Braid Group Relations Research Articles

Askey–Wilson Braid Algebra and Centralizer of $$U_q(\mathfrak {sl}_2)$$

A presentation of the centralizer of the threefold tensor product of the spin s representation of the quantum group $$U_q(\mathfrak {sl}_2)$$ is provided. It is expressed as a quotient of the Askey–Wilson braid algebra. This newly defined algebra combines the Askey–Wilson relations with the braid group relations, on three strands, together with a characteristic equation of degree $$2s+1$$ for the braid generators. Explicit bases are given for the centralizer.

LUSZTIG SYMMETRIES AND AUTOMORPHISMS OF QUANTUM SUPERALGEBRAS Uq(osp(1|2r))

We first construct algebra automorphisms Ti (1 ≤ i ≤ r) of ℤ2-graded Hopf algebra Uq( osp (1|2r)), which are called Lusztig's symmetries. It is shown that they satisfy braid group relations. The algebraic automorphisms of Uq( osp (1|2r)) are also described.

Orbifold Riemann surfaces and geodesic algebras

We study the Teichmüller theory of Riemann surfaces with orbifold points of order 2 using the fat graph technique. The previously developed technique of quantization, classical and quantum mapping-class group transformations, and Poisson and quantum algebras of geodesic functions is applicable to the surfaces with orbifold points. We describe classical and quantum braid group relations for particular sets of geodesic functions corresponding to An and Dn algebras and describe their central elements for the Poisson and quantum algebras.

Teichmüller Theory of Bordered Surfaces

We propose the graph description of Teichmuller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formu- late this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of Thurston variables (foliation-shear coordi- nates), mapping-class group invariance (both classical and quantum), Poisson and quantum algebra of geodesic functions, and classical and quantum braid-group relations. These new algebras can be defined on the double of the corresponding graph related (in a novel way) to a double of the Riemann surface (which is a Riemann surface with holes, not a smooth Riemann surface). We enlarge the mapping class group allowing transformations relating different Teichmuller spaces of bordered surfaces of the same genus, same number of boun- dary components, and same total number of marked points but with arbitrary distributions of marked points among the boundary components. We describe the classical and quantum algebras and braid group relations for particular sets of geodesic functions corresponding to An and Dn algebras and discuss briefly the relation to the Thurston theory.

Mickelsson algebras and Zhelobenko operators

We construct a family of automorphisms of Mickelsson algebra, satisfying braid group relations. The construction uses ‘Zhelobenko cocycle’ and includes the dynamical Weyl group action as a particular case.

Ringel–Hall algebras and Lusztig's symmetries

Let D (Λ) be the double Ringel–Hall algebra of a finite dimensional hereditary algebra Λ . The present paper shows that the BGP-reflection operators of D (Λ) coincide with Lusztig's symmetries (up to canonical isomorphisms) and satisfy the braid group relations on the whole double Ringel–Hall algebra D (Λ) . This answers a question of Sevenhant and Van den Bergh [J. Algebra 221 (1999) 135–160].

BGP-Reflection Functors and Lusztig's Symmetries: A Ringel–Hall Algebra Approach to Quantum Groups

According to the canonical isomorphisms between the Ringel–Hall algebras (composition algebras) and the quantum groups, we deduce Lusztig's symmetries T″i,1,i∈I, by applying the Bernstein–Gelfand–Ponomarev reflection functors to the Drinfeld doubles of Ringel–Hall algebras. The fundamental properties of T″i,1 including the following can be obtained conceptually. (1) T″i,1,i∈I induce automorphisms of the quantum groups Uq(g) and on the integrable modules. (2) T″i,1,i∈I satisfy the braid group relations. This extends and completes the results of B. Sevenhant and M. Van den Bergh (1999, J. Algebra221, 135–160).

Link invariants of finite type and perturbation theory

The Vassiliev-Gusarov link invariants of finite type are known to be closely related to perturbation theory for Chern-Simons theory. In order to clarify the perturbative nature of such link invariants, we introduce an algebra V_infinity containing elements g_i satisfying the usual braid group relations and elements a_i satisfying g_i - g_i^{-1} = epsilon a_i, where epsilon is a formal variable that may be regarded as measuring the failure of g_i^2 to equal 1. Topologically, the elements a_i signify crossings. We show that a large class of link invariants of finite type are in one-to-one correspondence with homogeneous Markov traces on V_infinity. We sketch a possible application of link invariants of finite type to a manifestly diffeomorphism-invariant perturbation theory for quantum gravity in the loop representation.

Statistics of spinning particles in 2+1 gravity.

We examine the dynamics of {ital N} spinning particles in 2+1 gravity, and obtain an effective Lagrangian description for the particles by eliminating the dreibeins and spin connections from the theory. The resulting equations of motion show that locally the particles are free, but globally their conserved momenta and angular momenta are not well defined. Conditions are found under which the effective action is invariant under particle exchanges. Ambiguities exist in passing to the quantum theory and these ambiguities can be exploited to obtain exotic statistics for the particles. We construct exchange operators which give exotic statistics, but which do not in general satisfy the braid group relations when {ital N}{gt}2.

Deformed SU(2) Heisenberg chain

The general Hamiltonian for the SU(2)q-invariant arbitrary-spin Heisenberg chain is presented. Some of these interactions are shown to satisfy braid group relations and the Temperley-Lieb algebra relations. Spin 3/2 is explored in more detail and, in this case, a general solution to the braid conditions is known in the generic sense.

STRANGE STATISTICS, BRAID GROUP REPRESENTATIONS AND MULTIPOINT FUNCTIONS IN THE N-COMPONENT MODEL

The statistics of fields in low dimensions is studied from the point of view of the braid group Bn of n strings. Explicit representations MR for the N-component model, N=2 to 5, are derived by solving the Yang-Baxter-like braid group relations for the statistical matrix R, which describes the transformation of the bilinear product of two N-component fields under the transposition of coordinates. When R2≠1 the statistics is neither Bose-Einstein nor Fermi-Dirac; it is strange. It is shown that for each N, the N+1 parameter family of solutions obtained is the most general one under a given set of constraints including “charge” conservation. Extended Nth order (N>2) Alexander-Conway relations for link polynomials are derived. They depend nonhomogeneously only on one of the N+1 parameters. The N=3 and 4 ones agree with those previously derived by Akutsu et al. Flat connections ω defining integrable systems of the N-component model are derived from the representations. The monodromy of the solution of such a system also carries a representation Mω of Pn⊂Bn. For N=2, Mω=MR, but the equality may not hold in general. The connections also lead directly to solutions of the classical Yang-Baxter equation. A generalization of Kohno’s monodromy representation of Bn associated with the algebra sl(N,C) is given. Applications of the braid group representations to statistical models, conformal field theory and many-body systems of extended objects are briefly discussed.