Properties of noninteger moments in a first passage time problem

We calculate the moments t ^q , whereq is not necessarily an integer, of the first passage time to trapping for a simple diffusion problem in one dimension. If a characteristic length of the system isL and t _q ~L (q) asL, then we show that there is a phase transition atq=q _c such that whenq<q _c ,(g)=0, and forq>q _c , (q) is a linear function ofq. These analytical results can be used to explain results for large moments for diffusion on a hierarchic structure. We also show how to calculate noninteger moments in terms of characteristic functions.     

Spin physics with internal targets and the blast detector

The aim of this paper is to give a set of central elements of the algebras U_q(so_m) and U _q(iso _m ) when q is a root of unity. They are surprisingly arise from a single polynomial Casimir element of the algebra U_q(so₃). It is conjectured that the Casimir elements of these algebras under any values of q (not only for q a root of unity) and the central elements for q a root of unity derived in this paper generate the centers of U_q(so_m) and U _q(iso _m ) when q is a root of unity.     

Representations of the cyclically symmetric q-deformed algebra U q(so3)

An algebra homomorphism from the nonstandard q-deformed (cyclically symmetric) algebra U _q(so₃) to the extension Û _q(sl₂) of the Hopf algebra U _q(sl₂) is constructed. Not all irreducible representations (IR) of U _q(sl₂) can be extended to representations of Û _q(sl₂). Composing the homomorphism with irreducible representations of Û _q(sl₂) we obtain representations of U _q(so₃). Not all of these representations of U _q(so₃) are irreducible. Reducible representations of U _q(so₃) are decomposed into irreducible components. In this way we obtain all IR of U _q(so₃) when q is not a root of unity. A part of these representations turn into IR of the Lie algebra so₃ when q 1.     

Method of action-angle variables and the classical dynamics of a nonlinear lagrangian

The method of action-angle variables is used to obtain the complete periodic solutions of a nonlinear chiral Lagrangian system with the Lagrangian of the formL=1/2 {q² [(q · q)²/(1 - q²)] - [k ₀q²/(1 - q²)]q=(q ₁,q ₂,q ₃) by making suitable canonical transformations. Usual semiclassical quantization procedure may then be applied to obtain the energy levels, which is shown to be in good agreement with exact results.     

On a nonstandard two-parametric quantum algebra and its connections withU p,q (gl(2)) andU p,q(gl(1/1))

A quantum algebraU _{p, q} (,H,X _±) associated with a nonstandardR-matrix with two deformation parameters (p, q) is studied and, in particular, its universal -matrix is derived using Reshetikhin's method. Explicit construction of the (p, q)-dependent nonstandardR-matrix is obtained through a coloured generalized boson realization of the universal -matrix of the standardU _{p, q}(gl(2)) corresponding to a nongeneric case. General finite dimensional coloured representation of the universal -matrix ofU _{p, q}(gl(2)) is also derived. This representation, in nongeneric cases, becomes a source for various (p, q)-dependent nonstandardR-matrices. Superization ofU _{p, q}(,H,X _±) leads to the super-Hopf algebraU _{p, q}(gl(1/1)). A contraction procedure then yields a (p, q)-deformed super-Heisenberg algebraU _{p, q}(sh(1)) and its universal -matrix.     

Disentangling q-Exponentials: A General Approach

We revisit the q-deformed counterpart of the Zassenhaus formula, expressing the Jackson q-exponential of the sum of two non-q-commuting operators as an (in general) infinite product of q-exponential operators involving repeated q-commutators of increasing order, E_q(A+B) = E_q0(A)E_q1 (B) _i=2 E_qi. By systematically transforming the q-exponentials into exponentials of series and using the conventional Baker–Campbell–Hausdorff formula, we prove that one can make any choice for the bases ^qi, i=0, 1, 2, ..., of the q-exponentials in the infinite product. An explicit calculation of the operators C _i in the successive factors, carried out up to sixth order, also shows that the simplest q-Zassenhaus formula is obtained for ₀ = ₁ =1, and ₂ = 2, and ₃ = 3. This confirms and reinforces a result of Sridhar and Jagannathan, on the basis of fourth-order calculations.     

