Compact matrix pseudogroups

The compact matrix pseudogroup is a non-commutative compact space endowed with a group structure. The precise definition is given and a number of examples is presented. Among them we have compact group of matrices, duals of discrete groups and twisted (deformed)SU(N) groups. The representation theory is developed. It turns out that the tensor product of representations depends essentially on their order. The existence and the uniqueness of the Haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived. The form of these relations differs from that in the group case. This is due to the fact that the Haar measure on pseudogroups is not central in general. The corresponding modular properties are discussed. The Haar measures on the twistedSU(2) group and on the finite matrix pseudogroup are found.     

QuantumSU(2) andE(2) groups. Contraction procedure

In [3] it was shown (in the framework of deformed enveloping algebras) that quantumSU(2) andE(2) groups are related by the contraction procedure. We consider the same problem on theC ^*-level. As a result we find a number of formulae coupling the comultiplications in quantumSU(2) andE(2). In particular we show that the comultiplications in both groups are implemented by partial isometries. An unexpected feature of quantumE(2) is discovered and the corresponding strange behavior of quantumSU(2) is described.Supported by CNRS, France     

Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory II: <Emphasis Type="Italic">SU</Emphasis>(3)

We complete the computation of spectral measures for SU(3) nimrep graphs arising in subfactor theory, namely the SU(3) ADE{SU(3) \mathcal{ADE}} graphs associated with SU(3) modular invariants and the McKay graphs of finite subgroups of SU(3). For the SU(2) graphs the spectral measures distill onto very special subsets of the semicircle/circle, whilst for the SU(3) graphs the spectral measures distill onto very special subsets of the discoid/torus. The theory of nimreps allows us to compute these measures precisely. We have previously determined spectral measures for some nimrep graphs arising in subfactor theory, particularly those associated with all SU(2) modular invariants, all subgroups of SU(2), the torus \mathbbT², SU(3){\mathbb{T}^2,\,SU(3)}, and some SU(3) graphs.     

Space symmetry in light nuclei. I Application to the 2s−1d shell

The Wigner supermultiplet scheme is studied in the 2s?1d shell using the spectral distribution methods of French. Expressions for the average expectation value of H and H² over states belonging to a definite SU(4) symmetry are derived. Numerical results for these averages are given for the 2s?1d shell. These averages are then used to predict ground state energies and low energy spectra for nuclei in this shell, using Ratcliff's procedure; a comparison with shell model and observed spectra is made. Mixing intensities of SU(4) representations near the ground state are also evaluated. These provide us with a measure of symmetry breaking. It appears from our calculations that the symmetry is strongly mixed. Finally a preliminary application to the theory of level densities is made.     

Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory

We determine spectral measures for some nimrep graphs arising in subfactor theory, particularly those associated with SU(3) modular invariants. Our methods also give an alternative approach to deriving the results of Banica and Bisch for ADE graphs and subgroups of SU(2) and explain the connection between their results for affine ADE graphs and the Kostant polynomials. We also look at the Hilbert generating series of associated pre-projective algebras.     

Symmetries of quantum spaces. Subgroups and quotient spaces of quantumSU(2) andSO(3) groups

We prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group. We give the formula for the corresponding multiplicities in the case of the quotient quantum spaces. We describe the subgroups and the quotient spaces of quantumSU(2) andSO(3) groups.     

Dimensional regularisation and instantons

Dimensional regularisation is applied to the calculation of the quantum corrections to the instanton tunnelling amplitude in an SU(2) gauge theory. The principal feature is the introduction of an n-dimensional field configuration (a “quasi-instanton”), which generalises the O(5) invariance of the instanton and allows a coordinatisation of the function space of fields in its neighbourhood. This enables the functional integral measure to be factorised, with integrations over the translation and dilatation degrees of freedom being extracted. It is shown that a conformally invariant definition of orthogonality must be used in relation to the zero-mode eigenfunctions of the small oscillations expansion, irrespective of regularisation. An O(n + 1) covariant formalism is employed. An unconventional choice of gauge fixing term, which is not a perfect square, is made and is shown to allow the important freedom of calculating in a gauge specified by an arbitrary parameter α.     

