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.     

Localization of Unitary Braid Group Representations

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.     

From Braid Group Representations with Four Eigenvalues to Solutions of Yang-Baxter Equation

A recursive formula of Jimbo-type trigonometric Yang-Baxterization is presented. The consistency conditions of the Yang-Baxterization for four eigenvalues are obtained. Several 4 × 4 solutions of Yang-Baxter equations are obtained from a braid group representation.     

General Eight-Vertex Model and Braid Group Representation

The braid group representations (BGRs) corresponding to general eight-vertex model are obtained for the case in which the total spins are not conserved. A Yang-Baxterization scheme is applied to obtain from these BGRs.     

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 q = 2 and 3. The BGRB obtained here are constant matrices.     

Baxter-Bazhanox Model,Frenkel-Moore Equation and the Braid Group

In this paper the three-dimensional vertex model is given, which is the duality of the threedimensional Baxter-Bazhanov (BE) model. The braid group corresponding to Frenkel-Moore equation is constructed and the transformations R, I are found. These maps act on the group and denote the rotations of the braids through the angles π about some special axes. The weight function of another three-dimensional .vertex model related the 3D laettice integrable model proposed by Boos, Mangazeev, Sergeev and Stroganov is presented also, which can be interpreted as the deformation of the vertex model corresponding to the BB model.     

A New Class of Representations of Braid Groups

In this letter, a new class of represen tations of the braid groups B_N, I_i-1⊕ T ⊕ I_N-i-1 is constructed. It is proved that those represen tations contain three kinds of irreducible representations: the trivial (identity) one, the Burau one, and a new N-dimensional one. The explicit form of the N-dimensional irreducible representation of the braid group B_N is given.     

Enhanced Gauge Symmetry and Braid Group Actions

 Enhanced gauge symmetry appears in Type II string theory (as well as F- and M-theory) compactified on Calabi–Yau manifolds containing exceptional divisors meeting in Dynkin configurations. It is shown that in many such cases, at enhanced symmetry points in moduli a braid group acts on the derived category of sheaves of the variety. This braid group covers the Weyl group of the enhanced symmetry algebra, which itself acts on the deformation space of the variety in a compatible way. Extensions of this result are given for nontrivial B-fields on K3 surfaces, explaining physical restrictions on the B-field, as well as for elliptic fibrations. The present point of view also gives new evidence for the enhanced gauge symmetry content in the case of a local A _2n -configuration in a threefold having global ℤ/2 monodromy. Received: 28 October 2002 / Accepted: 9 December 2002 Published online: 28 May 2003 Communicated by R.H. Dijkgraaf     

Solutions to Braid Group,Yang-Baxter Equation and Integrable Fermion Systems

Braid group representations are found in ferrnion systems in one space dimension. Explicit baxterization is performed to find corresponding new trigonometric solu tions of Yang-Baxter equation. The quantum algebra structures implied in the new solutions are discovered. Algebraic Bethe ansatz method is applied to solving these systems. The relationships between these fermion systems and polaron model, spin-1/2 Heisenberg spin chain are discussed.     

Poisson Algebras of Block-Upper-Triangular Bilinear Forms and Braid Group Action

In this paper we study a quadratic Poisson algebra structure on the space of bilinear forms on ${\mathbb{C}^{N}}$ C N with the property that for any ${n, m \in \mathbb{N}}$ n , m ∈ N such that n m = N, the restriction of the Poisson algebra to the space of bilinear forms with a block-upper-triangular matrix composed from blocks of size ${m \times m}$ m × m is Poisson. We classify all central elements and characterise the Lie algebroid structure compatible with the Poisson algebra. We integrate this algebroid obtaining the corresponding groupoid of morphisms of block-upper-triangular bilinear forms. The groupoid elements automatically preserve the Poisson algebra. We then obtain the braid group action on the Poisson algebra as elementary generators within the groupoid. We discuss the affinisation and quantisation of this Poisson algebra, showing that in the case m = 1 the quantum affine algebra is the twisted q-Yangian for ${\mathfrak{o}_{n}}$ o n and for m = 2 is the twisted q-Yangian for ${(\mathfrak{sp}_{2n})}$ ( sp 2 n ) . We describe the quantum braid group action in these two examples and conjecture the form of this action for any m > 2. Finally, we give an R-matrix interpretation of our results and discuss the relation with Poisson–Lie groups.     

