动力学对称群方法对三原子分子高激发振动态的理论研究

利用动力学对称群方法研究了弯曲三原子分子的振动高激发态能谱-该方法显示：三原子分子的动力学对称性为U₁(4)U₂(4),则三原子分子的Hamiltonian量可写成代数的各元素之和,通过李代数处理而求得分子代数Hamiltonian量的本征值,进而得到分子的振动能谱-并具体计算了O₃分子- 关键词：     

SU(1, 1) and SU(2) Perelomov number coherent states: algebraic approach for general systems

We study some properties of the SU(1, 1) Perelomov number coherent states. The Schrödinger's uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized for the standard coherent states. We obtain the time evolution of the number coherent states by supposing that the Hamiltonian is proportional to the third generator K₀ of the su(1, 1) Lie algebra. Analogous results for the SU(2) Perelomov number coherent states are found. As examples, we compute the Perelomov coherent states for the pseudoharmonic oscillator and the two-dimensional isotropic harmonic oscillator.     

Dynamical symmetries for superintegrable quantum systems

We study the dynamical symmetries of a class of two-dimensional superintegrable systems on a 2-sphere, obtained by a procedure based on the Marsden-Weinstein reduction, by considering its shape-invariant intertwining operators. These are obtained by generalizing the techniques of factorization of one-dimensional systems. We firstly obtain a pair of noncommuting Lie algebras su(2) that originate the algebra so(4). By considering three spherical coordinate systems, we get the algebra u(3) that can be enlarged by “reflexions” to so(6). The bounded eigenstates of the Hamiltonian hierarchies are associated to the irreducible unitary representations of these dynamical algebras. The text was submitted by the authors in English.     

Nonlinear integrable couplings of a nonlinear Schrödinger—modified Korteweg de Vries hierarchy with self-consistent sources

By means of the Lie algebra B₂, a new extended Lie algebra F is constructed. Based on the Lie algebras B₂ and F, the nonlinear Schrödinger-modified Korteweg de Vries (NLS-mKdV) hierarchy with self-consistent sources as well as its nonlinear integrable couplings are derived. With the help of the variational identity, their Hamiltonian structures are generated.     

Nonstandard q-deformation of the Euclidean Lie algebra and its representations

A nonstandard q-deformed Euclidean algebra U _q(iso _n ), based on the definition of the twisted q-deformed algebra U _qso_n) (different from the Drinfeld–Jimbo algebra U _q(so _n )), is defined. Infinite dimensional representations R of U _q(iso _n ) are described. Explicit formulas for operators of these representations in the orthonormal basis are given. The spectra of the operators R(T _n) corresponding to a q-analogue of the infinitesimal operator of shifts along the n-th axis are described. Contrary to the case of the classical Euclidean Lie algebra iso _n , these spectra are discrete and spectral points have one point of accumulation.     

Double Quantization on Some Orbits in the Coadjoint Representations of Simple Lie Groups

Let ? be the function algebra on a semisimple orbit, M, in the coadjoint representation of a simple Lie group, g, with the Lie algebra ?. We study one and two parameter quantizations ? _h and ? _t,h of ? such that the multiplication on the quantized algebra is invariant under action of the Drinfeld–Jimbo quantum group, U _{h (?)}. In particular, the algebra ? _t,h specializes at h= 0 to a U(?)-invariant ($G$-invariant) quantization, %Ascr; _{t ,0}. We prove that the Poisson bracket corresponding to ? _h must be the sum of the so-called r-matrix and an invariant bracket. We classify such brackets for all semisimple orbits, M, and show that they form a dim H ²(M) parameter family, then we construct their quantizations. A two parameter (or double) quantization, $? _t,h , corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on M. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization of these pairs in some of the cases. Received: 15 August 1998 / Accepted: 13 January 1999     

Embedding euclidean lie algebras into quantum structures

We show that it is possible to express the basis elements of the Lie algebra of the Euclidean group,E(2), as simple irrational functions of certainq deformed expressions involving the generators of the quantum algebraU _q (so(2, 1)). We consider implications of these results for the representation theory of the Lie algebra ofE(2). We briefly discess analogous results forU _q (so(2, 2)). Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

Logarithmic Intertwining Operators and Vertex Operators

This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg and lattice vertex algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra M(1) _a , of central charge 1 − 12a ². We classify these operators in terms of depth and provide explicit constructions in all cases. Our intertwining operators resemble puncture operators appearing in quantum Liouville field theory. Furthermore, for a = 0 we focus on the vertex operator subalgebra L(1, 0) of M(1)₀ and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of hidden logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1, 0)-module.     

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.     

The quantum super-Yangian and Casimir operators of Uq(gl(M |N))

TheZ ₂ graded Yangian Yq(gl(M |N)) associated with the Perk-SchultzR matrix is introduced. Its structural properties, the central algebra in particular, are studied. AZ ₂-graded associative algebra epimorphism Yq(gl(M |N)) Uq (gl(M |N)) is obtained in explicit form. Images of central elements of the quantum super-Yangian under this epimorphism yield the Casimir operators of the quantum supergroup Uq(gl(M |N)) constructed in an earlier publication.     

Does accidental degeneracy imply a symmetry group?

The relation between the appearance of accidental degeneracy in the energy levels of a given Hamiltonian and its symmetry group is probed. This is done by analyzing the very simple problem of an oscillator to which a particular spin-orbit and centrifugal force are added. The operators that connect all the states of given energy as well as their corresponding observables in the classical limit are found. The Poisson bracket relations between these observables leads to a Lie algebra U(3) × SU(2), but it does not translate into a Lie algebra for the commutators of the corresponding operators, as some matrix elements of commutators, corresponding to Poisson brackets that are zero, do not vanish. Thus while accidental degeneracy in the quantum problem may lead to a larger group in the classical limit, it is not always given by the dimensions of the irreducible representations of this group.     

Decomposition of symmetric tensor fields in the presence of a flat contact projective structure

Abstract

Let M be an odd-dimensional Euclidean space endowed with a contact 1-form α. We investigate the space of symmetric contravariant tensor fields over M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up of those vector fields that preserve the contact structure defined by a. If we consider symmetric tensor fields with coefficients in tensor densities (also called symbols), the vertical cotangent lift of the contact form a defines a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symbols. This generalized Hamiltonian operator on the space of symbols is invariant with respect to the action of the projective contact algebra sp(2n+2) the algebra of vector fields which preserve both the contact structure and the projective structure of the Euclidean space. These two operators lead to a decomposition of the space of symbols, except for some critical density weights, which generalizes a splitting proposed by V. Ovsienko in [18].     

Discrete series of unitary irreducible representations of the Uq(u(3, 1)) and Uq(u(n, 1)) noncompact quantum algebras

The structure of all discrete series of unitary irreducible representations of the U _q (u(3, 1)) and U _q (u(n, 1)) noncompact quantum algebras are investigated with the aid of extremal projection operators and the q-analog of the Mickelsson-Zhelobenko algebra Z(g, g′) _q . The orthonormal basis constructed in the infinite-dimensional space of irreducible representations of the U _q (u(n, 1)) ⊇ U _q (u(n)) algebra is the q-analog of the Gelfand-Graev basis in the space of the corresponding irreducible representations of the u(n, 1) ⊇ u(n) classical algebra.     

Hamiltonian and Super-Hamiltonian Extensions Related to Broer-Kaup-Kupershmidt System

Based on the Lie algebra A ₁, the integrable Broer-Kaup-Kupershmidt (BKK) system is revisited. The bi-Hamiltonian structure is constructed by the trace identity. Two extensions of the Lie algebra A ₁ are considered, i.e., the non-semi-simple Lie algebra of 4×4 matrix and the super-Lie algebra of 3×3 matrix, from which two hierarchies of soliton equations related to BKK system are given. With the aid of the generalized trace identity and the super-trace identity, the Hamiltonian and super-Hamiltonian structures of the resulting systems are constructed.     

IRREDUCIBLE REPRESENTATIONS OF THE GRADED LIE ALGEBRA SU(m/n)

Using the definition of irreducible tensor operators of the Lie group SU(n), we ive a method of calculating the irreducible representations of the graded Lie algebra U(m/n). We also give some simple examples of irreducible representations of U(3/2) and SU(m/n).     

q-Deformed orthogonal and pseudo-orthogonal algebras and their representations

Deformed orthogonal and pseudo-orthogonal Lie algebras are constructed which differ from deformations of Lie algebras in terms of Cartan subalgebra and root vectors and which make it possible to construct representations by operators acting according to Gel'fand-Tsetlin-type formulas. Unitary representations of the q-deformed algebras U _q (so _n,1) are found.     

Solvability of the Hamiltonians Related to Exceptional Root Spaces: Rational Case

Solvability of the rational quantum integrable systems related to exceptional root spaces G₂,F₄ is re-examined and for E_6,7,8 is established in the framework of a unified approach. It is shown that Hamiltonians take algebraic form being written in certain Weyl-invariant variables. It is demonstrated that for each Hamiltonian the finite-dimensional invariant subspaces are made from polynomials and they form an infinite flag. A notion of minimal flag is introduced and minimal flag for each Hamiltonian is found. Corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for arbitrary values of the coupling constants. The Hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite-dimensional but finitely-generated Lie algebra of differential operators, taken in a finite-dimensional representation.Alexander V. Turbiner: On leave of absence from the Institute for Theoretical and Experimental Physics, Moscow 117259, Russia.