Extension of K.M.S. states with respect to some locally compact groups

Given a state ω on the C^*-system { , R×G, α} which is K.M.S. with respect to time on the sub-algebra of of fixed points under the action of some locally compact group G, we study its K.M.S.-extension to the whole algebra. In this work, we study more precisely the cases where G is a class of locally compact abelian groups (such that { , G, α|G} is almost periodic) or a product of these groups with a compact group.     

Theory of Gravitational-Inertial Field of Universe. IV. The Universe and the Microcosm

On basis of principle of discreteness of the space and time the following relations are obtained Λ_oM_oc = 2π?, τ_oE_o = 2π? and c² = 2GM_o/Λ_o giving the values of fundamental elements of length Λ_o ≈ (?G/c³)^1/2, mass M_o ≈ (?c/G)^1/2, time τ_o ≈ (?G/c⁵)^1/2 and energy E_o ≈ (?c⁵/G)^1/2. The geon crown of any critical system and the crown of the Universe must have a thickness equal to the fundamental length Λ_o = 2(π?G/c³)^1/2 = 5.74. 10^?33 cm. Each critical system has its specific (most probable) quantum with an average invariant mass which in the case of the Universe is equal to (2π²?H_u/Gc)^1/3 ≈ 300 m_e where H_u is Hubble's constant. There are all reasons to consider the universal virtual quanta of an invariant mass m_u ≈ 300 m_e as carriers of gravitational, electromagnetic and nuclear fields in the Universe.     

Photoconductivity derived from an exact modelling of the uncompensated one-donor model. II—computer simulation of the steady-state exact model

Abstract

Stationary photoconductivity is treated with the model of Part I, free from any of the usual simplifying approximations of the related equations. This simulation leads to set forth a new concept of characteristic temperature T ₀, at which the donor population is independent of the illumination intensity G. T ₀ separates two intervals of temperature over which G either partially empties (T< T ₀) or fills (T > T ₀) the level. Also T ₀ has, in particular, an influence on the ratio n/p of free electrons and holes. The effective isothermal behaviour of n(G) shows that n(G) ∞ G ^1/2 on the lower side of the G range, and n(G) ∞ G at higher G. Variations of n(T) at constant G also display original, T ₀ dependent, characteristics. Finally, a qualitative comparison is made of the 1D model with the two 1D_ai approximate models described in I, in order to distinguish their most prominent behaviour differences.     

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     

Activation energy for elastic interaction between Frank dislocation loops and glide dislocations in f.c.c. Metals

The activation energyG for the thermally activated passing of glide dislocations close to Frank dislocation loops which do not intersect the slip planes is calculated by means of the linear theory of elasticity as a function of the applied stress, of the distances loop centre-slip plane, and of the orientations and radii of the loops. The influence of anisotropy onG is investigated for some f.c.c. metals, especially for copper. For sufficiently high stresses the dependence ofG on stress is shown to approach theG-stress expression suggested by Seeger in his theory of radiation hardening.On leave fromFaculty of Mathematics and Physics, Charles University, Praha, Ke Karlovu 5, Czechoslovakia.     

The enveloping algebra of a covariant system

In an earlier work, Doplicher, Kastler and Robinson have examined a mathematical structure consisting of a pair (A, G), whereA is aC*-algebra andG is a locally compact automorphism group ofA. We call such a structure a covariant system. The enveloping von Neumann algebraA(A, G) of (A, G) is defined as a *-algebra of operator valued functions (called options) on the space of covariant representations of (A, G). The system (A, G) is canonically embedded in, and in fact generates, the von Neumann algebraA(A, G). Further we show there is a natural one-to-one correspondence between the normal *-representations ofA(A, G) and the proper covariant representations of (A, G). The relation ofA(A, G) to the covarainceC*-algebraC*(A, G) is also examined.     

Group duality and the Kubo-Martin Schwinger condition. II

Let be an invariant state of theC*-system { ,G, } on a locally compact noncommutative groupG. Assume further that is extremal -invariant for an action of an amenable groupH which is -asymptotically abelian and commutes with . Denoting byF _AB,G _AB the corresponding two point functions, we give criteria for the fulfillment of the KMS condition with respect to some one parameter subgroup of the center ofG based on the existence of a closable mapT such thatTF _AB=G _AB for allA,B . Closability is either inL (G),B(G) orC (G), according to clustering properties for . The basic mathematical technique is the duality theory for noncompact, noncommutative locally compact groups.This work is supported in part by the National Science Foundation, Grant MCS 79-03041     

