended Birman-Wenzl algebra and Yang-Baxterization

A new algebra A is presented by directly extending the Birman-Wenzl algebra. If a braid group representation with four distinct eigenvalues satisfies A then it can be Yang-Baxterized to derive the solution of the Yang-Baxter equation in terms of a standard scheme.

New Braid Group Representations of C3(D3) and C4(D4) and Their Correspondent Link Polynomials

The new braid group representations of C3(D3) and C4(D4) are obtained by solving the defining relations of braid groups directly. And by introducing the appropriate diagonal matrices h, the associated link polynomials are derived also.

Multiparameter solutions of Yang–Baxter equation from braid group representations

On the basis of weight conservation of the An case, Yang–Baxter relations are solved without the assumption that the upper‐left triangle of each sub‐block of S matrix vanishes. A family of braid group representations having no transpose symmetry are obtained and shown to construct link polynomials. Finally, a sequence of solutions of the Yang–Baxter equation with multiparameters are obtained via the Yang‐Baxterization procedure.

Yang Baxterization

On the basis of the Yang–Baxter equation and the initial condition of Ř matrix, Yang Baxterization of representations of braid group and that of Temperley–Lieb algebra are discussed systematically. It is shown that there exist many Yang-Baxterization formulas beyond the knowns. New developments of Yang-Baxterization approach are presented. As examples some new solutions of the Yang–Baxter equation are given.

Baxterization of ther-matrix for the adjoint representation of U q [D(2, 1;?)

The Baxterization procedure is applied to the Braid group representation arising from the adjoint representation of the quantum supergroup U(q)[D(2, 1: alpha)]. This yields a two-parameter family of solutions to the (parameter dependent) Yang-Baxter equation of interest in supersymmetric lattice models.

Representations of a braid group with transpose symmetry and the related link invariants

Under a weight conservation condition when q is not a root of unity, Yang-Baxter relations are solved without the assumption that the upper-left triangle of each sub-block of the S matrix of the braid group representation (BGR) vanishes. A general form of BGR associated with Bn, Cn and Dn is obtained and shown to construct link invariants. The many-to-one correspondence between BGRs and link invariants is given.

Braid group representations arising from quantum supergroups with arbitrary q and link polynomials

The theory of quantum supergroups is applied to the construction of braid group representations and link invariants. A method developed in an earlier publication is reviewed, which produces link polynomials from braid group representations generated by quantum supergroup invariant Ř matrices. When q is a generic complex parameter, this method, applied to the universal R matrices, yields hierarchies of link polynomials associated with different irreps of quantum supergroups. When q is a root of unity, suitable constraints are imposed on quantum supergroups to turn them into finite-dimensional quasitriangular Z2-graded Hopf algebras. The corresponding universal R matrices are obtained, which, in turn, lead to series of link polynomials upon using the above mentioned method. As concrete examples, the link polynomials related to the vector irreps of the quantum supergroups Uq(osp(M/2n)), all the irreps of Uq(osp(1/2)) with generic q, and a class of doubly atypical irreps of Uq(gl(2/1)) with q a root of unity are studied in detail.

Finite Dimensional Representations of Braid Groups11The project partially supported by National Natural Science Foundation of China.

In this letter the general forms of the finite dimensional representations of braid groups are obtained. The equivalent conditions and the irreducibility of the representations are also discussed.

Link Polynomials and Hamiltonian Corresponding to Braid Group Representations of ZN Model11The project supported in part by National Natural Science Foundation of China.

The more general braid group representations of ZN model are given. Topologically invariant link polynomials related to the model are constructed. Here we present the simplest Hamiltonian with arbitrary coefficients of 1D quantum chain.

Braid Group Representation and Solutions of the Quantum Yang-Baxter Equation Associated with 10-Dimensional Representation of SU(4)

The braid group representation associated with the 10-dimensional representation of SU(4) is derived, based on the weight conservation and extended Kauffman diagrammatic technique. The corresponding trigonometric and rational solutions of the quantum Yang-Baxter equations are constructed in terms of the Yang-Baxterization procedure.

Yang-Baxterization and New Solutions Associated with SU(2) Tensorial Space **The project supported in part by National Natural Science Foundation of China.

By applying extended diagrammatic technique to SU(2) tensorial space new solutions of the correspondent braid group representations and their Yang-Baxterization are established. The explicit forms of the new solutions of Yang-Baxter equation for j = 1 are given. Through this example the general feature of this new type of solutions is figured out.

Irreducible representations of braid groups

The irreducible representations of the braid groups, described by the Young patterns, are obtained in terms of a set of commutant operators. The relations between our representations and the monodromy representations based on the minimal representation of the quantum enveloping algebra Uqsl(2) are discussed briefly.

On quantum groups forZNmodels

The quantum group for the ZN model is studied from the braid group representation. The fundamental representation is constructed from the Weyl relation ZX= omega XZ with omega being an Nth root of unity. In the case of N=2 (the eight vertex model), the quantum group is shown to be a homomorphic image of the GLq(2) with q2=1.

Quantum groups constructed from the nonstandard braid group representations in the Faddeev-Reshetikhin-Takhtajan approach. II

For pt.I see ibid., vol.24, p.L725 (1991). From the nonstandard braid group representation for the fundamental representation of slq(n), the authors obtain the corresponding algebra in the Faddeev-Reshetikhin-Takhtajan approach. The main features of the algebra consist of (X+or-k)2=0, k=1, 2,. . ., n-1, which obliterates half of the generalized Serre relations; an unusual scalar product normalization of the underlying root vectors, besides the presence of two parameters, one of which is a root of unity. Explicit detail is given for the n=5 case. An alternative interpretation suggests that such nonstandard solutions are closely connected with the quantum superalgebras.

Yang-Baxterization of Spin-1 L Operator11The project supported in part by National Natural Science Foundation of China.

The Yang-Baxterization of L operator is performed for spin-1 model, namely, for given braid group representation (BGR) the x-dependent L(x) operator can be determined where x is the spectral parameter. It is pointed out that in order to establish such an x-dependence besides the usual L± operators in accordance with the Faddeev-Reshetikhin-Takhtajan (FRT) approach an additional operator L0 should be introduced. The explicit expression of L0 is given. This Yang-Baxterization prescription preserves the FRT and the YBE is satisfied by L(x) and R(x).

Braid group representations arising from the generalized chiral Potts models

A class of braid group representations are construced for each non-singular bilinear form B : (Z/NZ) l ×(Z/(NZ) l →Z/NZ with N odd. Associated link invariants are given as a Gauss sum involving the Seifert matrix and B. With a special choice of B these representations are Yang-Baxterized to the sl(n) generalizations of the chiral Potts model discovered recently

On the topological phases in a planar N-point vortex system

A topological phase is given for the planar N-point vortex problem. This phase arises from an element in the pure braid group, and is independent of the evolution equations. It is deduced from a regular representation of the group of diffeomorphisms on the plane. This resulting representation of the pure braid group is the monodromy of the hypergeometric functions in N variables. The relations between this topological phase and the usual geometrical phase are discussed.