On the representation theory of quantum Heisenberg group and algebra

We show that the quantum Heisenberg groupH _q (1) and its *-Hopf algebra structure can be obtained by means of contraction from quantumSU _q (2) group. Its dual Hopf algebra is the quantum Heisenberg algebraU _q (h(1)). We derive left and right regular representations forU _q (h(1)) as acting on its dualH _q (1). Imposing conditions on the right representation, the left representation is reduced to an irreducible holomorphic representation with an associated quantum coherent state. Realized in the Bargmann-Hilbert space of analytic functions the unitarity of regular representation is also shown. By duality, left and right regular representations for quantum Heisenberg group with the quantum Heisenberg algebra as representation module are also constructed. As before reduction of group left representations leads to finite dimensional irreducible ones for which the intertwinning operator is also investigated.     

Random matrix ensembles with random interactions: Results for EGUE(2)-SU(4)

We introduce in this paper embedded Gaussian unitary ensemble of random matrices, for m fermions in Ω number of single particle orbits, generated by random twobody interactions that are SU(4) scalar, called EGUE(2)-SU(4). Here the SU(4) algebra corresponds to Wigner’s supermultiplet SU(4) symmetry in nuclei. Formulation based on Wigner-Racah algebra of the embedding algebra U(4Ω) ? U(Ω) ? SU(4) allows for analytical treatment of this ensemble and using this analytical formulas are derived for the covariances in energy centroids and spectral variances. It is found that these covariances increase in magnitude as we go from EGUE(2) to EGUE(2)-s to EGUE(2)-SU(4) implying that symmetries may be responsible for chaos in finite interacting quantum systems.     

A Locally Compact Quantum Group Analogue of the Normalizer of SU(1,1) in SL(2, ℂ)

 S.L. Woronowicz proved in 1991 that quantum SU(1,1) does not exist as a locally compact quantum group. Results by L.I. Korogodsky in 1994 and more recently by Woronowicz gave strong indications that the normalizer of SU(1,1) in SL(2,ℂ) is a much better quantization candidate than SU(1,1) itself. In this paper we show that this is indeed the case by constructing , a new example of a unimodular locally compact quantum group (depending on a parameter 0<q<1) that is a deformation of . After defining the underlying von Neumann algebra of we use a certain class of q-hypergeometric functions and their orthogonality relations to construct the comultiplication. The coassociativity of this comultiplication is the hardest result to establish. We define the Haar weight and obtain simple formulas for the antipode and its polar decomposition. As a final result we produce the underlying C ^* -algebra of . The proofs of all these results depend on various properties of q-hypergeometric ₁ϕ₁ functions. Received: 28 June 2001 / Accepted: 25 July 2002 Published online: 10 December 2002 RID="*" ID="*" Post-doctoral researcher of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.) Communicated by L. Takhtajan     

Gauge theory ofSU(2) Weyl fermion: Is it consistent?

Gauge theory ofSU(2) Weyl fermions was alleged by Witten to be inconsistent due to global anomaly. Evidences of inconsistency were also reported from contradictions between the anomalousU(1) symmetry and the fact that theSU(2) group is free of local anomaly. Here we show how the zero modes of Dirac operator, ignored by the authors of these arguments, play a decisive role and saveSU(2) Weyl fermions from inconsistency in each case. The symmetric chiral current, obtained by adding the Chern-Simons current to the fermionic chiral current, fails to be conserved precisely due to the contributions of zero modes to the ABJ anomaly equation. The Jacobian of the fermion measure under rigid chiralU(1) transformation is, however, guaranteed to be trivial by the Atiyah-Singer index theorem. Finally, a zero mode is the point of bifurcation of eigenvalue trajectory in the homotopy space. In its neighbourhood the hypothesis of adiabaticity made by Witten breaks down due to violent oscillations between levels, which makes his allegation of global anomaly untenable.     

Octonionic Non-Abelian Gauge Theory

We have made an attempt to describe the octonion formulation of Abelian and non-Abelian gauge theory of dyons in terms of 2×2 Zorn vector matrix realization. As such, we have discussed the U(1) _e ×U(1) _m Abelian gauge theory and U(1)×SU(2) electroweak gauge theory and also the SU(2) _e ×SU(2) _m non-Abelian gauge theory in term of 2×2 Zorn vector matrix realization of split octonions. It is shown that SU(2) _e characterizes the usual theory of the Yang Mill’s field (isospin or weak interactions) due to presence of electric charge while the gauge group SU(2) _m may be related to the existence of ’t Hooft-Polyakov monopole in non-Abelian Gauge theory. Accordingly, we have obtained the manifestly covariant field equations and equations of motion.     