Eigenvalues of graphs with twofold symmetry

There are many cases in which the spectrum of a graph contains the complete spectrum of a smaller graph. The larger (composite) graph and the smaller (component) graph are said to be subspectral. It is shown here that whenever a composite graph G has a twofold symmetry operation which defines two equivalent sets of vertices r and s, it is possible to construct two subspectral components G ₊ and G _-, whose eigenvalues, taken jointly, comprise the full spectrum of G. The following rules are given for constructing the components. (1) Draw the r set of vertices and all the edges connecting the members of the set. Then examine in G the vertices through which r and s are connected (the so-called bridging vertices). (2) If a bridging vertex r ₁ is connected to its symmetry-equivalent partner s ₁, then r ₁ is weighted +1 in G ₊ and -1 in G _-. (3) If r ₁ is connected to a vertex s ₂ which is symmetry-equivalent to a second bridging vertex r ₂ in r, then the weight of the edge between r ₁ and r ₂ in G (+1 if they are connected, zero if they are not) is increased by one unit in G ₊ and decreased by one unit in G _-. The derivation of these rules is shown, and the relationship between the spectrum of G and the spectra of G ₊ and G _- is explained in terms of the symmetry properties of the adjacency matrix of G.     

Yang-Mills Flows on Nearly Kähler Manifolds and G 2-Instantons

We consider Lie(G)-valued G-invariant connections on bundles over spaces ${G/H,\, \mathbb{R}\times G/H\, {\rm and}\, \mathbb{R}^2\times G/H}We give a geometric construction of the ${\mathcal{W}_{1+\infty}}We consider Lie(G)-valued G-invariant connections on bundles over spaces G/H, \mathbbR×G/H and \mathbbR²×G/H{G/H,\, \mathbb{R}\times G/H\, {\rm and}\, \mathbb{R}^2\times G/H}, where G/H is a compact nearly K?hler six-dimensional homogeneous space, and the manifolds \mathbbR×G/H{\mathbb{R}\times G/H} and \mathbbR²×G/H{\mathbb{R}^2\times G/H} carry G ₂- and Spin(7)-structures, respectively. By making a G-invariant ansatz, Yang-Mills theory with torsion on \mathbbR×G/H{\mathbb{R}\times G/H} is reduced to Newtonian mechanics of a particle moving in a plane with a quartic potential. For particular values of the torsion, we find explicit particle trajectories, which obey first-order gradient or hamiltonian flow equations. In two cases, these solutions correspond to anti-self-dual instantons associated with one of two G ₂-structures on \mathbbR×G/H{\mathbb{R}\times G/H}. It is shown that both G ₂-instanton equations can be obtained from a single Spin(7)-instanton equation on \mathbbR²×G/H{\mathbb{R}^2\times G/H}.     

Boundary Scattering,Symmetric Spaces and the Principal Chiral Model on the Half-Line

We investigate integrable boundary conditions (BCs) for the principal chiral model on the half-line, and rational solutions of the boundary Yang-Baxter equation (BYBE). In each case we find a connection with (type I, Riemannian, globally) symmetric spaces G/H: there is a class of integrable BCs in which the boundary field is restricted to lie in a coset of H; these BCs are parametrized by G/H×G/H; there are rational solutions of the BYBE in the defining representations of all classical G parametrized by G/H; and using these we propose boundary S-matrices for the principal chiral model, parametrized by G/H×G/H, which correspond to our boundary conditions.An erratum to this article can be found at     

Bounds on completely convergent Euclidean Feynman graphs

LetG be a Euclidean Feynman graph containingL(G) lines. We prove that ifG has massive propagators and does not contain any divergent subgraphs its value is bounded byK ^L(G). We also prove the infrared analogue of this bound.     

Modular Categories and Orbifold Models

In this paper, we try to answer the following question: given a modular tensor category ? with an action of a compact group G, is it possible to describe in a suitable sense the “quotient” category ?/G? We give a full answer in the case when ?=?ℯ? is the category of vector spaces; in this case, ?ℯ?/G turns out to be the category of representation of Drinfeld's double D(G). This should be considered as the category theory analog of the topological identity {pt}/G=BG. This implies a conjecture of Dijkgraaf, Vafa, E. Verlinde and H. Verlinde regarding so-called orbifold conformal field theories: if ? is a vertex operator algebra which has a unique irreducible module, ? itself, and G is a compact group of automorphisms of ?, and some not too restrictive technical conditions are satisfied, then G is finite, and the category of representations of the algebra of invariants, ? ^G , is equivalent as a tensor category to the category of representations of Drinfeld's double D(G). We also get some partial results in the non-holomorphic case, i.e. when ? has more than one simple module. Received: 27 August 2001 / Accepted: 1 March 2002     

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.     

