Noncommutative symmetric functions and laplace operators for classical Lie algebras

New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the use of some properties of the noncommutative symmetric functions associated with a matrix. The decomposition of the Sklyanin determinant into a product of quasi-determinants play the main role in the construction. Analogous decomposition for the quantum determinant provides an alternative proof of the known construction for the Lie algebra gl(N).     

Dynamical Lie algebra method for highly excited vibrational state of asymmetric linear tetratomic molecules

The highly excited vibrational states of asymmetric linear tetratomic molecules are studied in the framework of Lie algebra. By using symmetric groupU ₁(4)U ₂(4)⊗U ₃(4), we construct the Hamiltonian that includes not only Casimir operators but also Majorana operators M₁₂, M₁₃ and M₂₃, which are useful for getting potential energy surface and force constants in Lie algebra method. By Lie algebra treatment, we obtain the eigenvalues of the Hamiltonian, and make the concrete calculation for molecule C₂HF.     

Micu-type invariants for the casimir operators of the graded Lie algebrasosp(1, 2n) application to the algebraosp(1, 4)

A method which represents a generalization of the method proposed by Micu for the construction of the Casimir operators of the Lie algebras, namely the algebrasSp(2n), is presented for the construction of the Casimir operators of higher degrees of the graded Lie algebrasosp(1, 2n). Explicite expressions for the Casimir operators are obtained for the graded Lie algebraosp(1,4).This paper appeared as a preprint of the Institute of Physics, Czechoslovak Academy of Sciences, FZU 80-1 and will be published.Abstract of the paper presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.     

On the 1-cohomology of Lie groups

Given a continous representation of a Lie group in a Banach space we study its 1-cohomology. We prove that the computation of the 1-cocycles can be reduced to that of the 1-cocycles of the differential of the representation. When the group is semi-simple and the representation is K-finite, we prove that the cohomology is equivalent to the cohomology of the Lie algebra representation on K-finite vectors. We prove, using Casimir operators, that there exist only a finite number of irreducible representation of a semi-simple Lie group with a non-trivial cohomology. Exemples of such representations are given.     

Casimir Functions for Five-Dimensional Lie Groups with a Non–Semi-Hausdorff Space of Orbits

In the present paper, Lie groups with the multivalued Casimir functions are examined, in particular, a definition of the multivalued Casimir functions is given. It is demonstrated that when a Lie group consists of the essentially multivalued Casimir functions, the space of orbits of the coadjoint representation is non–semi-Hausdorff one, which allows a criterion for identification of these groups to be formulated. As an example, complete involute sets of the Casimir functions are retrieved for all real five-dimensional Lie algebras, and two Lie algebras with a non-Hausdorff space of orbits are identified by this criterion.     

On the Casimir operators associated with any Lie algebra

We give a characterization of the Casimir operators of a Lie algebra by polynomial solutions of a system of first-order partial differential equations. Further an upper bound is stated for the number of independent Casimir operators of a nilpotent Lie algebra.     

Infinite dimensional groups and Riemann surface field theories

We show how to obtain positive energy representations of the groupG of smooth maps from a union of circles toU(N) from geometric data associated with a Riemann surface having these circles as boundary. Using covering spaces we can reduce to the case whereN=1. Then our main result shows that Mackey induction may be applied and yields representations of the connected component of the identity ofG which have the form of a Fock representation of an infinite dimensional Heisenberg group tensored with a finite dimensional representation of a subgroup isomorphic to the first cohomology group of the surface obtained by capping the boundary circles with discs. We give geometric sufficient conditions for the correlation functions to be positive definite and derive explicit formulae for them and for the vacuum (or cyclic) vector. (This gives a geometric construction of correlation functions which had been obtained earlier using tau functions.) By choosing particular functions inG with non-zero winding numbers on the boundary we obtain analogues of vertex operators described by Segal in the genus zero case. These special elements ofG (which have a simple interpretation in terms of function theory on theRiemann surface) approximate fermion (or Clifford algebra) operators. They enable a rigorous derivation of a form of boson-fermion correspondence in the sense that we construct generators of a Clifford algebra from the unitaries representing these elements ofG.     

Boson realizations of quantum algebras and some physical spaces with quantum algebra symmetry

The generators ofq-boson algebra are expressed in terms of those of boson algebra, and the relations among the representations of a quantum algebra onq-Fock space, on Fock space, and on coherent state space are discussed in a general way. Two examples are also given to present concrete physical spaces with quantum algebra symmetry. Finally, a new homomorphic mapping from a Lie algebra to boson algebra is presented.This work is supported by the National Foundation of Natural Science of China.     

Integrable Euler equations on SO(4) and their physical applications

For the Lie algebra SO(4) (and other six dimensional Lie algebras) we find some Euler's equations which have an additional fourth order integral and are algebraically integrable (in terms of elliptic functions) in a one parameter set of orbits. Integrable Euler's equations having an additional second order integral and generalizing Steklov's case are presented. Equations for rotation of a rigid body havingn ellipsoid cavities filled with the ideal incompressible fluid being in a state of homogeneous vortex motion are derived. It is shown that the obtained equations are Euler's equations for the Lie algebra of the groupG _n+1=SO(3) × ... × SO(3). New physical applications of Euler's equations on SO(4) are discussed.     

