gl q (n) and quantum monodromy

A generalized Toda lattice based on gl(n) is considered. The Poisson brackets are expressed in terms of a Lax connection, L=–() and a classical r-matrix, {₁,₂}=[r,₁+₂}. The essential point is that the local lattice transfer matrix is taken to be the ordinary exponential, T=e; this assures the intepretation of the local and the global transfer matrices in terms of monodromy, which is not true of the T-matrix used for the sl(n) Toda lattice. To relate this exponential transfer matrix to the more manageable and traditional factorized form, it is necessary to make specific assumptions about the equal time operator product expansions. The simplest possible assumptions lead to an equivalent, factorized expression for T, in terms of operators in (an extension of) the enveloping algebra of gl(n). Restricted to sl(n), and to multiplicity-free representations, these operators satisfy the commutation relations of sl _q (n), which provides a very simple injection of sl _q (n) into the enveloping algebra of sl(n). A deformed coproduct, similar in form to the familiar coproduct on sl _q (n), turns gl(n) into a deformed Hopf algebra gl _q (n). It contains sl _q (n) as a subalgebra, but not as a sub-Hopf algebra.     

Spectra of Hamiltonians with generalized single-site dynamical disorder

Starting from the deformed commutation relationsa _q (t) a ^q (s)–q a ^q (s) a _q (t)=(t–s)1, –1q1 with a covariance (t–s) and a parameterq varying between –1 and 1, a stochastic process is constructed which continuously deforms the classical Gaussian and classical compound Poisson process. The moments of these distinguished stochastic processes are identified with the Hilbert space vacuum expectation values of products of with fixed parametersq, and . Thereby we can interpolate between dichotomic, random matrix and classical Gaussian and compound Poisson processes. The spectra of Hamiltonians with single-site dynamical disorder are calculated for an exponential covariance (coloured noise) by means of the time convolution generalized master equation formalism (TC-GME) and the partial cumulants technique. The final result for the spectral function is given as aq-dependent infinite continued fraction. In the case of the random matrix processes the infinite continued fraction can be summed up yielding a self-consistent equation for the one-particle Green function.     

A generalized consistency condition for massless fields

Using a generalized consistency condition (gcc), we construct couplings between massless scalar fields and the spin 2 gravitational field. Specifically, we consider all possible third-order interaction terms for scalar fields and a and use the gcc to single out one. We also find three generalized current identities associated with the massless gauge fields.     

Z 3-Graded exterior differential calculus and gauge theories of higher order

We present a possible generalization of the exterior differential calculus, based on the operator d such that d³=0, but d²0. The entities dx ⁱ and d² x ^k generate an associative algebra; we shall suppose that the products dx ⁱ dx ^k are independent of dx ^k dx ⁱ , while theternary products will satisfy the relation: dx ⁱ dx ^k dx ^m =jdx ^k dx ^m dx ⁱ =j ²dx ^m dx ^m dx ⁱ dx ^k , complemented by the relation dx ⁱ d² x ^k =jd² x ^k dx ⁱ , withj:=e^2i/3.We shall attribute grade 1 to the differentials dx ⁱ and grade 2 to the second differentials d² x ^k ; under the associative multiplication law the grades add up modulo 3.We show how the notion ofcovariant derivation can be generalized with a 1-formA so thatD:=d+A, and we give the expression in local coordinates of thecurvature 3-form defined as :=d² A+d(A ²)+AdA+A ³.Finally, the introduction of notions of a scalar product and integration of theZ ₃-graded exterior forms enables us to define the variational principle and to derive the differential equations satisfied by the 3-form . The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensorF _ik and its covariant derivativesD _i F _km .     

Wave equations onq-Minkowski space

We give a systematic account of a component approach to the algebra of forms onq-Minkowski space, introducing the corresponding exterior derivative, Hodge star operator, coderivative, Laplace-Beltrami operator and Lie-derivative. Using this (braided) differential geometry, we then give a detailed exposition of theq-d'Alembert andq-Maxwell equation and discuss some of their non-trivial properties, such as for instance, plane wave solutions. For theq-Maxwell field, we also give aq-spinor analysis of theq-field strength tensor.     

