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.     

Localization of Unitary Braid Group Representations

Governed by locality, we explore a connection between unitary braid group representations associated to a unitary R-matrix and to a simple object in a unitary braided fusion category. Unitary R-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary R-matrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q = e ^πi/6 specialization of the unitary Jones representation of the braid groups can be localized by a unitary 9 × 9 R-matrix. Actually this Jones representation is the first one in a family of theories (SO(N), 2) for an odd prime N > 1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science.     

Universal Covering Group of U(n) and Projective Representations

Using fiber bundle theory, we construct the universal covering group of U(n),U(n), and show that U(n) is isomorphic to the semidirect product SU(n) ∝.We give a bijection between the set of projective representations of U(n) and theset of equivalence classes of certain unitary representations of SU(n) ∝.Applying Bargmann's theorem, we give explicit expressions for the liftings ofprojective representations of U(n) to unitary representations of SU(n) ∝. Forcompleteness, we discuss the topological and group theoretic relations betweenU(n), SU(n), U(t), and Z _n .     

Contractions of representations of de Sitter groups

In order to construct the quantum field theory in a curved space with no old infinities as the curvature tends to zero, the problem of contraction of representations of the corresponding group of motions is studied. The definitions of contraction of a local group and of its representations are given in a coordinate-free manner. The contraction of the principal continuous series of the de Sitter groupsSO ₀(n, 1) to positive mass representations of both the Euclidean and Poincaré groups is carried out in detail. It is shown that all positive mass continuous unitary irreducible representations of the resulting groups can be obtained by this method. For the Poincaré groups the contraction procedure yields reducible representations which decompose into two non-equivalent irreducible representations.On leave of absence from the Institute of Physics of the Czechoslovak Academy of Sciences, Prague, Czechoslovakia.     

Unitary representations of the quantum group SU q (1, 1): II — Matrix elements of unitary representations and the basic hypergoemetric functions

Some series of unitary representations of the quantum group SU _q (1, 1) are introduced. Their matrix elements are expressed in terms of the basic hypergeometric functions. Operator realization of the coordinate elements of SU _q (1, 1) and aq-analogue of some classical identities are discussed.     

u q(2,1) noncompact quantum algebra: Intermediate discrete series of unitary irreducible representations

The unitary irreducible representations of the u _q(2,1) quantum algebra that belong to the intermediate discrete series are considered. The q analog of the Mickelsson-Zhelobenko algebra is developed. Use is made of the U basis corresponding to the reduction u _q(2,1) ? u _q(2). Explicit formulas for the matrix elements of the generators are obtained in this basis. The projection operator that projects an arbitrary vector onto the extremal vector of the intermediate-series representation is found.     

Noncommutative Instantons on the 4-Sphere¶from Quantum Groups

We describe an approach to the noncommutative instantons on the 4-sphere based on quantum group theory. We quantize the Hopf bundle ?⁷→?⁴ making use of the concept of quantum coisotropic subgroups. The analysis of the semiclassical Poisson–Lie structure of U(4) shows that the diagonal SU(2) must be conjugated to be properly quantized. The quantum coisotropic subgroup we obtain is the standard SU _q (2); it determines a new deformation of the 4-sphere ∑⁴ _q as the algebra of coinvariants in ? _q ⁷. We show that the quantum vector bundle associated to the fundamental corepresentation of SU _q (2) is finitely generated and projective and we compute the explicit projector. We give the unitary representations of ∑⁴ _q , we define two 0-summable Fredholm modules and we compute the Chern–Connes pairing between the projector and their characters. It comes out that even the zero class in cyclic homology is non-trivial. Received: 3 January 2001 / Accepted: 14 November 2001     

1- Products in the method of orbits for nilpotent groups

On each orbit W of the coadjoint representation of any nilpotent (connected, simply connected) Lie group G, we construct 1-products and associated Von Neumann algebras $\text{G}$. G acts canonically on $\text{G}$ by automorphisms. In the unique faithful, irreducible representation of $\text{G}$, this action is implemented by the unitary irreducible representation of G corresponding to W by the Kirillov method. This construction is uniquely determined by W and gives the classification of all unitary irreducible representations of G.     

Quantum graph vertices with permutation-symmetric scattering probabilities

Boundary conditions in quantum graph vertices are generally given in terms of a unitary matrix U. Observing that if U has at most two eigenvalues, then the scattering matrix S(k) of the vertex is a linear combination of the identity matrix and a fixed Hermitian unitary matrix, we construct vertex couplings with this property: For all momenta k, the transmission probability from the j-th edge to ?-th edge is independent of (j,?), and all the reflection probabilities are equal. We classify these couplings according to their scattering properties, which leads to the concept of generalized δ- and δ^′-couplings.     