Fusion Rings of Loop Group Representations

We compute the fusion rings of positive energy representations of the loop groups of the simple, simply connected Lie groups.     

Finite- and Infinite-Dimensional Representations of the Lorentz Group

Finite- and infinite-dimensional representations of the Lorentz group are discussed and various topics in which this group is currently in use are mentioned. The infinitesimal approach of finding representations is reviewed and all finite-dimensional spinor representations of the Lorentz group are obtained. Infinite-dimensional representations are then discussed, including the principal, complementary, and complete series of representations. A generalized Fourier transformation is introduced which enables one to use the global approach to representation theory so as to express infinite-dimensional representations in terms of matrices. This method is shown to lead to a generalization of the spinor form of finite-dimensional representation to the infinite-dimensional case. However, whereas the usual spinor representations are nonunitary, the obtained new form describes both unitary and non-unitary representations, depending on the choice of certain parameters appearing in the representation formula.     

Hypercomplex Representations of the Heisenberg Group and Mechanics

In the spirit of geometric quantisation we consider representations of the Heisenberg(–Weyl) group induced by hypercomplex characters of its centre. This allows to gather under the same framework, called p-mechanics, the three principal cases: quantum mechanics (elliptic character), hyperbolic mechanics and classical mechanics (parabolic character). In each case we recover the corresponding dynamic equation as well as rules for addition of probabilities. Notably, we are able to obtain whole classical mechanics without any kind of semiclassical limit ħ→0.     

The Calogero-Sutherland Model and Generalized Classical Polynomials

Multivariable generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of certain Schr?dinger operators for Calogero-Sutherland-type quantum systems. For the generalized Hermite and Laguerre polynomials the multidimensional analogues of many classical results regarding generating functions, differentiation and integration formulas, recurrence relations and summation theorems are obtained. We use this and related theory to evaluate the global limit of the ground state density, obtaining in the Hermite case the Wigner semi-circle law, and to give an explicit solution for an initial value problem in the Hermite and Laguerre case. Received: 16 August 1996 / Accepted: 21 January 1997     

Induced Representations of the One-Dimensional Quantum Galilei Group

We study the representations of the quantum Galilei group by a suitable generalization of the Kirillov method on spaces of noncommutative functions. On these spaces, we determine a quasi-invariant measure with respect to the action of the quantum group by which we discuss unitary and irreducible representations. The latter are equivalent to representations on ², i.e. on the space of square summable functions on a one-dimensional lattice.     

A New Approach to Spinors and Some Representations of the Lorentz Group on Them

We give a geometric realization of space-time spinors and associated representations, using the Jordan triple structure associated with the Cartan factors of type 4, the so-called spin factors. We construct certain representations of the Lorentz group, which at the same time realize bosonic spin-1 and fermionic spin- wave equations of relativistic field theory, showing some unexpected relations between various low-dimensional Lorentz representations. We include a geometrically and physically motivated introduction to Jordan triples and spin factors.     

Geometrical Aspects of Lie Group Representations and Their Optical Applications

A new procedure to obtain unitary and irreducible representations of Lie groups starting from the cotangent bundle of the group (the cotangent group) is presented. Some applications of the construction in quantumoptics problems are discussed. The notion of phase space of a Lie group is studied. The possibility of describing the quadrature components of a photon, in view of the Lie group phase space, is pointed out. Examples of two and threedimensional Lie groups including Heisenberg–Weyl group are considered.     

Hamiltonian Approach to Meson Structure in Extended Nambu-Jona-Lasinio Model

The Nambu-Jona-Lasinio model,is extended to inclusion of the nonlocal effective quark interaction with confinement. An effective Hamiltonian in the instantaneous approximation is used to study meson properties in the random phse approximation.     

Representations of the algebra sl(2, R) and integration of Hamiltonian systems

A new representation of the sl(2, R) is given, which is related to the integrable N-particle system with inversely quadratic potential. The Hamiltonian plays the role of a raising operator and integrals correspond to highest weight vectors.