Absence of the partially ordered phase in theq-state models

We investigate theq-state models called (N ,N ) model using an infinitesimal Migdal-Kadanoff renormalization-group method. We distinguish two cases namely the isotropic model and the anisotropic model. The first one presents a critical value ofq,q _c such that forq _c we obtain an Ashkin-Teller phase diagrams while forq>q _c the partially ordered phase disappears then the model exhibits only phase transition between ferromagnetic phase and disordered one. The phase diagrams in the second case are qualitatively similar to one obtained forZ(6) model for all values ofq.     

The cohomology of the space of magnetic monopoles

Denote byX _q the reduced space ofSU ₂ monopoles of chargeq in ³. In this paper the cohomology ofX _q , the cohomology with compact supports ofX _q , and the image of the latter in the former are all calculated as representations of /q which acts onX ₂. This provides a non-trivial lower bound for theL ² cohomology ofX _q which is compatible with some conjectures of Sen. It is also shown that, granted some assumptions about the metric onX _q , itsL ² cohomology does not exceed this bound in the situation referred to in the paper as the coprime case.The work described here was carried out partly at the University of Texas at Austin.     

Classification of bicovariant differential calculi on quantum groups

Suppose thatq is not a root of unity. We classify all bicovariant differential calculi of dimension greater than one on the quantum groupsGL _q (N),O _q (N) andSp _q (N) for which the differentials du _j ⁱ of the matrix entriesu _j ⁱ generate the left module of first order forms. Our first classification theorem asserts that there are precisely two one-parameter families of such calculi onGL _q (N) forN3. In the limitq1 only two of these calculi give the ordinary differential calculus onGL(N). Our second main theorem states that apart from finitely manyq there exist precisely two differential calculi with these properties onO _q (N) andSp _q (N) forN4. This strengthens the corresponding result proved in our previous paper [SS2]. There are four such calculi onO _q (3). We introduce two new 4-dimensional bicovariant differential calculi onO _q (3).     

Nonclassical type representations of the q-deformed algebra U′q(son)

The nonstandard q-deformation U_q(so_n) of the universal enveloping algebra U(so _n ) has irreducible finite dimensional representations which are a q-deformation of the well-known irreducible finite dimensional representations of U(so _n ). But U_q(so_n) also has irreducible finite dimensional representations which have no classical analogue. The aim of this paper is to give these representations which are called nonclassical type representations. They are given by explicit formulas for operators of the representations corresponding to the generators of U_q(so_n).     

Quantized universal enveloping superalgebra of type Q and a super-extension of the Hecke algebra

The quantized universal enveloping algebra U _q(q(n)) of the strange Lie superalgebra q(n) and a super-analogue HC _q (N) of the Hecke algebra H _q (N) are constructed. These objects are in a duality similar to the known duality between U _q (gl(n)) and H _q (N).     

Aq-deformation of Wakimoto modules,primary fields and screening operators

Theq-vertex operators of Frenkel and Reshetikhin are studied by means of aq-deformation of the Wakimoto module for the quantum affine algebraU _q at an arbitrary levelk0, –2. A Fock-module version of theq-deformed primary field of spinj is introduced, as well as the screening operators which (anti-)commute with the action ofU _q up to a total difference of a field. A proof of the intertwining property is given for theq-vertex operators corresponding to the primary fields of spinj1/2Z ₀. A sample calculation of the correlation function is also given.This is a revised version of the preprint distributed in December, 1992, with the title Free Field Realization ofq-deformed Primary Fields forU _q (sl ₂)     

Coordinates of the quantum plane as q-tensor operators in (\mathfrak{s}\mathfrak{u}(2)*\mathfrak{s}\mathfrak{u}(2))

The relation between the set of transformations of the quantum plane and the quantum universal enveloping algebra U _q (u(2)) is investigated by constructing representations of the factor algebra U _q (u(2))* . The noncommuting coordinates of , on which U _q (2) * U _q (2) acts, are realized as q-spinors with respect to each U _q (u(2)) algebra. The representation matrices of U _q (2) are constructed as polynomials in these spinor components. This construction allows a derivation of the commutation relations of the noncommuting coordinates of directly from properties of U _q (u(2)). The generalization of these results to U _q (u(n)) and is also discussed.     

