On the tensor product of representations of semisimple Lie groups

All topologically irreducible representations involved in the tensor product of finite and infinite dimensional representations of the Principal nonunitary series (including the unitary series) of a semisimple Lie group are defined.     

SO q (n+1,n−1) as a real form of SO q (2n,ℂ)

Quantum pseudo-orthogonal groups SO _q (n+1,n–1) are defined as real forms of quantum orthogonal groups SO _q (n+1,n–1) by means of a suitable antilinear involution. In particular, the casen=2 gives a quantized Lorentz group.     

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.     

Canonical representations of the Lie superalgebraosp(1,4)

The method for constructing infinite-dimensional representations of Lie superalgebras proposed by the authors recently is applied to the superalgebraosp (1, 4). Explicit formulae for its generators in terms of two or three pairs of operators fulfilling the canonical commutation relations, at most one pair of operators fulfilling the canonical anticommutation relations and at most one real parameter are obtained. The generators of the Lie subalgebrasp (4, ) osp (1,4) are represented skew-symmetrically and both the Casimir operators are equal to multiples of the unity operator.Dedicated to Professor Ivan Úlehla on the occasion of his sixtieth birthday.     

Matrix elements and casimir operators of the discrete series representations of the groupU(p,q)

Explicit form of matrix elements of the discrete series representations of the groupU(p, q) are given. Casimir operators of these representations are defined.     

On the kinematics of (e, e′p) reactions

The problem of interpreting experimental data on quasielastic electron scattering on nuclei in A(e, e′p)(A?1) reactions is considered. It is shown that the existing discrepancies in experimental data on the reaction ⁴He(e, e′p)T are associated with the fact that the residual-nucleus momentum p _m as determined from the law of energy-momentum conservation cannot be treated as that which is equal to the momentum p of the primary intranuclear nucleon. Under the assumption that the momentum transferred from the electron to the intranuclear nucleon is redistributed during the divergence of the products of the (e, e′p) reaction in question, the method for extracting p is modified by introducing a kinematical correction, whereby the situation is considerably improved. For a first approximation, the correction can be evaluated on the basis of data on A(e, e′) inclusive reactions. The argument behind this evaluation is illustrated by considering the example of the reaction ⁴He(e, e′p)T.     

Fock representations of the affine Lie algebraA 1 (1)

The aim of this note is to show that the affine Lie algebraA ₁ ⁽¹⁾ has a natural family _, ,v of Fock representations on the spaceC[x _i,y _j;i andj ], parametrized by (,v) C ². By corresponding the highest weight _, of _, to each (,), the parameter spaceC ₂ forms a double cover of the weight spaceC₀C –₁ with singularities at linear forms of level –2; this number is (–1)-times the dual Coxeter number. Our results contain explicit realizations of irreducible non-integrable highest wieghtA ₁ ⁽¹⁾ -modules for generic (,v).     

A unified treatment of the groups SO(4) and SO(3,1)

The irreducible representations of the group SO(4) in which the SO(3) subgroup is reduced are studied by an explicit construction of the operators and the basis in the spinor representation. The basis function which is formally identical with that for the coupling of two angular momentaj ₁ andj ₂ is expressible in terms of a hypergeometric function and strongly resembles the one for the irreducible representations of the groups SO(3,1). For the Lorentz group, the bases for the unitary representations which require unphysical values ofj ₁ andj ₂ are found to be analytic continuation of those for SO(4). The realization of the unitary irreducible representations of the group SO(4) in the Hilbert space of these functions leads, for appropriate unphysical values ofj ₁,j ₂, to the Gelfand-Naimark formula for the principal and complementary series of the representations of SO(3;1). The matrix elements for finite transformations of SO(4) and SO(3,1) can be evaluated, in this approach, in a unified manner by using standard properties of the hypergeometric function. These turn out to be a finite sum of₃ F ₂-functions which, as expected, are polynomials for SO(4) and infinite series for SO(3,1). A number of special matrix elements are calculated from the general formula and these agree with the results obtained previously.The authors are deeply indebted to Professor S.Dutta Majumdar fo many important suggestions and clarifications.     

The Multicomponent Coherent States of the Lie Algebra SO(4)

The multicomponent coherent states associated with the Lie algebra SO(4) are presented. An inhomogeneous differential realization of SO(4) in this multicomponent coherent state space is obtained.     

Topological representations of the quantum groupU q(sl 2)

We define a topological action of the quantum groupU _q(sl ₂) on a space of homology cycles with twisted coefficients on the configuration space of the punctured disc. This action commutes with the monodromy action of the braid groupoid, which is given by theR-matrix ofU _q(sl ₂).Currently visiting the Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA. Supported in part by the NSF under Grant No. PHY89-04035, supplemented by funds from the NASA     

A classification of the unitary irreducible representations ofSU(N, 1)

All inequivalent continuous unitary irreducible representations ofS U(N, 1) (N2) have been determined and classified. The matrix elements of the infinitesimal generators realized on a certain Hilbert space have been derived. Representations of the groups ,S U(N, 1)/Z _N+1, andU(N, 1) are classified in a similar manner.     

R-matrix formulation of the quantum inhomogeneous groups ISO q,r (N) and ISp q,r (N)

The quantum commutationsRTT=TTR and the orthogonal (symplectic) conditions for the inhomogeneous multiparametricq-groups of theB _n ,C _n ,D _n type are found in terms of theR-matrix ofB _n+1 ,C _n+1 ,D _n+1 .A consistent Hopf structure on these inhomogeneousq-groups is constructed by means of a projection fromB _n+1 ,C _n+1 ,D _n+1 .Real forms are discussed; in particular, we obtain theq-groups ISO _q,r (n+1,n–1), including the quantum Poincaré group. The inhomogeneusq-groups do not contain dilatations when the parameters satisfy certain conditions. For example, we find a dilatation-freeq-Poincaré group depending on one real parameterq.     

Representation of Spin Group Spin（p, q）

The representation η（P, q） of spin group Spin（p, q） in any dimensional space is given by induction, and the relation between two representations, which are obtained in two kinds of inductions from Spin（p, q） to Spin（p ＋ 1, q ＋ 1） are studied.     

A (p,q)-Deformation of Quantum Mechanics

On a one-dimensional lattice with a geometric sequence spacing, the Hermitian conjugation of a (p, q)-derivative operator is discussed by means of (p, q)-integration. Then a (p, q)-deformation of both the Heisenberg algebra for the canonical coordinates and the Heisen berg-Weyl algebra for the harmonic oscillator is presented. It is shown that although in the algebraic aspect the (p, q)-deformation discussed here is identical with 9-deformation given by Truong, the (p, q)-deformed SchrGdinger picture is in fact different from the q-deformed one.     

A classification of the unitary irreducible representations ofSO 0(N, 1)

All inequivalent continuous unitary irreducible representations of the groupSO ₀(N, 1),N3, and its universal covering group are classified.     

A remarkable connection between the representations of the Lie superalgebras osp(1, 2n) and the Lie algebras o(2n+1)

We show that there is a one-to-one correspondence between the graded representations of osp(1, 2n) and the non-spinorial representations of o(2n+1). The Clebsch-Gordan series for osp(1, 2n) reduce to the corresponding series for o(2n+1) and the properly defined Casimir operators of order at least up to four have the same eigenvalues.Supported by the Deutsche Forschungsgemeinschaft