On the linearization of second-order ordinary differential equations

In this Letter, the problem of characterizing all second-order ordinary differential equations y=f(x, y, y) which are locally linearizable by a change of dependent and independent variables (x, y)(X, Y) is considered. Since all second-order linear equations are locally equivalent to y=0, the problem amounts to finding necessary and sufficient conditions for y=f(x, y, y) to be locally equivalent to y=0. It turns out that two apparently different criteria for linearizability have been formulated in the literature: the one found by M. Tresse and later rederived by É. Cartan, and the criterion recently given by Arnol'd [Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983]. It is shown here that these two sets of linearizability conditions are actually equivalent. As a matter of fact, since Arnol'd's criterion is stated without proof in the latter reference, this work can be alternatively considered as a proof of Arnol'd's linearizability conditions based on Cartan-Tresse's. Some further points in connection with the relationship between Arnol'd's and Cartan-Tresse's treatment of the linearization problem are also discussed and illustrated with several examples.     

Symmetries of the heat equation on the lattice

Discrete versions of the heat equation on two-dimensional uniform lattices are shown to possess the same symmetry algebra as their continuum limits. Solutions with definite symmetry properties are presented.     

Action variables for Toda Lattices

This Letter contains constructions of complex action variables for both the full Kostant-Toda Lattice in sl(n, ) and the generalized nonperiodic tridiagonal Toda lattice associated to an arbitrary complex semisimple Lie algebra g. The main tool is the explicit factorization solution for certain Hamiltonian flows. The Letter also contains a generalization of the standard factorization solution theorem necessary for the analysis of the full Kostant-Toda lattice.     

Representation theory of generalized deformed oscillator algebras

The representation theory of the generalized deformed oscillator algebras (GDOA's) is developed. GDOA's are generated by the four operators {1, a, a , N}. Their commutators and Hermiticity properties are those of the boson oscillator algebra, except for [a, a ] _q = G(N), where [a, b] _q = ab – q ba and G(N) is a Hermitian, analytic function. The unitary irreductible representations are obtained by means of a Casimir operator C and the semi-positive operator a a. They may belong to one out of four classes: bounded from below (BFB), bounded from above (BFA), finite-dimentional (FD), unbounded (UB). Some examples of these different types of unirreps are given.     

Asymptotics for Bosonic atoms

It was proved by Benguria and Lieb that for an atom where the electrons do not satisfy the exclusion principle, the critical electron number N _c, i.e., the maximal number of electrons the atom can bind, satisfies lim inf_zN_c/Z 1 + , where Z is the nuclear charge. Here is a positive constant derived from the Hartree model. We complete this result by proving that the correct asymptotics for N _c(Z) is indeed _zN_c/Z = 1 + .This work was done while the author was a graduate student at Princeton University supported by a Danish Research Academy fellowship and U.S. National Science Foundation grant PHY-85-15288-A03.     

Energy spectrum of the two-dimensionalq-hydrogen atom

The discrete energy spectrum of theq-analog of the two-dimensional hydrogen atom is derived by deforming the Pauli equation. It contracts to that of the ordinary two-dimensional hydrogen atom in the limitq ± 1. The degeneracy is discussed generally and some properties of theq-energy spectrum are studied both for realq and for complexq of magnitude unity.     

Berenstein-Zelevinsky triangles,elementary couplings,and fusion rules

We present a general scheme for describing (N) _k fusion rules in terms of elementary couplings, using Berenstein-Zelevinsky triangles. A fusion coupling is characterized by its corresponding tensor product coupling (i.e. its Berenstein-Zelevinsky triangle) and the threshold level at which it first appears. We show that a closed expression for this threshold level is encoded in the Berenstein-Zelevinsky triangle and an explicit method to calculate it is presented. In this way, a complete solution of (4) _k fusion rules is obtained.Work supported by NSERC (Canada).Work supported by NSERC (Canada) and FCAR (Québec).     

