Existence and invariance of twisted products on a symplectic manifold

It is now well-known [1] that the twisted product on the functions defined on a symplectic manifold, play a fundamental role in an invariant approach of quantum mechanics. We prove here a general existence theorem of such twisted products. If a Lie group G acts by symplectomorphisms on a symplectic manifold and if there is a G-invariant symplectic connection, the manifold admits G-invariant Vey twisted products. In particular, if a homogeneous space G/H admits an invariant linear connection, T ^*(G/H) admits a G-invariant Vey twisted product. For the connected Lie group G, the group T ^*G admits a symplectic structure, a symplectic connection and a Vey twisted product which are bi-invariant under G.     

Non-compact quantum groups associated with Abelian subgroups

LetG be a Lie group. For any Abelian subalgebra of the Lie algebra g ofG, and any , the difference of the left and right translates ofr gives a compatible Poisson bracket onG. We show how to construct the corresponding quantum group, in theC ^*-algebra setting. The main tool used is the general deformation quantization construction developed earlier by the author for actions of vector groups onC ^*-algebras.The research reported on here was supported in part by National Science Foundation grant DMS-9303386.     

An explicit *-product on the cotangent bundle of a Lie group

We give explicit formulas for a ^*-product on the cotangent bundle T ^* G of a Lie group G; these formulas involve on the one hand the multiplicative structure of the universal enveloping algebra U(G) of the Lie algebra G of G and on the other hand bidifferential operators analogous to the ones used by Moyal to define a ^*-product on IR²ⁿ.Chargé de recherches au FNRS, on leave of absence from Université libre de Bruxelles.     

A Unified Approach to Poisson Reduction

Given any Poisson action G×PP of a Poisson–Lie group G we construct an object =T ^*G*T^* P which has both a Lie groupoid structure and a Lie algebroid structure and which is a half-integrated form of the matched pair of Lie algebroids which J.-H. Lu associated to a Poisson action in her development of Drinfeld's classification of Poisson homogeneous spaces. We use to give a general reduction procedure for Poisson group actions, which applies in cases where a moment map in the usual sense does not exist. The same method may be applied to actions of symplectic groupoids and, most generally, to actions of Poisson groupoids.     

Classical and quantum duality in Jordanian quantizations

We prove that for the first order coboundary deformation of a Lie bialgebra (g, g₁ ^*) (g, g₁ ^* + g₂ ^*) one can always get the quantized Lie bialgebra A(g, g₂ ^*) as a limit of the sequence of quantizations of the type A(g, g₁ ^*).     

Unitary positive energy representations of the gauge group

We investigate the positive energy representations (also called highest weight representations) of the gauge groupC (T ^v,G ₀),G ₀ being a compact simple Lie group, and discuss their unitarity, using the technique of Verma modules constructed from generalized loop algebras (a simple generalization of Kac-Moody affine Lie algebras). We show that the unitarity of the representation imposes severa restrictions in it. In particular, we show, as a part of a more general result, that the gauge group does not admit faithful unitary positive energy representations.Allocataire du MRT.     

A Duality Theorem for Ergodic Actions of Compact Quantum Groups on <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

The spectral functor of an ergodic action of a compact quantum group G on a unital C ^*-algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation, and the inclusion maps satisfy natural properties. We show that any quasitensor ^*-functor from Rep(G) to the category of Hilbert spaces is the spectral functor of an ergodic action of G on a unital C ^*-algebra. As an application, we associate an ergodic G-action on a unital C ^*-algebra to an inclusion of Rep(G) into an abstract tensor C ^*-category . If the inclusion arises from a quantum subgroup K of G, the associated G-system is just the quotient space K\G. If G is a group and has permutation symmetry, the associated G-system is commutative, and therefore isomorphic to the classical quotient space by a subgroup of G. If a tensor C ^*-category has a Hecke symmetry making an object ρ of dimension d and μ-determinant 1, then there is an ergodic action of S _μ U(d) on a unital C ^*-algebra having the as its spectral subspaces. The special case of is discussed.     

On the integrability of the Lie algebra of the conformal group in quantum field theory

It is shown that the infinitesimal conformal symmetry implies (in any quantum field theory which satisfies the Wightman axioms without invoking locality and global Poincaré symmetry) that there exists a uniquely defined unitary representation of the universal (-sheeted) covering group of the Minkowskian conformal groupSO _e (4,2)/ ₂. Proof was obtained using sufficient conditions for the integrability of a representation of a Lie algebra given by [8].     

