Bethe subalgebras in twisted Yangians

We study analogues of the Yangian of the Lie algebra for the other classical Lie algebras and . We call them twisted Yangians. They are coideal subalgebras in the Yangian of and admit homomorphisms onto the universal enveloping algebras U( ) and U( ) respectively. In every twisted Yangian we construct a family of maximal commutative subalgebras parametrized by the regular semisimple elements of the corresponding classical Lie algebra. The images in U( ) and U( ) of these subalgebras are also maximal commutative.     

On the support of the Ashtekar-Lewandowski measure

We show that the Ashtekar-Isham extension of the configuration space of Yang-Mills theories is (topologically and measure-theoretically) the projective limit of a family of finite dimensional spaces associated with arbitrary finite lattices.These results are then used to prove that is contained in a zero measure subset of with respect to the diffeomorphism invariant Ashtekar-Lewandowski measure on . Much as in scalar field theory, this implies that states in the quantum theory associated with this measure can be realized as functions on the extended configuration space .     

Covariance algebras in field theory and statistical mechanics

Starting from aC*-algebra and a locally compact groupT of automorphisms of we construct a covariance algebra with the property that the corresponding *-representations are in one-to-one correspondence with covariant representations of i.e. *-representations of in which the automorphisms are continuously unitarily implemented. We further construct for relativistic field theory an algebra yielding the *-representations of in which the space time translations have their spectrum contained inV. The problem of denumerable occurence of superselection sectors is formulated as a condition on the spectrum of . Finally we consider the covariance algebra built with space translations alone and show its relevance for the discussion of equilibrium states in statistical mechanics, namely we restore in this framework the equivalence of uniqueness of the vacuum, irreducibility and a weak clustering property.On leave of absence from Istituto di Fisica G. Marconi — Roma.     

Level-rank duality of WZW models in conformal field theory

We consider the decomposition of the conformal blocks under the conformal embeddings. The case (â is an affine extension of the abelian subalgebra of the central elements ofgl(lr)) is studied in detal. The reciprocal decompositions of -modules induce a pairing between the spaces of conformal blocks of and Wess-Zumino-Witten models on the Riemann sphere. The completeness of the pairing is shown. Hence it defines aduality between two spaces.Dedicated to Professor Masahisa Adachi on his 60th birthday     

Quantum principal commutative subalgebra in the nilpotent part ofU_q \widehat{sl}_2 and lattice KdV variables

We propose a quantum lattice version of B. Feigin and E. Frenkel's constructions, identifying the KdV differential polynomials with functions on a homogeneous space under the nilpotent part of . We construct an action of the nilpotent part of on their lattice counterparts, and embed the lattice variables in a , coinduced from a quantum version of the principal commutative subalgebra, which is defined using the identification of with its dual algebra.     

N=2 structures on solvable Lie algebras: Thec=9 classification

Let be a finite-dimensional Lie algebra (not necessarily semisimple). It is known that if is self-dual (that is, if it possesses an invariant metric) then it admits anN=1 (affine) Sugawara construction. Under certain additional hypotheses, thisN=1 structure admits anN=2 extension. If this is the case, is said to possess anN=2 structure. It is also known that anN=2 structure on a self-dual Lie algebra is equivalent to a vector space decomposition , where are isotropic Lie subalgebras. In other words,N=2 structures on in one-to-one correspondence with Manin triples . In this paper we exploit this correspondence to obtain a classification of thec=9N=2 structures on solvable Lie algebras. In the process we also give some simple proofs for a variety of Lie algebras. In the process we also give some simple proofs for a variety of Lie algebraic results concerning self-dual Lie algebras admitting symplectic or Kähler structures.     

Regularized,continuum Yang-Mills process and Feynman-Kac functional integral

Giving an ultraviolet regularization and volume cut off we construct a nuclear Riemannian structure on the Hilbert manifold of gauge orbits. This permits us to define a regularized Laplace-Beltrami operator on and an associated global diffusion in governed by . This enables us to define, via a Feynman-Kac integral, a Euclidean, continuum regularized Yang-Mills process corresponding to a suitable regularization (of the kinetic term) of the classical Yang-Mills Lagrangian onT .On leave of absence from Zaragoza University (Spain)Laboratoire associé au CNRS     

Fields,observables and gauge transformations II

We wish to study the construction of charge-carrying fields given the representation of the observable algebra in the sector of states of zero charge. It is shown that the set of those covariant sectors which can be obtained from the vacuum sector by acting with localized automorphisms has the structure of a discrete Abelian group. An algebra of fields can be defined on the Hilbert space of a representation of the observable algebra which contains each of the above sectors exactly once. The dual group of acts as a gauge group on in such a way that is the gauge invariant part of is made up of Bose and Fermi fields and is determined uniquely by the commutation relations between spacelike separated fields.     

