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.     

Completely integrable classical systems connected with semisimple lie algebras

The complete integrability of a class of dynamical systems, more general than those considered recently by Moser and Calogero, is proved. It is shown that these systems are connected with semisimple Lie algebras.Some of the results of this paper were announced in [1].     

Factorization of the ground-state wavefunctions of quantum systems related to semisimple lie algebras

Conditions are considered under which the ground-state wavefunctions of quantum systems connected with a semisimple Lie algebra are factorizable and may be found explicitly.     

Boson expansion of lie algebras: The fermion pair algebra

A unitary boson expansion theory for Lie algebras with ladder representations is discussed and applied to the fermion pair algebra SO(2n).     

Embedding Q-deformed Heisenberg algebras into undeformed ones

Any deformation of a Weyl or Clifford algebra can be realized through a change of generators in the undeformed algebra. Here we briefly describe and motivate our systematic procedure for constructing all such changes of generators for those particular deformations where the original algebra is covariant under some Lie group and the deformed algebra is covariant under the corresponding quantum group.     

On the role of semisimple subalgebras in the deformation theory of lie algebras and their homomorphisms general theory and applications

Formal deformations of Lie algebras are determined by sequences of bilinear alternating maps, and those of their homomorphisms by sequences of linear maps. The question of the existence, in any equivalence class of formal deformations of Lie algebras and of their homomorphisms, of elements determined by well-behaved sequences is investigated in this paper. A satisfactory affirmative answer is given provided the Lie algebra to be deformed has a semisimple subalgebra different from {0}. The meaning of this result in the geometric approach to deformation theory is pointed out. Applications to the problem of coupling the Poincaré group and an internal symmetry group in a nontrivial way and to the study of deformations of irreducible finite-dimensional representations of $\text{E}$(3) are given.     

Orthogonal polynomials in the lie algebras

We present a method of constructing orthogonal polynomials generated by pairs of Hermitian operators in representations of Lie algebras. All known classical polynomials of both discrete and continuous argument are generated naturally by the simplest Lie algebras.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 60–66, May, 1986.     

Regular * representations of lie algebras

We prove the existence of a * product on the cotangent bundle of a parallelizable manifold M. When M is a Lie group the properties of this * product allow us to define a linear representation of the Lie algebra of this group on L ²(G), which is, in fact, the one corresponding to the usual regular representation of G.Chargé de recherches au FNRS.     

Structural parameters of semi-simple lie algebras

Based on the exploitations of properties of the Killing forms of semi-simple Lie algebras, we set out in a readily programmable form, the structural analysis and the Iwasawa-type decompositions of semi-simple Lie algebras. As an example, the case ofSO(3,1) and its covering groupSL(2,C) is worked out in some detail.     

On the exponentiation of the affine metaplectic lie algebra

This paper concerns the infinite polynomials with maximal degree 2 of creation and annihilation operators, which give a Fock space representation of the complexification of the affine symplectic group. We study exponentiability of these operators, and obtain explicitly the local connection between the complexification of the affine metaplectic representation and the corresponding Lie algebra.     

Canonical realizations of classical lie algebras

We construct sets of canonical realizations for all classical Lie algebras (A _n ,B _n ,C _n ,D _n ). These realizations depend ond parameters,d=1, 2, 3,...,n; all Casimir operators are realized by multiples of identity. For most of the real forms of these algebras we give sets of realizations which are, moreover, in well-defined sense skew-Hermitian. Further we study extremal cases of the presented realizations. The realizations with minimal numbers of canonical pairs are discussed from the point of view of general results concerning minimal realizations. On the other hand, a connection is found between our maximal realizations ofA _n and the Gel'fand-Kirillov Conjecture.The authors would like to thank Prof. A.Uhlmann for his kind interest in this work. They are very grateful to Prof. A. A.Kirillov and Prof. D. P.Zhelobenko for helpful discussions and to Prof. J.Dixmier for his informative letter concerning the problem mentioned in Sect. 5.One of the authors (W. L.) thanks Prof. I.Úlehla for the hospitality at the Nuclear Center of the Charles University, Praha.     

Dynamical realization of l-conformal Galilei algebra and oscillators

The method of nonlinear realizations is applied to the l-conformal Galilei algebra to construct a dynamical system without higher derivative terms in the equations of motion. A configuration space of the model involves coordinates, which parametrize particles in d spatial dimensions, and a conformal mode, which gives rise to an effective external field. It is shown that trajectories of the system can be mapped into those of a set of decoupled oscillators in d dimensions.     

Constructing lie algebras of first-order differential operators

Formulas for calculating vector fields — generators of groups of transformations to a uniform space — from specified structural constants are obtained. The problem of vector-field continuation — the construction of Lie algebras of inhomogeneous first-order differential operators — is considered. It is also shown that the existence of a nontrivial continuation is closely associated with the structure of the isotopic subalgebra and, in particular, that no nontrivial continuation exists for semisimple algebras. Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 25–32, June, 1997.     

Remark on the integrability of some representations of the semisimple Lie algebras

It is discussed how boundedness of the quadratic Casimir operator in skew-symmetric representation τ of semisimple Lie algebra can simplify the proof of integrability of τ.     

Graded contractions of lie algebras and central extensions

We generalize the methods of graded contractions in order to determine, using grading arguments only, the existence of central charges within the limit Lie algebras. As an illustration we show how this formalism allows one to recover the u(n)-bosons limits of the classical Lie algebras. Presented by Marc de Montigny at the DI-CRM Workshop held in Prague, 18–21 June 2000.     

An interesting representation of lie algebras of linear groups

I have presented a means of getting a representation space of a general linear group ofn dimensions in terms of homogeneous functions ofn,n-dimensional vectors. Except in particular cases, the representation is of the Lie algebra, rather than the group. A general formalism is set up to evaluate the Casimir operators of the Lie algebra of the group in terms of the degrees of homogeneity of the functions (which are eigenfunctions of the Casimir operators) in then variables. It is noticed that the Casimir operators exhibit certain symmetries in these degrees of homogeneity which relate different representations having the same eigenvalues for the Casimir operators. Contour integral formulas that enable one to pass from one such representation to another are presented. An expression for the eigenvalues of a general Casimir operator in terms of the degree of homogeneity is presented.     

Quadratic algebras applied to noncommutative integration of the Klein-Gordon equation: Four-dimensional quadratic algebras containing three-dimensional nilpotent lie algebras

The study is continued on noncommutative integration of linear partial differential equations [1] in application to the exact integration of quantum-mechanical equations in a Riemann space. That method gives solutions to the Klein-Gordon equation when the set of noncommutative symmetry operations for that equation forms a quadratic algebra consisting of one second-order operator and of first-order operators forming a Lie algebra. The paper is a continuation of [2], where a single nontrivial example is used to demonstrate noncommutative integration of the Klein-Gordon equation in a Riemann space not permitting variable separation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 89–94, March, 1995.     

Contraction-based classification of supersymmetric extensions of kinematical lie algebras

We study supersymmetric extensions of classical kinematical algebras from the point of view of contraction theory. It is shown that contracting the supersymmetric extension of the anti-de Sitter algebra leads to a hierarchy similar in structure to the classical Bacry—Lévy-Leblond classification.