The su q(2) algebra in the off-diagonal basis and applications to quantum optics

We consider a new exactly solvable nonlinear quantum model as a Hamiltonian defined in terms of the generators of the su _q(2) algebra. The corresponding matrix elements of finite rotations (the q-deformed Wigner d functions) are introduced. It is shown that the quantum optical model of the three-wave interaction has an approximate su _q(2) dynamical symmetry given by this Hamiltonian. Such q symmetry allows us to investigate the spectral and dynamical properties of the three wave model through new perturbation techniques.     

Indecomposable representations of the algebra su(2,1)

Representations of the Lie algebra sl(3) with highest weight are analyzed. Invariant subspaces of indecomposable representations are determined. We study the decomposition of these representations with respect to the subalgebras su(2) and su(1,1) (in their obvious imbedding in su(2,1)).For special cases this decomposition gives indecomposable non multiplicity free representations (indecomposable pairs) with highest weight. These were discussed in [1] and appear also in the decomposition so(3,2) su(1,1) of the Rac representation, [7].     

Quantum deformedsu(m/n) algebra and superconformal algebra on quantum superspace

We study a deformedsu(m/n) algebra on a quantum superspace. Some interesting aspects of the deformed algebra are shown. As an application of the deformed algebra we construct a deformed superconformal algebra. From the deformedsu(1/4) algebra, we derive deformed Lorentz, translation of Minkowski space,iso(2, 2) and its supersymmetric algebras as closed subalgebras with consistent automorphisms.     

On a general analytic formula for U q(su(3)) Clebsch-Gordan coefficients

We present the projection operator method in combination with the Wigner-Racah calculus of the subalgebra U _q(su(2)) for calculation of Clebsch-Gordan coefficients (CGCs) of the quantum algebra U _q(su(3)). The key formulas of the method are couplings of the tensor and projection operators and also a tensor form for the projection operator of U _q(su(3)). We obtain a very compact general analytic formula for the U _q(su(3)) CGCs in terms of the U _q(su(2)) Wigner 3nj symbols.     

Non-periodic Ishibashi states: the su(2) and su(3) affine theories

We consider the su(2) and su(3) affine theories on a cylinder, from the point of view of their discrete internal symmetries. To this end, we adapt the usual treatment of boundary conditions leading to the Cardy equation to take the symmetry group into account. In this context, the role of the Ishibashi states from all (non-periodic) bulk sectors is emphasized. This formalism is then applied to the su(2) and su(3) models, for which we determine the action of the symmetry group on the boundary conditions, and we compute the twisted partition functions. Most if not all data relevant to the symmetry properties of a specific model are hidden in the graphs associated with its partition function, and their subgraphs. A synoptic table is provided that summarizes the many connections between the graphs and the symmetry data that are to be expected in general.     

The su(n) Hubbard model

The one-dimensional Hubbard model is known to possess an extended su(2) symmetry and to be integrable. I introduce an integrable model with an extended su(n) symmetry. This model contains the usual su(2) Hubbard model and has a set of features that makes it the natural su(n) generalization of the Hubbard model. Complete integrability is shown by introducing the L-matrix and showing that the transfer matrix commutes with the Hamiltonian. While the model is integrable in one dimension, it provides a generalization of the Hubbard Hamiltonian in any dimension.     

The representations and coupling coefficients of su(n); application to su(4)

Analytical expressions for the matrices and an explicit algorithm for computing Clebsch-Gordan coupling coefficients are given forsu(4) in au(3)-coupled basis as an example of the construction for anysu(n) in au(n−1) basis. The results areinduced from the known results foru(3) by means of the vector-coherent-state (VCS) theory of induced representations. The important recent result that makes this possible is the discovery that a complete set of shift tensors for the finitedimensional representations of reductive Lie algebras can be induced, by VCS methods, from those of suitably defined subalgebras.     

Kac-Moody symmetries of critical ground states

The symmetries of critical ground states of two-dimensional lattice models are investigated. We show how mapping a critical ground state to a model of a rough interface can be used to identify the chiral symmetry algebra of the conformal field theory that describes its scaling limit. This is demonstrated in the case of the six-vertex model, the three-coloring model on the honeycomb lattice, and the four-coloring model on the square lattice. These models are critical and they are described in the continuum by conformal field theories whose symmetry algebras are the su(2)_k=1, su(3)_k=1, and the su(4)_k=1 Kac-Moody algebra, respectively. Our approach is based on the Frenkel-Kac-Segal vertex operator construction of level-one Kac-Moody algebras.     

Fundamental Fermions Fit Inside One <Emphasis Type="Italic">su</Emphasis>(1|5) Irreducible Representation

The Lie superalgebra su(1|5) has irreducible representations of dimension 32, in which the 32 fundamental fermions of one generation (leptons and quarks, of left and right chirality, and their antiparticles) can be accommodated. The branching of these su(1|5) representations with respect to its subalgebra su(3)× su(2)× u(1) reproduces precisely the classification of these fundamental fermions according to the gauge group su(3)^c× su(2)^w× u(1)^w of the Standard Model. Furthermore, a simple construction of the relevant representations is given, and some consequences are discussed.     