Operator representations of U q (sl2(ℝ))

Integrable (well-behaved) operator representations of the *-algebra U _q (sl₂()), |q|= 1, q ² 1, |q|=1, q² 1, in Hilbert space are defined and classified up to unitary equivalence.     

The computation of characteristic classes of lattice gauge fields

AGL(p,C)-valued lattice gauge fieldu on a simplicial complex determines a principalGL(p,C)-bundle if the plaquette products are sufficiently small with respect to the maximum distortion coefficient of the transporters. A representative cocyclec _q for theq ^th Chern class of can be computed on each 2q-simplex by takingc _q() to be the intersection number of a certain singular 2q-cubeM with a Schubert-type variety _q in the space of allp×p matrices. This reduces to the solution of polynomial equations with coefficients coming fromu and thus avoids numerical integration or cooling-type procedures. An application of this method is suggested for the computation of the topological charge of anSU(3)-valued lattice gauge field on a 4-complex.Partially supported by NSF grant DMS 8607168Partially supported by PSC-CUNY and by NSF grant DMS 8805485     

q-Weyl coefficients forU q (3) andq-Racah coefficients forSU q (2)

We show that theq-Weyl coefficients of the quantum algebraSU _q (3) are equal to theq-Racah coefficients of the quantum algebraSU _q (2) (up to a simple phase factor). Using aq-analog of the resummation procedure we obtain also theq-analogues of all known general analytical expressions for the 6j-symbols (or the Racah coefficients) of the quantum algebraSU _q (2) starting from one such formula.Presented at the 4th Colloquium Quantum groups and integrable systems, Prague, 22–24 June 1995.The research described in this publication was supported in part by Grants No. MB1000 and No. NRC000 from International Science Foundation.     

Radiactiv corrections to deeply virtual compton scattering

The non-commuting matrix elements of matrices from the quantum group GL _q(2;C) with q = being the n-th root of unity are given a representation as operators in Hilbert space with help of C ₄ ⁽ⁿ⁾ generalized Clifford algebra generators.The case of q C, |q| = 1 is treated parallelly.     

Algebra of Deformed Differential Operators and Induced Integrable Toda Field Theory

We build in this paper the algebra of q-deformed pseudo-differential operators, shown to be an essential step toward setting a q-deformed integrability program. In fact, using the results of this q-deformed algebra, we derive the q-analogues of the generalized KdV hierarchy. We focus in particular on the first leading orders of this q-deformed hierarchy, namely the q-KdV and q-Boussinesq integrable systems. We also present the q-generalization of the conformal transformations of the currents u _n ,n 2, and discuss the primary condition of the fields W _n , n 2, by using the Volterra gauge group transformations for the q-covariant Lax operators. An induced su(n)-Toda(su(2)-Liouville) field theory construction is discussed and other important features are presented.     

Generalized q-Modified Laguerre Functions

In this paper we define a new q-special function A _n (x, b, c; q). The new function is a generalization of the q-Laguerre function and the Stieltjes–Wigert function. We deduced all the properties of the function A _n (x, b, c; q). Finally, lim _q1 A _n ((1 – q)x, –, 1;q) gives L _n ^(,)(x,q), which is a -modification of the ordinary Laguerre function.     

Differential calculus onISO q(N), quantum Poincaré algebra and q-gravity

We present a general method to deform the inhomogeneous algebras of theB _n,C_n,D_n type, and find the corresponding bicovariant differential calculus. The method is based on a projection fromB _n+1,C_n+1,D_n+1. For example we obtain the (bicovariant) inhomogeneousq-algebraISO _q(N) as a consistent projection of the (bicovariant)q-algebraSO _q(N=2). This projection works for particular multiparametric deformations ofSO(N+2), the so-called minimal deformations. The case ofISO _q(4) is studied in detail: a real form corresponding to a Lorentz signature exists only for one of the minimal deformations, depending on one parameterq. The quantum Poincaré Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains theclassical Lorentz algebra. Only the commutation relations involving the momenta depend onq. Finally, we discuss aq-deformation of gravity based on the gauging of thisq-Poincaré algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian.