Volume 33, Issue 2, February 1992
Index of content:

The q‐boson realization of parametrized cyclic representations of quantum algebras at q ^{ p } = 1
View Description Hide DescriptionA parametrized cyclic representation of the q‐deformed Heisenberg–Weyl algebras is constructed. The cyclic representations of any quantum algebras are studied in terms of their q‐boson realizations. A general method to construct the q‐boson realizations of quantum algebras from their Verma representations is proposed. Two explicit examples sl(2)_{ q } and sl(3)_{ q } are studied in detail. Some cyclic representations of quantum algebras sl(n)_{ q } and U _{ qC } _{ n } are presented by using their q‐boson realizations.

Tensor operators for quantum groups and applications
View Description Hide DescriptionTensor operators are defined for Hopf algebras. If the Hopf algebra is quasitriangular, the properly defined tensor product of tensor operators is a tensor operator. In one application, the problem of defining invariant correlation functions for the U _{ q } (sl(2))‐symmetric spin 1/2 Heisenberg chain is considered. In another application, it is shown that suitably modified bosonic and fermionic q‐creation and annihilation operators are tensor operators of U _{ q } (sl(n)).

Quantum algebras on phase space
View Description Hide DescriptionQuantum algebras are represented by functions on phase space in terms of *‐product, and explicit examples for U_{η} (sl(n+↑ 1), R ) with n≥1 are given.

Primary operators of coset conformal field theories
View Description Hide DescriptionThe nature of primary operators of coset conformal field theories is discussed. Some primary states of typical coset conformal field theory are explicitly calculated that have correct conformal dimensions and dynamics. The nature of modular invariant partition functions from the diagonal sum of absolute squares of characters of coset primary states is studied.

Matrix representations of SO_{ n+2} in an SO_{ n }×SO_{2} basis and some isoscalar factors for SO_{ n+2}⊇SO_{ n }×SO_{2}
View Description Hide DescriptionVector coherent state (VCS) theory is applied to the group chain SO_{ n+2}⊇SO_{ n }×SO_{2}. Matrix elements of SO_{ n+2} generators in the SO_{ n+2}⊇SO_{ n }×SO_{2} basis are derived. A new formula for the evaluation of some isoscalar factors for SO_{ n+2}⊇SO_{ n }×SO_{2} with branching multiplicity is derived in the VCS framework. As a simple example, a new expression of some isoscalar factors for SO_{5}⊇SO_{3}×SO_{2}, which involves only 6j coefficients and K‐normalization factors, are obtained by using this formula.

On the composition factors of Kac modules for the Lie superalgebras sl(m/n)
View Description Hide DescriptionIn the classification of finite‐dimensional modules of Lie superalgebras, Kac distinguished between typical and atypical modules. Kac introduced an induced module, the so‐called Kac module V̄(Λ) with highest weight Λ, which was shown to be simple if Λ is a typical highest weight. If Λ is an atypical highest weight, the Kac module is indecomposable and the simple module V(Λ) can be identified with a quotient module of V̄(Λ). In the present paper the problem of determining the composition factors of the Kac modules for the Lie superalgebra sl(m/n) is considered. An algorithm is given to determine all these composition factors, and conversely, an algorithm is given to determine all the Kac modules containing a given simple module as a composition factor. The two algorithms are presented in the form of conjectures, and illustrated by means of detailed examples. Strong evidence in support of the conjectures is provided. The combinatorial way in which the two algorithms are intertwined is both surprising and interesting, and is a convincing argument in favor of the solution to the composition factor problem presented here.

On the representation of operator algebras of the quantum conformal field theory
View Description Hide DescriptionIt is established in this paper that an arbitrary operator algebra of the quantum conformal field theory admits the strict representation by matrices, whose elements belong to the special operator algebra−the algebra of the vertex operators in the model of the Verma modules over the Virasoro algebra.

Symmetries of the octonionic root system of E _{8}
View Description Hide DescriptionOctonionic root system of E _{8} is decomposed as the 9 sets of Hurwitz integers, each set satisfying the binary tetrahedral group structure, and the 12 sets of quaternionic units, each set obeying the dicylic group structure of order 12. This fact is used to prove that the automorphism group of the octonionic root system of E _{7} is the finite subgroup of G _{2}, of order 12 096, an explicit 7×7 matrix realization of which is constructed. Possible use of the octonionic representation of the E _{6} root system as orbifolds and the relevance of the binary tetrahedral structures with the statistical mechanics models are suggested.

Bundle realizations and invariant connections in an Abelian principal bundle
View Description Hide DescriptionIn this paper the concept of bundle realization of a Lie group on an Abelian principal bundle is defined. This definition is based on the theory of locally operating realizations of Lie groups. Afterward the bundle realizations are studied and characterized into pseudoequivalence classes. This theory is applied to a systematic study and classification of invariant connections under a Lie group. In particular, some examples of gauge invariant potentials under some subgroups of the Poincaré group are worked out.

Generating relations for Tratnik’s multivariable biorthogonal continuous Hahn polynomials
View Description Hide DescriptionMultiple series generating relations for the generalized continuous Hahn polynomials are obtained. Special cases consist of results involving products of several confluent hypergeometric functions.

Solvable nonlinear integrodifferential equations of Boltzmann type. I
View Description Hide DescriptionA technique suitable to manufacture solvable nonlinear integrodifferential equations of Boltzmann type is introduced. Some examples of solvable homogeneous one‐dimensional Boltzmann‐like equations are exhibited.

A classification of pseudospherical surface equations of type u _{ t }=u _{ xxx }+G(u,u _{ x },u _{ xx })
View Description Hide DescriptionThe equations of type u _{ t } = u _{ xxx } + G(u,u _{ x },u _{ xx }) which describe η‐pseudospherical surfaces are characterized. A new class of such equations is obtained. This class together with the Korteweg–de Vries (KdV) equation, the modified Korteweg–de Vries (MKdV) equation, and the linear equation completes the classification.

Simple spinors and real structures
View Description Hide DescriptionA concept of a real index associated with any maximal totally null subspace in a complexified vector space endowed with a scalar product, and also with any complex simple (pure) spinor, is introduced and elaborated. It is shown that the real Pin group acts transitively on the projective space of simple spinors with any given real index. The connection between the real index and multivectors associated with a simple spinor and its charge conjugate is established. The simple spinors with extreme values of the real index deserve a special attention: The generic simple spinors for which the real index is minimal and simple‐r spinors for which it is maximal.

On the first tangency of stable and unstable manifolds in C ^{ r } planar diffeomorphisms
View Description Hide DescriptionTwo problems in two‐dimensional C ^{ r } diffeomorphisms are discussed. In doing this, the intersection and tangency of the stable and unstable manifolds are strictly discriminated and the notion of the first direct and asymptotic tangencies is introduced. The first problem is what happens if neither stable and unstable manifolds transversely nor nontransversely intersect with each other nor are their closures disjoint. It is shown that there are cases in which the existence of homoclinic tangency of the usual sense is inhibited. The second problem is what is expected as a consequence of the first direct homoclinic tangency. Several consequences are shown. Among others, an extension of Palis’s lambda lemma to cases of the first direct tangency is obtained. Finally, the numerical examples on the first asymptotic tangency are given using two maps.

Noisy fractals
View Description Hide DescriptionAn analytical study of affine contractive iterated function systems (IFS’s) is done in which noise is put in between each iteration of the maps. Since any contractive Poincaré map can be approximated arbitrarily well by IFS’s, this is also a study of how noise modifies repellers or attractors of dissipative systems. In particular the scale at which the noise erases the fractal detail is found.

Analytical evaluation of elastic Coulomb integrals
View Description Hide DescriptionIndefinite integrals over the product of two Coulomb wave functions and a factor r ^{−λ−1}, λ=1,2,3,..., have been evaluated analytically. The results for these multipole integrals could be expressed again in terms of Coulomb wave functions, except for the electric quadrupole (λ=2) integral at zero angular momentum in both the incident and final channels.

Lagrangians for differential equations of any order
View Description Hide DescriptionIn this work the inverse problem of the variational calculus for systems of differential equations of any order is analyzed. It is shown that, if a Lagrangian exists for a given regular system of differential equations, then it can be written as a linear combination of the equations of motion. The conditions that these coefficients must satisfy for the existence of an S‐equivalent Lagrangian are also exhibited. A generalization is also made of the concept of Lagrangian symmetries and they are related with constants of motion.

Representations of one‐dimensional Hamiltonians in terms of their invariants
View Description Hide DescriptionA general formalism for representing the Hamiltonian of a system with one degree of freedom in terms of its invariants is developed. Those Hamiltonians H(q,p,t) are derived for which any particular function I(q,p,t) is an invariant. For each of those Hamiltonians, a function canonically conjugate to I(q,p,t) is derived which is also an invariant of H(q,p,t). The formalism is also presented for the case in which I(q,p,t) is expressed as a function of two canonically conjugate functions. The formalism is illustrated by applying it to the case of a particle moving in a time‐dependent potential. Some earlier results are recovered and an invariant is found for a new potential. Lines for further study are outlined that may be fruitful for finding more examples of integrable systems.

The Hamiltonization problem from a global viewpoint
View Description Hide DescriptionOwing to the so‐called Lie–Koenigs theorem, any (finite‐dimensional) dynamical system can be put into Hamiltonian form locally. To that end, the introduction of dummy variables is not necessary, provided that the original system is even dimensional and that we allow for an explicitly time‐dependent symplectic structure. In this paper, inquiry is made as to which conditions are needed for a universality statement similar to the Lie–Koenigs theorem to hold true globally. In addition, the passage to the Lagrangian formalism is discussed, and the results are illustrated by three examples.

Hamiltonian formalism for nonregular Lagrangian theories in fibered manifolds
View Description Hide DescriptionA Hamiltonian formalism corresponding to the nonregular Lagrangian systems on jet bundles is constructed. The notion of a Hamiltonian on the Legendre bundle is introduced. The relation between Lagrangian and Hamiltonian formalisms is investigated.