Equilibrium states of gravitational systems

We formulate the equilibrium correlation functions for local observables of an assembly of non-relativistic, neutral gravitating fermions in the limit where the number of particles becomes infinite, and in a scaling where the region , to which they are confined, remains fixed. We show that these correlation functions correspond, in the limit concerned, to states on the discrete tensor product , where the are copies of the gauge invariantC*-algebra of the CAR overL ²(R ³). The equilibrium states themselves are then given by , where , is the Gibbs state on for an infinitely extended ideal Fermi gas at density , and where ₀ is the normalised density function that minimises the Thomas-Fermi functional, obtained in [2], governing the equilibrium thermodynamics of the system.     

Relativistic electronic structure of the Pb (001) surface

The surface electronic band structure of the Pb (001) was calculated using the self-consistent, first-principles linear-augmented-plane-wave method and the norm-conserving pseudopotentiai method. In the nonrelativistic case, forbidden gaps appear above and below the Fermi levelin the bulk projected band structure of lead. An occupied surface state at the point and two surface states in a wide forbidden gap above E_F are found. A characteristic feature of the electronic structure of the Pb (001) surface is the absence of a surface state within the forbidden S-P gap in the vicinity of the point. The inclusion of scalar-relativistic effects leads to the merger of several S-P gaps into one wide gap extending throughout the entire Brillouin zone. At the same time, the occupied state at point extends to point and its energy decreases by 2 eV. New, relatively weak surface states in the direction and unoccupied states in the vicinity of the point appear. An unoccupied surface state is found at the bottom of the forbidden gap at point . Including the contribution of the spinorbit pseudopotentiai leads to the appearance of two-spin orbit gas; however, the surface level structure is practically unchanged (except for the disappearance of the unoccupied surface state of P_z-symmetry at point ).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 47–53, October, 1991.     

On a new derivation of the Navier-Stokes equation

Given a net of finite-dimensional real Lie algebras contracting into a Lie algebra, a representation of is constructed explicitly as limit of a net ( _l ) of representations, each _l being a representation of on a complex Hilbert space _l . Conditions are imposed on the net ( _l ) implying that the carrier space of contain a-stable set of vectors which are analytic for all, where is a basis of. As a corollary, the corresponding result for contractions of representations of simply connected finite-dimensional real Lie groups is derived.Supported by the Swiss National Science Foundation     

Slepton and squark production and decay inep collisions at the LHC

We present a detailed study of and and the subsequent decays of , , and at the LHC . We explore the relevant supersymmetry parameter range. We calculate the cross sections as well as the rates for interesting signatures such as the production of one or two leptons on the lepton side.     

Superconformal current algebras and their unitary representations

A natural supersymmetric extension is defined of the current (= affine Kac-Moody Lie) algebra ; it corresponds to a superconformal and chiral invariant 2-dimensional quantum field theory (QFT), and hence appears as an ingredient in superstring models. All unitary irreducible positive energy representations of are constructed. They extend to unitary representations of the semidirect sumS (G) of with the superconformal algebra of Neveu-Schwarz, for , or of Ramond, for =0.On leave of absence from the Institute for Nuclear Research and Nuclear Energy of the Bulgarian Academy of Sciences, BG-1184 Sofia, Bulgaria     