Fusion of Symmetric D-Branes and Verlinde Rings

We explain how multiplicative bundle gerbes over a compact, connected and simple Lie group G lead to a certain fusion category of equivariant bundle gerbe modules given by pre-quantizable Hamiltonian LG-manifolds arising from Alekseev-Malkin-Meinrenken’s quasi-Hamiltonian G-spaces. The motivation comes from string theory namely, by generalising the notion of D-branes in G to allow subsets of G that are the image of a G-valued moment map we can define a ‘fusion of D-branes’ and a map to the Verlinde ring of the loop group of G which preserves the product structure. The idea is suggested by the theorem of Freed-Hopkins-Teleman. The case where G is not simply connected is studied carefully in terms of equivariant bundle gerbe modules for multiplicative bundle gerbes.     

A duality theorem for the automorphism group of a covariant system

In this paper we examine the covariant representation theory of a covariant system (A, G) introduced by Doplicher, Kastler and Robinson. (A is aC*-algebra andG is a locally compact group of automorphisms ofA.) We define the concept of left tensor product of two covariant representations. Loosely stated, our duality theorem says thatG is canonically isomorphic to the set of bounded operator valued maps on the set of covariant representations of the covariant system (A, G) which preserve direct sums, unitary equivalence and left tensor products. We further show that the enveloping von Neumann algebraA(A, G) of the covariant system (A, G) admits a (not necessarily injective) comultiplicationd such that (A(A, G),d) is a Hopf von Neumann algebra. The intrinsic group of this Hopf von Neumann algebra is canonically isomorphic and (strong operator topology) homeomorphic toG.     

Grafting of maleic anhydride onto polyethylene wax by melt ultrasound and solid co-irradiation

This paper presents a model study to prepare maleic anhydride (MAH)-grafted polyethylene wax (PEW) by using melt ultrasound followed by solid co-irradiation without any initiator or solvent. The MAH graft degree (G) of the product was determined by titration using a solution of organic alkali tetrabutyl ammonium hydroxide in ethanol, which had good accuracy and reproducibility. The experimental results showed that MAH was successfully grafted onto PEW by this method, and the G was 1.57%. The co-irradiation processes showed that G and the graft efficiency (G _E) first increased and then decreased with the increase of MAH concentration, and G and G _E gradually increased with the increase of the total co-irradiation dose in experimental conditions. In addition, a product with a certain G can be obtained by using sole melt ultrasonic treatment.     

Group duality and the Kubo-Martin-Schwinger condition

We consider clusteringG-invariant states of aC*-algebraU endowed with an action of a locally compact abelian groupG. Denoting as usual byF _AB,G _AB, the corresponding two-point functions, we give criteria for the fulfillment of the KMS condition (w.r.t. some one-parameter subgroup ofG) based upon the existence of a closable mapT such thatTF _AB =G _AB for allA,BU. Closability is either inL (G),B(G), orC (G), according to clustering assumptions. Our criteria originate from the combination of duality results for the groupG (phrased in terms of functions systems), with density results for the two-point functions.Supported in part by the National Science Foundation     

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.     

The many levels pairing Hamiltonian for two pairs

We address the problem of two pairs of fermions living on an arbitrary number of single-particle levels of a potential well (mean field) and interacting through a pairing force in the framework of the Richardson equations. The associated solutions are classified in terms of a number v_l, which reduces to the seniority v in the limit of a large pairing strength G and yields the number of pairs not developing a collective behaviour, their energy remaining finite in the G limit. We express analytically, through the moments of the single-particle levels distribution, the collective mode energy and the two critical values G_cr⁺ and G_cr^- of the coupling which can exist on a single-particle level with no pair degeneracy. Notably G_cr⁺ and G_cr^-, when the number of single particle levels goes to infinity, merge into the critical coupling of a one-pair system G_cr (when it exists), which is not envisioned by the Richardson theory. In correspondence of G_cr, the system undergoes a transition from a mean-field- to a pairing-dominated regime. We finally explore the behaviour of the excitation energies, wave functions and pair transfer amplitudes versus G finding out that the former, for G > G_cr^-, come close to the BCS predictions, whereas the latter display a divergence at G_cr, signaling the onset of a long-range off-diagonal order in the system.