Matter as Spectrum of Spacetime Representations

Bound and scattering state Schrödinger functions of nonrelativistic quantum mechanics as representation matrix elements of space and time are embedded into residual representations of spacetime as generalizations of Feynman propagators. The representation invariants arise as singularities of rational representation functions in the complex energy and complex momentum plane. The homogeneous space GL(²)U(2) with rank 2, the orientation manifold of the unitary hypercharge-isospin group, is taken as model of nonlinear spacetime. Its representations are characterized by two continuous invariants whose ratio will be related to gauge field coupling constants as residues of the related representation functions. Invariants of product representations define unitary Poincaré group representations with masses for free particles in tangent Minkowski spacetime.     

Unitary representations of some infinite dimensional groups

We construct projective unitary representations of (a) Map(S ¹;G), the group of smooth maps from the circle into a compact Lie groupG, and (b) the group of diffeomorphisms of the circle. We show that a class of representations of Map(S ¹;T), whereT is a maximal torus ofG, can be extended to representations of Map(S ¹;G),     

∗-Representations of the Quantum Algebra U q(sl(3))

Abstract

Studied in this paper are real forms of the quantum algebra U _q(sl(3)). Integrable operator representations of ?-algebras are defined. Irreducible representations are classified up to a unitary equivalence.     

New rational solutions of Yang-Baxter equation and deformed Yangians

In this paper a new class of quantum groups, deformed Yangians, is used to obtain new matrix rational solutions of the Yang-Baxter equation (YBE). The deformed Yangians arise from rational solutions of the classical Yang-Baxter equation of the form c ₂/u + const. The image of the universal quantum R-matrix for the deformed Yangian in finite-dimensional representations gives these new matrix rational solutions of YBE.     

Quasi-1 structure on q-Poincaré algebras

We use braided groups to introduce a theory of 1-structures on general inhomogeneous quantum groups, which we formulate as quasi-1 Hopf algebras. This allows the construction of the tensor product of unitary representations up to a quantum cocycle isomorphism, which is a novel feature of the inhomogeneous case. Examples include q-Poincaré quantum group enveloping algebras in R-matrix from appropriate to the previous q-Euclidean and q-Minkowski space-time algebras R₂₁x₁x₂ = x₂x₁R and R₂₁u₁Ru₂ = u₂R₂₁u₁R. We obtain unitarity of the fundamental differential representations. We further show that the Euclidean and Minkowski-Poincaré quantum groups are twisting equivalent by another quantum cocycle.     

The real symplectic groups in quantum mechanics and optics

We present a utilitarian review of the family of matrix groups Sp(2n, ℛ), in a form suited to various applications both in optics and quantum mechanics. We contrast these groups and their geometry with the much more familiar Euclidean and unitary geometries. Both the properties of finite group elements and of the Lie algebra are studied, and special attention is paid to the so-called unitary metaplectic representation of Sp(2n, ℛ). Global decomposition theorems, interesting subgroups and their generators are described. Turning ton-mode quantum systems, we define and study their variance matrices in general states, the implications of the Heisenberg uncertainty principles, and develop a U(n)-invariant squeezing criterion. The particular properties of Wigner distributions and Gaussian pure state wavefunctions under Sp(2n, ℛ) action are delineated.     

Algebraic Coset Conformal Field Theories

All unitary Rational Conformal Field Theories (RCFT) are conjectured to be related to unitary coset Conformal Field Theories, i.e., gauged Wess–Zumino–Witten (WZW) models with compact gauge groups. In this paper we use subfactor theory and ideas of algebraic quantum field theory to approach coset Conformal Field Theories. Two conjectures are formulated and their consequences are discussed. Some results are presented which prove the conjectures in special cases. In particular, one of the results states that a class of representations of coset W _N (N≥ 3) algebras with critical parameters are irreducible, and under the natural compositions (Connes' fusion), they generate a finite dimensional fusion ring whose structure constants are completely determined, thus proving a long-standing conjecture about the representations of these algebras. Received: 5 November 1998 / Accepted: 18 October 1999     

Geometry of 2d spacetime and quantization of particle dynamics

We analyze classical and quantum dynamics of a relativistic particle in 2d spacetimes with constant curvature. We show that global symmetries of spacetime specify the symmetries of physical phase-space and the corresponding quantum theory. To quantize the systems we parametrize the physical phase-space by canonical coordinates. Canonical quantization leads to unitary irreducible representations of SO_↑(2.1) group.     

Diagonalization of the braid generator on unitary irreps of quantum supergroups

It is shown that the braid generator associated with the universalR-matrix is diagonalizable on all unitary representations of quantum supergroups. An example is considered using U _q (gl(2|1)) and a family of eight-dimensional typical representations.     

A Coset-Type Construction for the Deformed Virasoro Algebra

An analog of the minimal unitary series representations for the deformed Virasoro algebra is constructed using vertex operators of the quantum affine algebra U_q(sl₂). A similar construction is proposed for the elliptic algebra A_q,p(sl₂).