Fundamental representations of quantum groups at roots of 1

To every finite-dimensional irreducible representation V of the quantum group U(g) where is a primitive lth root of unity (l odd) and g is a finite-dimensional complex simple Lie algebra, de Concini, Kac and Procesi have associated a conjugacy class C _V in the adjoint group G of g. We describe explicitly, when g is of type A _n , B _n , C _n , or D _n , the representations associated to the conjugacy classes of minimal positive dimension. We call such representations fundamental and prove that, for any conjugacy class, there is an associated representation which is contained in a tensor product of fundamental representations.     

Specific heat behavior of high temperature superconductors in the pseudogap regime

We consider the competition between the one dimensionalization effect due to a magnetic field and the hopping parameters in quasi-one-dimensional conductors. Our study is based on a perturbative renormalization group method with three cut-off parameters, the bandwidth E₀, the 1D-2D crossover temperature T^*₁, which is related to the hopping process t₁, and the magnetic energy . We have found that the renormalized crossover temperatures T^*₁ and T^*₂, at which the respectively hopping processes t₁ and t₂ become coherent, are reduced compared to the bare values as the field is increased. We discuss the consequences of these renormalization effects on the temperature-field phase diagram of the organic conductors.Received: 5 March 2003, Published online: 23 July 2003PACS: 64.60.-i General studies of phase transitions - 75.30.Fv Spin-density waves - 72.15.Gd Galvanomagnetic and other magnetotransport effects - 74.70.Kn Organic superconductors     

Highly mobile Einstein spaces in the large

We consider an Einstein spaceV of the Petrov type II or III admitting a group of motionsG of high order. First we calculate the composition law and topological structure ofG. ThenV (or its submanifolds of transitivity) is represented as the homogeneous spaceG/H ofG,H being a subgroup ofG, and the actionG onV and the topology ofV are determined. The topologies of the spacesV are as follows: ⁴ (spaceT*₂), ⁴ of ³ T¹ (spaceT ₂), ⁴ (spaceT*₃), ³ (submanifolds of transitivity in spaceT ₃).In two cases (spacesT ₂ andT ₃) we have obtained metrics free of singularities.     

Affine Orbifolds and Rational Conformal Field Theory Extensions of W 1+∞

Chiral orbifold models are defined as gauge field theories with a finite gauge group Γ. We start with a conformal current algebra associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group Γ of inner automorphisms or (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra of local observables invariant under Γ. A set of positive energy modules is constructed whose characters span, under some assumptions on Γ, a finite dimensional unitary representation of . We compute their asymptotic dimensions (thus singling out the nontrivial orbifold modules) and find explicit formulae for the modular transformations and hence, for the fusion rules. As an application we construct a family of rational conformal field theory (RCFT) extensions of W _1+∞ that appear to provide a bridge between two approaches to the quantum Hall effect. Received: 5 December 1996 / Accepted: 1 April 1997     

Embedding of Dynamical Symmetry Groups U(1, 1) and U(2) of a Free Particle on AdS 2 and S 2 into Parasupersymmetry Algebra

Using two different types of the laddering equations realized simultaneously by the associated Gegenbauer functions, we show that all quantum states corresponding to the motion of a free particle on AdS ₂ and S ² are splitted into infinite direct sums of infinite-and finite-dimensional Hilbert subspaces which represent Lie algebras u(1, 1) and u(2) with infinite- and finite-fold degeneracies, respectively. In addition, it is shown that the representation bases of Lie algebras with rank 1, i.e., gl(2, C), realize the representation of nonunitary parasupersymmetry algebra of arbitrary order. The realization of the representation of parasupersymmetry algebra by the Hilbert subspaces which describe the motion of a free particle on AdS ₂ and S ² with the dynamical symmetry groups U(1, 1) and U(2) are concluded as well.     

Operator Algebras and Poisson Manifolds¶Associated to Groupoids

It is well known that a measured groupoid G defines a von Neumann algebra W ^*(G), and that a Lie groupoid G canonically defines both a C ^*-algebra C ^*(G) and a Poisson manifold A ^*(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C ^*-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects. Subsequently, we show that the maps G↦W ^*(G), G↦C ^*(G), and G↦A ^*(G) are functorial between the categories in question. It follows that these maps preserve Morita equivalence. Received: 6 December 2000 / Accepted: 19 April 2001     