Generalized two-mode harmonic oscillator model: squeezed number state solutions and nonadiabatic Berry’s phase

A generalized two-mode harmonic oscillator model is investigated within the framework of its general dynamical algebra so(3,2). Two types of eigenstates, formulated as extended su(1,1), su(2) squeezed number states are found respectively. The nonadiabatic Berrys phase for this system with the cranked time-dependent Hamiltonian is also given.Received: 16 January 2004, Published online: 10 August 2004PACS: 42.50.Dv Nonclassical states of the electromagnetic field, including entangled photon states; quantum state engineering and measurements - 03.65.Fd Algebraic methods - 03.65.Vf Phases: geometric; dynamic or topological     

Realizations ofU q(su(1, 1)) by deformed quantum harmonic oscillator algebras and associated deformed Heisenberg algebras

Forsu(1, 1)-symmetric Hamiltonians of quantum mechanical systems (e.g. single-mode quantum harmonic oscillator, radial Schrödinger equation for Coulomb problem or isotropic quantum harmonic oscillator, etc.), the Heisenberg algebra of phase-space variables in two dimensions satisfy the bilinear commutation relation [ip,x]=1 (in normal units). Also there are different realizations ofsu(1, 1) by the generators of quantum harmonic oscillator algebra. We seek here the forms of deformed Heisenberg algebras (bilinear in deformedx and ip) associated with deformedsu(1, 1)-symmetric Hamiltonians. These forms are not unique in contrast to the undeformed case; and these forms are obtained here by considering different realizations of the deformedsu(1, 1) algebra by deformed oscillator algebras (satisfying different bilinear relations in deformed creation and annihilation operators), and then imposing different conditions (e.g. the deformed Heisenberg algebra of the form of the undeformed one, the form of realizations of the deformedsu(1, 1) algebra by deformed phase-space variables being the same as that ofsu(1, 1) algebra by undeformed phase-space variables, etc.), assuming linear relations between deformed phase-space variables and deformed creation-annihilation operators (as it is done in the undeformed case), we get different Heisenberg algebras. These facts are revealed in the case of a two-body Calogero model in its centre of mass frame (and for no other integrable systems in one-dimension having potential of the formV(x _i ? x_j).     

Noncompact <Emphasis Type="Italic">u</Emphasis><Subscript>q</Subscript>(2, 1) quantum algebra: Discrete series of highest weight irreducible representations

The structure of unitary irreducible representations of the noncompact u_q(2, 1) quantum algebra that are related to a negative discrete series is examined. With the aid of projection operators for the su_q(2) subalgebra, a q analog of the Gelfand-Graev formulas is derived in the basis corresponding to the reduction u_q(2, 1) → su_q(2)×u(1). Projection operators for the su_q(1, 1) subalgebra are employed to study the same representations for the reduction u_q(2, 1) → u(1)×su_q(1, 1). The matrix elements of the generators of the u_q(2, 1) algebra are computed in this new basis. A general analytic expression for an element of the transformation brackets <U∣T>_q between the bases associated with the above two reductions (the elements of this matrix are referred to as q Weyl coefficients) is obtained for a general case where the deformation parameter q is not equal to a root of unity. It is shown explicitly that, apart from a phase, the q Weyl coefficients coincide with the q Racah coefficients for the su_q(2) quantum algebra.     

Integrable chiral theories in 2 + 1 dimensions

Following a recent proposal for integrable theories in higher dimensions based on zero curvature, new Lorentz invariant submodels of the principal chiral model in 2 + 1 dimensions are found. They have infinite local conserved currents, which are explicitly given for the su(2) case. The construction works for any Lie algebra and in any dimension, and it is given explicitly also for su(3). We comment on the application to supersymmetric chiral models.     

On the classification of diagonal coset modular invariants

We relate in a novel way the modular matrices of GKO diagonal cosets without fixed points to those of WZNW tensor products. Using this we classify all modular invariant partition functions ofsu(3) _k su(3)₁/su(3) _k+1 for all positive integer levelk, andsu(2) _k su(2) _l /su(2) _k+1 for allk and infinitely manyl (in fact, for eachk a positive density ofl). Of all these classifications, only that forsu(2) _k su(2)₁/su(2) _k+1 had been known. Our lists include many new invariants.Supported in part by NSERC.     

Ternary Virasoro–Witt algebra

A 3-bracket variant of the Virasoro–Witt algebra is constructed through the use of su(1,1)$\mathit{su}(1,1)$ enveloping algebra techniques. The Leibniz rules for 3-brackets acting on other 3-brackets in the algebra are discussed and verified in various situations.     

Graded Hopf maps and fuzzy superspheres

We argue supersymmetric generalizations of fuzzy two- and four-spheres based on the unitary-orthosymplectic algebras, uosp(N|2) and uosp(N|4), respectively. Supersymmetric version of Schwinger construction is applied to derive graded fully symmetric representation for fuzzy superspheres. As a classical counterpart of fuzzy superspheres, graded versions of 1st and 2nd Hopf maps are introduced, and their basic geometrical structures are studied. It is shown that fuzzy superspheres are represented as a “superposition” of fuzzy superspheres with lower supersymmetries. We also investigate algebraic structures of fuzzy two- and four-superspheres to identify su(2|N) and su(4|N) as their enhanced algebraic structures, respectively. Evaluation of correlation functions manifests such enhanced structure as quantum fluctuations of fuzzy supersphere.     

Invariant differential operators for non-compact Lie groups: Summary of su(4,4) multiplets

The present paper is part of the project of systematic construction of invariant differential operators of noncompact semisimple Lie algebras. Here we give a summary of all multiplets containing physically relevant representations including the minimal ones for the algebra su(4, 4). Due to the recently established parabolic relations the results are valid also for the algebras sl(8, R) and su*(8)