Morse and Melnikov functions for NLS Pde's

The theory of the focusing NLS equation under periodic boundary conditions, together with the Floquet spectral theory of its associated Zakharov-Shabat linear operator , is developed in sufficient detail for later use in studies of perturbations of the NLS equation. Counting lemmas for the non-selfadjoint operator , are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite discD in the complex eigenvalue plane. The radius of the discD is controlled by theH ¹ norm of the potential . For this integrable NLS Hamiltonian system, unstable tori are identified, and Backlund transformations are then used to construct global representations of their stable and unstable manifolds-whiskered tori for the NLS pde.The Floquet discriminant is used to introduce a natural sequence of NLS constants of motion, [ , where _c ^j denotes thej ^th critical point of the Floquet discriminant ()]. A Taylor series expansion of the constants , with explicit representations of the first and second variations, is then used to study neighborhoods of the whiskered tori. In particular, critical tori with hyperbolic structure are identified through the first and second variations of , which themselves are expressed in terms of quadratic products of eigenfunctions of . The second variation permits identification, within the disc D, of important bifurcations in the spectral configurations of the operator . The constant , as the height of the Floquet discriminant over the critical point _c ^j , admits a natural interpretation as a Morse function for NLS isospectral level sets. This Morse interpretation is studied in some detail. It is valid globally for the infinite tail, , which is associated with critical points outside the discD. Within this disc, the interpretation is only valid locally, with the same obstruction to its global validity as to a global ordering of the spectrum. Nevertheless, this local Morse theory, together with the Backlund representations of the whiskered tori, produces extremely clear pictures of the stratification of NLS invariant sets near these whiskered tori-pictures which are useful in the study of perturbations of NLS. Finally, a natural connection is noted between the constants of the integrable theory and Melnikov functions for the theory of perturbations of the NLS equation. This connection generates a simple, but general, representations of the Melnikov functions.Funded in part by AFOSR-90-0161 and by NSF DMS 8922717 A01     

Limiting behavior of asymptotically flat gravitational fields

Couch and Torrence suggest that the vacuum Einstein equations admit a larger class of asymptotically flat solutions than those exhibiting the peeling property. Starting with the assumption that , (d/dr) and (/x ^A ) , wherex ^A (A = 2, 3) are angular coordinates, they show that , where ₁ 2 and ₁<₀; , where ₂ 1 and ₁< ₁; and ₄ and ₃ peel as they would under the stronger peeling conditions. The Winicour-Tamburino energy-momentun and angular momentum integrals for these solutions, in general, diverge. In fact, since Couch and Torrence determine only the radial dependence of the solution, it is not clear that the solutions are well defined. We find that the stronger assumption , (d/dr) , and (/x ^A ) does result in well-defined solutions for which both the energy-momentum and angular momentum intergrals are not only finite but result in the same expressions as are obtained for peeling space-times. This assumption appears to be the minimal assumption that is necessary for investigating outgoing radiation at null infinity.In part based on a dissertation by Stephanie Novak and submitted to Syracuse University in partial fulfillment of the requirement for the Ph.D. degree.     

The microstructure of electrodeposited amorphous or microcrystalline Fe-P alloys

Dependent on is produced through electrodeposition in three modifications: microcrystalline if , amorphous if and in a mixed state containing both components if . The interpretation of results of large and small angle X-ray scattering, of measurements of the coercivity and of the mechanical density shows that the three modifications consist of two phases. The structure of the grain boundary phase in the microcrystalline alloys differs considerably from that of the amorphous phase. This arises from the dependence of the atomic volume on the average phosphorus content . The volume fraction of the crystalline component decreases with increasing up to . But even in alloys revealing X-ray patterns without any peaks coming from crystalline particles small angle X-ray scattering indicates the existence of metal-rich inclusions occupying some percent of the total volume.     

Quantum Riemann surfaces I. The unit disc

We construct a non-commutative ^*-algebra which is a quantum deformation of the algebra of continuous functions on the closed unit disc . is generated by the Toeplitz operators on a suitable Hilbert space of holomorphic functions onU.Supported in part by the National Science Foundation under grant DMS/PHY 88-16214     

Derivations and automorphisms of operator algebras

The theorem that each derivation of aC*-algebra extends to an inner derivation of the weak-operator closure ( )^– of in each faithful representation of is proved in sketch and used to study the automorphism group of in its norm topology. It is proved that the connected component of the identity in this group contains the open ball of radius 2 with centerl and that each automorphism in extends to an inner automorphism of ( )^–.Research conducted with the partial support of the NSF and ONR.     

Modules over\mathfrak{U}_q (\mathfrak{s}\mathfrak{l}_2 )

The restricted quantum universal enveloping algebra decomposes in a canonical way into a direct sum of indecomposable left (or right) ideals. They are useful for determining the direct summands which occur in the tensor product of two simple . The indecomposable finite-dimensional are classified and located in the Auslander-Reiten quiver.     

Symmetry breaking in Landau gauge a comment to a paper by T. Kennedy and C. King

For the non-compact abelian lattice Higgs model in Landau gauge Kennedy and King (Princeton preprint, 1985) showed that the two point function does not decay in the Higgs phase. We generalize their methods to show that for the same range of parameters there are states parametrized by an angle [0, 2) such that and 0$$ " align="middle" border="0"> .