Drinfeld-Sokolov gravity

A lagrangian euclidean model of Drinfeld-Sokolov (DS) reduction leading to generalW-algebras on a Riemann surface of any genus is presented. The background geometry is given by the DS principal bundleK associated to a complex Lie groupG and anSL(2,) subgroupS. The basic fields are a hermitian fiber metricH ofK and a (0, 1) Koszul gauge fieldA ^* ofK valued in a certain negative graded subalgebrar ofg related tos. The action governing theH andA ^* dynamics is the effective action of a DS field theory in the geometric background specified byH andA ^*. Quantization ofH andA ^* implements on one hand the DS reduction and on the other defines a novel model of 2d gravity, DS gravity. The gauge fixing of the DS gauge symmetry yields an integration on a moduli space of DS gauge equivalence classes ofA ^* configurations, the DS moduli space. The model has a residual gauge symmetry associated to the DS gauge transformations leaving a given fieldA ^* invariant. This is the DS counterpart of conformal symmetry. Conformal invariance and certain non-perturbative features of the model are discussed in detail.     

Solvability of the Hamiltonians Related to Exceptional Root Spaces: Rational Case

Solvability of the rational quantum integrable systems related to exceptional root spaces G₂,F₄ is re-examined and for E_6,7,8 is established in the framework of a unified approach. It is shown that Hamiltonians take algebraic form being written in certain Weyl-invariant variables. It is demonstrated that for each Hamiltonian the finite-dimensional invariant subspaces are made from polynomials and they form an infinite flag. A notion of minimal flag is introduced and minimal flag for each Hamiltonian is found. Corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for arbitrary values of the coupling constants. The Hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite-dimensional but finitely-generated Lie algebra of differential operators, taken in a finite-dimensional representation.Alexander V. Turbiner: On leave of absence from the Institute for Theoretical and Experimental Physics, Moscow 117259, Russia.     

Braided m-Lie Algebras

Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of End _F M, where M is a Yetter–Drinfeld module over B with dimB < . In particular, generalized classical braided m-Lie algebras sl _{q, f} (GM _G (A), F) and osp _{q, t} (GM _G (A), M, F) of generalized matrix algebra GM _G (A) are constructed and their connection with special generalized matrix Lie superalgebra sl _{s, f} (GM _{Z_2}(A ^s ), F) and orthosymplectic generalized matrix Lie super algebra osp _{s, t} (GM _{Z_2}(A ^s ), M ^s , F) are established. The relationship between representations of braided m-Lie algebras and their associated algebras are established.This revised version was published online in March 2005 with corrections to the cover date.     

Does the G·G*syn DNA mismatch containing canonical and rare tautomers of the guanine tautomerise through the DPT? A QM/QTAIM microstructural study

We have established that the asynchronous concerted double proton transfer (DPT), moving with a time gap and without stable intermediates, is the underlying mechanism for the tautomerisation of the G·G^*_syn DNA base mispair (C₁ symmetry), formed by the keto and enol tautomers of the guanine in the anti- and syn-configurations, into the G^*·G^*_syn base mispair (C₁), formed by the enol and imino tautomers of the G base, using quantum-mechanical calculations and Bader's quantum theory of atoms in molecules. By constructing the sweeps of the geometric, electron-topological, energetic, polar and natural bond orbital properties along the intrinsic reaction coordinate of the G·G^*_syn?G^*·G^*_syn DPT tautomerisation, the nine key points, that are critical for the atomistic understanding of the tautomerisation reaction, were set and comprehensively analysed. It was found that the G·G^*_syn mismatch possesses pairing scheme with the formation of the O6···HO6 (7.01) and N1H···N7 (6.77) H-bonds, whereas the G^*·G^*_syn mismatch – of the O6H···O6 (10.68) and N1···HN7 (9.59 kcal mol^?1) H-bonds. Our results highlight that these H-bonds are significantly cooperative and mutually reinforce each other in both mismatches. The deformation energy necessary to apply for the G·G^*_syn base mispair to acquire the Watson–Crick sizes has been calculated. We have shown that the thermodynamically stable G^*·G^*_syn base mispair is dynamically unstable structure with a lifetime of 4.1 × 10^?15 s and any of its six low-lying intermolecular vibrations can develop during this period of time. These data exclude the possibility to change the tautomeric status of the bases under the dissociation of the G·G^*_syn mispair into the monomers during DNA replication. Finally, it has been made an attempt to draw from the physico-chemical properties of all four incorrect purine–purine DNA base pairs a general conclusion, which claims the role of the transversions in spontaneous point mutagenesis.     

On Modules over Matrix Quantum Pseudo-Differential Operators

We classify all the quasifinite highest-weight modules over the central extension of the Lie algebra of matrix quantum pseudo-differential operators, and obtain them in terms of representation theory of the Lie algebra (, R _m ) of infinite matrices with only finitely many nonzero diagonals over the algebra R _m = [t]/(t ^m+1). We also classify the unitary ones.