Superalgebras in N = 1 gauge theories

N = 1 supersymmetric gauge theories with global flavor symmetries contain a gauge invariant W-superalgebra which acts on its moduli space of gauge invariants. With adjoint matter, this superalgebra reduces to a graded Lie algebra. When the gauge group is SO(n_c), with vector matter, it is a W-algebra, and the primary invariants form one of its representation. The same superalgebra exists in the dual theory, but its construction in terms of the dual fields suggests that duality may be understood in terms of a charge conjugation within the algebra. We extend the analysis to the gauge group E₆.     

Canonical realizations of the lie algebrasp(2n,R)

The generators of the Lie algebra of the symplectic groupsp(2n, R) are, recurrently, realized by means of polynomials in the quantum canonical variablesp _i andq _i. These realizations are skew-Hermitian, the Casimir operators are realized by constant multiples of identity elements, and, depending on the number of the canonical pairs used, they depend ond, d=1, 2, ...,n free real parameters.     

Strings fromN=2 gauged Wess-Zumino-Witten models

We present an algebraic approach to string theory. An embedding ofsl(2|1) in a super Lie algebra together with a grading on the Lie algebra determines a nilpotent subalgebra of the super Lie algebra. Chirally gauging this subalgebra in the corresponding Wess-Zumino-Witten model, breaks the affine symmetry of the Wess-Zumino-Witten model to some extension of theN=2 superconformal algebra. The extension is completely determined by thesl(2|1) embedding. The realization of the superconformal algebra is determined by the grading. For a particular choice of grading, one obtains in this way, after twisting, the BRST structure of a string theory. We classify all embeddings ofsl(2|1) into Lie super algebras and give a detailed account of the branching of the adjoint representation. This provides an exhaustive classification and characterization of both all extendedN=2 superconformal algebras and all string theories which can be obtained in this way.     

A Basis for Representations of Symplectic Lie Algebras

A basis for each finite-dimensional irreducible representation of the symplectic Lie algebra ¤(2n) is constructed. The basis vectors are expressed in terms of the Mickelsson lowering operators. Explicit formulas for the matrix elements of generators of ¤(2n) in this basis are given. The basis is natural from the viewpoint of the representation theory of the Yangians. The key role in the construction is played by the fact that the subspace of ¤(2nm 2) highest vectors in any finite-dimensional irreducible representation of ¤(2n) admits a natural structure of a representation of the Yangian Y(‹•(2)).     

Quantization of U q [so(2n + 1)] with deformed para-Fermi operators

The observation thatn pairs of para-Fermi (pF) operators generate the universal enveloping algebra of the orthogonal Lie algebra so(2n + 1) is used in order to define deformed pF operators. It is shown that these operators are an alternative to the Chevalley generators. With this background U _q [so(2n + 1)] and its Cartan-Weyl generators are written down entirely in terms of deformed para-Fermi operators.     

The principal series ofSp(n, ℝ)

The principal series of unitary representations of the noncompact symplectic groupSp(n, ) is constructed for alln. The Lie algebra ofSp(n, ) is isomorphic to the algebra of bilinear products of boson operators inn dimensions. The spectrum of the number operator for the principal series representations is shown to be unbounded, both from above and from below.     

Classification of irreducible super-unitary representations ofsu(p,q/n)

In this paper we classify all the irreducible super-unitary representations ofsu(p,q/n), which can be integrated up to a unitary representation ofS(U(p,q)×U(n)), a Lie group corresponding to the even part ofsu(p,q/n). Note that a real form of the Lie superalgebrasl(m/n;) which has non-trivial superunitary representations is of the formsu(p,q/n)(p+q=m) orsu(m/r,s)(r+s=n). Moreover, we give an explicit realization for each irreducible super-unitary representation, using the oscillator representation of an orthosymplectic Lie superalgebra.     

Non-linear gauge symmetry in R2-gravity in two dimensions

The dynamical symmetry ofR ²-gravity with torsion in two dimensions is investigated within Hamiltonian approach. The symmetry may be interpreted as quadratically deformed iso(2,1)-gauge algebra with the deformation given by the Casimir operators of the undeformed algebra.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June 1992.     

Characterization of Lie groups on the cotangent bundle of a Lie group

Characterization, in differential geometric terms, of the groups which can be interpreted as semidirect products of a Lie group G by the group of translations of the dual space of its Lie algebra. Study of the canonical cotangent group of G corresponding to the coadjoint representation. Applications.     

An Analogue of Holstein-Primakoff and Dyson Realizations for Lie Superalgebras. The Lie Superalgebra sl(1/n)

Abstract

An analogue of the Holstein-Primakoff and Dyson realizations for the Lie superalgebra sl(1/n) is written down. Expressions are the same as for the Lie algebra sl(n + 1), however in the latter, Bose operators have to be replaced with Fermi operators.