Unbounded elements affiliated withC*-algebras and non-compact quantum groups

The affiliation relation that allows to include unbounded elements (operators) into theC ^*-algebra framework is introduced, investigated and applied to the quantum group theory. The quantum deformation of (the two-fold covering of) the group of motions of Euclidean plane is constructed. A remarkable radius quantization is discovered. It is also shown that the quantumSU(1, 1) group does not exist on theC ^*-algebra level for real value of the deformation parameter.Supported by Japan Society for the Promotion of Science     

Spectral Measures Associated to Rank two Lie Groups and Finite Subgroups of {GL(2,\mathbb{Z})}

Spectral measures for fundamental representations of the rank two Lie groups SU(3), Sp(2) and G₂ have been studied. Since these groups have rank two, these spectral measures can be defined as measures over their maximal torus \({\mathbb{T}^2}\) and are invariant under an action of the corresponding Weyl group, which is a subgroup of \({GL(2,\mathbb{Z})}\). Here we consider spectral measures invariant under an action of the other finite subgroups of \({GL(2,\mathbb{Z})}\). These spectral measures are all associated with fundamental representations of other rank two Lie groups, namely \({\mathbb{T}^2=U(1) \times U(1)}\), \({U(1) \times SU(2)}\), U(2), \({SU(2) \times SU(2)}\), SO(4) and PSU(3).     

Calculation of the viscosity of SU(2) gluodynamics with the lattice simulation

The viscosity of SU(2) gluodynamics within the simulation of the lattice quantum chromodynamics at a temperature of T/T _c = 1.2 has been calculated with the Kubo formula relating the viscosity to the spectral function of the correlation function of the energy-momentum tensor. The correlation function of the energy-momentum tensor has been calculated using the numerical simulation of the lattice SU(2) gluodynamics on supercomputers.     

The SU(5) potential in desitter space

The one-loop effective potential for a minimal SU(5) theory is calculated on a curved DeSitter background spacetime. The stability of its extrema in the following subgroups is investigated: SU(4) × U(1), SU(3) × SU(2) × U(1), SU(3) × U(1) × U(1), SU(2) × SU(2) × U(1) × U(1). A combination of analytic and numerical methods is used to obtain phase diagrams for the model. In the inflationary universe, the curvature effects do not prevent a slide into the SU(4) × U(1) extremum.     

Algebraic realization of the quantum rotor — odd-A nuclei

An algebraic realization of the quantum rotor for non-zero spin values (integer as well as half-integer) is established by constructing a model Hamiltonian out of rotationally invariant functions of the generators ofSU(3). The eigenvalues of this Hamiltonian in the leading normal-SU(3) symmetry for²⁵Mg and the so-called leading pseudo-SU(3) symmetries for¹⁵⁹Dy and¹⁶⁵Er are compared with the corresponding rotor results. For spinfree systems the internal symmetry group of the rotor and itsSU(3) realization are known to be D₂, the Vierergruppe. This symmetry extends to integral spin values, while for half-integer spins the rotor and itsSU (3) realization are shown to display an internal quaternion group symmetry. The theory points to a microscopic (many-particle shell-model) picture of nuclear rotational motion with spin degrees of freedom taken fully into account. An algebraic realization of the many-particle Nilsson model for odd-A nuclei, with the orbit-orbit and spin-orbit terms included, is given and applied to²³Na.     

SO(N) supergravity and unification of all fundamental forces

From the group theoretical arguments, we find that among allSO(N) supergravitiesN=10 is the minimal supersymmetry group which unifies all fundamental forces of weak, electromagnetic, strong and gravitational interactions. The (super)symmetry is broken throughSO(10)→SU(3)?SU(2)?U(1)→SU(3)?U(1). All observed particles of the low energy physics (three generations of quarks and leptons, γ,Z, W ^± and gluons) and graviton can be minimally accomodated with the correctSU(3)?SU(2)?U(1) quantum numbers. Some characteristic predictions, which can be checked in the coming high energy experiments, are briefly discussed.