Some remarks on producing Hopf algebras

We report some observations concerning two well-known approaches to construction of quantum groups. Thus, starting from a bialgebra of inhomogeneous type and imposing quadratic, cubic or quartic commutation relations on a subset of its generators we come, in each case, to aq-deformed universal enveloping algebra of a certain simple Lie algebra. An interesting correlation between the order of initial commutation relations and the Cartan matrix of the resulting algebra is observed. Another example demonstrates that the bialgebra structure ofsl _q (2) can be completely determined by requiring theq-oscillator algebra to be its covariant comodule, in analogy with Manin's approach to defineSL _q (2) as a symmetry algebra of the bosonic and fermionic quantum planes.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.This work was supported in part by International Sciences Foundation (grant RFF-300) and by Russian Basic Research Foundation (grant 95-02-05679).I acknowledge helpful discussions with A. Isaev, P. Kulish, V. Lyakhovsky, O. Ogievetsky, P. Pyatov, and V. Tolstoy.     

External noise effects on the fluctuation line width

We calculate stationary state correlation functions of the anharmonic overdamped oscillator driven by multiplicative (white Gaussian) noise of strengthQ together with additive noise of relative strengthq. (i) We donot observe a particular slowing down at the so-called noise induced transition. But there is a region of the oscillator's stiffness parametera neara0 in which the decay time is typically enhanced. (ii) If the phase transition point is defined by the minimum of the decay rate, it lies within the ordered phasea>0 forq1 and shifts down toa=0 forq0. In our approximation it is even in the disordered regiona<0 for very smallq. (iii) As a function of the multiplicative noiseQ the decay ratedecreases with increasingQ if the system is well above or well below threshold. There seem to be experimental indications of this behavior. But within the proper threshold regime of smalla increasing noiseQ increases the decay rate. The valuesa _c where the cross-over occurs depend on the fluctuating variable and onq.     

Small representations of quantum affine algebras

We characterize the finite-dimensional representations of the quantum affine algebra U _q ( _n+1) (whereq ^× is not a root of unity) which are irreducible as representations of U _q (sl _n+1). We call such representations small. In 1986, Jimbo defined a family of homomorphismsev _a from U _q (sl _n+1) to (an enlargement of) U _q (sl_,n+1), depending on a parametera ^·. A second family,ev ^a can be obtained by a small modification of Jimbo's formulas. We show that every small representation of U _q ( _n+1) is obtained by pulling back an irreducible representation of U _q (sl _n+1) byev _a orev ^a for somea ^·.     

Deformed harmonic oscillators: Coherent states and Bargmann representations

Generalizing the case of the usual harmonic oscillator, we look for Bargmann representations corresponding to deformed harmonic oscillators. Deformed harmonic oscillator algebras are generated by four operators a, a , N and the unity 1 such as [a, N] = a, [a , N] = –a , a a = (N) and aa = (N + 1). We discuss the conditions of existence of a scalar product expressed with a true integral on the space spanned by the e igenstates of a (or a ). We give various examples, in particular we consider functions that are linear combinations of q ^N, q ^–N and unity and that correspond to q-oscillators with Fock-representations or with non-Fock-representations.     

λφ 4 q-Renormalization program

A regularization scheme for quantum field theories given in aq-mutator algebra for the internal momentum space in a loop integration is constructed. We show Feynman integrals that are finite forq 1but diverse asq 1. Using this regularization scheme, we propose a renormalization program in q-mutator space (q-renormalization program) for thef ⁴ theory as an example, up to some one-loop diagrams. This work paves the way to obtaining physically measurable quantities from quantum field theories over spaces that neither commute nor anticommute.     

On spectral properties ofq-oscillator operators

The property of self-adjointness of the operatorQ =a ₊ +a _- in three types ofq-oscillator algebras is considered. Spectral measures and generalized eigenfunctions ofQ are found in the cases when this operator is bounded. Generalized eigenvectors are expressed in terms ofq-Hermite polynomials. If the operatorQ is unbounded, then its closure is not self-adjoint. However, in this case, admits self-adjoint extensions. Deficiency subspaces are one-dimensional. These subspaces are explicitly found.     

Newq-Minkowski space-time and hierarchies ofq-difference conformal invariant equations

We propose several new infinite hierarchies ofq-conformal invariantq-difference equations, in particular, newq-Maxwell equations as the first member of one of these hierarchies. We use an indexless formulation in which all indices are traded for two conjugate variables,z, ¯z. We propose also newq-Minkowski coordinates which together withz, ¯z can be interpreted as the six local coordinates of aq-deformation of theSU (2, 2) flag manifold.Plenary lecture at the 4th Colloquium Quantum groups and integrable systems, Prague, 22–24 June 1995.     

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.