Anyonic lie algebras

We introduce anyonic Lie algebras in terms of structure constants. We provide the simplest examples and formulate some open problems. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997. This paper is in final form and no version of it will be published elsewhere.     

Coherent states and infinite-dimensional lie algebras: An outlook

Summary The possible extension of the notion of generalized coherent state to the case of infinite-dimesional affine Lie algebras is discussed with special attention to the resulting topological structure of the coherent states manifold, and to its connection with the structure of the algebra. The relevance for the solution of nonlinear dynamical systems equations of motion is briefly reviewed. To speed up publication, the authors have agreed not to receive proofs which have been supervised by the Scientific Committee.     

On trace representation of linear functionals on unbounded operator algebras

In this paper we consider the following problem: Given a *-algebraA of unbounded operators, under what conditions is every strongly positive linear functionalf onA a trace functional, i.e. of the formf(a)=Trtta,aA, wheret is an appropriate positive nuclear operator. Further, the linear functionalsf onA which can be represented asf(a)=Trta (f andt not necessarily positive) are characterized by their continuity in a certain topology. Some applications (canonical commutation relations on the Schwartz space, integrable representations of enveloping algebras) are discussed.     

Method of orbits of coassociated representation in thermodynamics of the lie noncompact groups

In the present paper, an efficient method of solving the main problem of thermodynamics of homogeneous spaces is presented. The method is based on the formalism of the noncommutative harmonic analysis relying on the method of orbits of coassociated representation. In the present work, the formula is derived that allows one to construct effectively the density matrix and the statistical sum in space of any arbitrary generally noncompact non-unimodular Lie group. To illustrate the method, an example is given of exact calculation of the statistical sum and density matrix in the Riemannian space of the noncompact Lie group with left-invariant metric. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 84–88, March, 2007.     

RationalR-matrices and exceptional lie algebras

We give a brief introduction to our work in math. QA/0608248: an explicit construction of certain rationalR-matrices associated with the exceptionale ₆ ande ₇ series of Lie algebras, which explains Westbury’s observation of their uniform spectral decompositions.     

Centrosymmetric lie algebras and boost-dilations

We define the Lie algebrac(n) of centrosymmetric matrices. It generates a noncompact and nonsemisimple local Lie group with the unusual property that expc(n) c(n). The group contains an invariant subgroup of Lorentz boost/ dilation transformations. Forn even, these form a subgroup of the conformal group of the Lorentzian metric with signature (– + – + – +).     

Nijenhuis tensors and lie algebras

Let M be a 2n-dimensional almost complex manifold: we construct a local almost complex structure starting from a Nijenhuis tensor given at a point. Moreover, we determine, on a 6-manifold, the conditions ensuring that its Nijenhuis tensor induces a Lie algebra structure on the tangent space. We give a class of examples for every kind.     

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.     

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.     

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.     

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.     

Contragradient algebras and lax operators associated with simple lie algebras

It is shown that the two-dimensional field theories connected with some representations of compact Lie groups admit the Lax pair in terms of contragradient algebras, associated with the corresponding simple Lie algebras.     

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.     

Graded differential lie algebras and model building

We develop a mathematical concept towards gauge field theories based upon a Hilbert space endowed with a representation of a skew-adjoint Lie algebra and an action of a generalized Dirac operator. This concept shares common features with the non-commutative geometry à la Connes/Lott, differs from that, however, by the implementation of skew-adjoint Lie algebras instead of unital associative *-algebras. We present the physical motivation for our approach and sketch its mathematical strategy. Moreover, we comment on the application of our method to the standard model and the flipped SU(5)×U(1)-grand unification model.     

Quantum integrable systems related to lie algebras

Some quantum integrable finite-dimensional systems related to Lie algebras are considered. This review continues the previous review of the same authors [83] devoted to the classical aspects of these systems. The dynamics of some of these systems is closely related to free motion in symmetric spaces. Using this connection with the theory of symmetric spaces some results such as the forms of spectra, wave functions, S-matrices, quantum integrals of motion are derived. In specific cases the considered systems describe the one-dimensional n-body systems interacting pairwise via potentials g²v(q) of the following 5 types: v_I(q) = q^?2, v_II(q) = sinh^?2q, v_III(q) = sin^?2q, $\text{v}{}_{\mathrm{<mtext>IV</mtext>}}\text{(q) =}\text{P}\text{(q)}$, v_V(q) = q^?2 + ω²q². Here $\text{P}\text{(q)}$ is the Weierstrass function, so that the first three cases are merely subcases of the fourth. The system characterized by the Toda nearest-neighbour potential exp(q_jq_{j+ 1}) is moreover considered.This review presents from a general and universal point of view results obtained mainly over the past fifteen years. Besides, it contains some new results both of physical and mathematical interest.     

Hydrodynamical poisson brackets and local lie algebras

Zakharov's and Gibbons' observations on the relation between the Benney equation in hydrodynamics and the Vlasov equation in kinetic theory is put into general framework of infinite-dimensional Lie algebras associated to Lie algebra structures on rings of functions on finite-dimensional manifolds.     

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.     

Spatial representation of groups of automorphisms of von Neumann algebras with properly infinite commutant

Theorem. Let a topological groupG be represented (a→φ _a ) by *-automorphisms of a von Neumann algebraR acting on a separable Hilbert spaceH. Suppose that

1. G is locally compact and separable,
2. R′ is properly infinite,
3. for anyT∈R,x,y∈H the function

$$a \to \left\langle {\phi _a (T)x,y} \right\rangle _H $$ is measurable onG. Then there exists a strongly continuous unitary representation ofG onH,a→U _a , such that forT∈R,a∈G, $$\phi _\alpha (T) = U_a TU_a *.$$ .     

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.     

Bilocal lie groups and solitons

A group theoretical scheme where solition equations are associated with the integrability conditions for differential system is proposed. These conditions ensure the existence of a bilocal Lie group structure which naturally generates a set of conserved currents for arbitrary space-time dimensions.