sh-Lie Algebras Induced by Gauge Transformations

Traditionally symmetries of field theories are encoded via Lie group actions, or more generally, as Lie algebra actions. A significant generalization is required when “gauge parameters” act in a field dependent way. Such symmetries appear in several field theories, most notably in a “Poisson induced” class due to Schaller and Strobl [SS94] and to Ikeda [Ike94], and employed by Cattaneo and Felder [CF99] to implement Kontsevich's deformation quantization [Kon97]. Consideration of “particles of spin > 2” led Berends, Burgers and van Dam [Bur85,BBvD84,BBvD85] to study “field dependent parameters” in a setting permitting an analysis in terms of smooth functions. Having recognized the resulting structure as that of an sh-Lie algebra (L _∞-algebra), we have now formulated such structures entirely algebraically and applied it to a more general class of theories with field dependent symmetries. Received: 14 December 2000 / Accepted: 8 February 2002?Published online: 2 October 2002     

Strongly Homotopy Lie Bialgebras and Lie Quasi-bialgebras

Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer–Cartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of Kosmann–Schwarzbach. This approach provides a definition of an L _∞-(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an L _∞-algebra structure as a particular case of our construction. The formal geometry interpretation leads to a definition of an L _∞ (quasi)bialgebra structure on V as a differential operator Q on V, self-commuting with respect to the big bracket. Finally, we establish an L _∞-version of a Manin (quasi) triple and get a correspondence theorem with L _∞-(quasi)bialgebras. This paper is dedicated to Jean-Louis Loday on the occasion of his 60th birthday with admiration and gratitude.     

On the Spectrum of the Generator of an Infinite System of Interacting Diffusions

We study the spectrum of the operator

Cohomologies and peripheric chains

The deformation properties of Lie subalgebrasLεg that are carriers for twistsF : U(g)→UF(g) are tightly connected with the possibility to apply additional twisting deformationsFR andFJ┴ toUF(g) and quantize new sets ofγ-matrices. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. This work has been partially supported by the Russian Foundation for Fundamental Research under the grant N 00-01-00500.     

Quantum deformation of lorentz group

A one parameter quantum deformationS _μ L(2,ℂ) ofSL(2,ℂ) is introduced and investigated. An analog of the Iwasawa decomposition is proved. The compact part of this decomposition coincides withS _μ U(2), whereas the solvable part is identified as a Pontryagin dual ofS _μ U(2). It shows thatS _μ L(2,ℂ) is the result of the dual version of Drinfeld's double group construction applied toS _μ U(2). The same construction applied to any compact quantum groupG _c is discussed in detail. In particular the explicit formulae for the Haar measures on the Pontryagin dualG _d ofG _c and on the double groupG are given. We show that there exists remarkable 1-1 correspondence between representations ofG and bicovariant bimodules (“tensor bundles”) overG _c . The theory of smooth representations ofS _μ L(2,ℂ) is the same as that ofSL(2,ℂ) (Clebsh-Gordon coefficients are however modified). The corresponding “tame” bicovariant bimodules onS _μ U(2) are classified. An application to 4D ₊ differential calculus is presented. The nonsmooth case is also discussed.     

A Characteristic Decay Semigroup for the Resonances of Trace Class Perturbations with Analyticity Conditions of Semibounded Hamiltonians

To asymptotic complete scattering systems {M ₊+V,M ₊} on H₊:=L²(R₊,K{\mathcal{H}}_{+}:=L^{2}(\mathbf{R}_{+},{\mathcal{K}}, d λ), where M ₊ is the multiplication operator on H₊{\mathcal{H}}_{+} and V is a trace class operator with analyticity conditions, a decay semigroup is associated such that the spectrum of the generator of this semigroup coincides with the set of all resonances (poles of the analytic continuation of the scattering matrix into the lower half plane across the positive half line), i.e. the decay semigroup yields a “time-dependent” characterization of the resonances. As a counterpart a “spectral characterization” is mentioned which is due to the “eigenvalue-like” properties of resonances.     

Experimental investigation of ligand effects on the conversion electron spectrum of the 22.5 keV M1 + E2 nuclear transition in <Superscript>149</Superscript>Sm

The conversion electron spectrum of the 22.5keV M1 + E2 nuclear transition in ¹⁴⁹Sm from the electron capture decay of ¹⁴⁹Eu was experimentally studied for the “Eu₂O₃” and “EuF₃” compounds in which ¹⁴⁹Eu ions have the same assumed oxidation number +3 . While the energies of the L, M, N, O, and P₁ conversion lines for “EuF₃” were lower, on average, by 1.7(1), 2.7(2), 2.3(3), 4.1(2), and 5.7(9)eV, respectively, than those for “Eu₂O₃”, no changes between the two ligand complexes were observed for relative intensities of the conversion lines and their natural widths within the error limits.     

Experimental study of the energy gap in the spin-wave spectrum in Co/Pd multilayer films

The energy gap appearing in the spin-wave spectrum as a result of Bragg scattering by the modulation period q=2π/(d ₁+d ₂) of a one-dimensional superlattice is observed by the method of spin-wave resonance in Co/Pd multilayer films. It is shown that this gap is asymmetric: The “positive” deviation is from two to three times greater than the “negative” deviation. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 12, 938–941 (25 June 1996)     

Dynamicalr-matrices on the affinizations of arbitrary self-dual Lie algebras

We associate a dynamicalr-matrix with any such subalgebraL of a finite dimensional self-dual Lie algebraA for which the scalar product ofA remains nondegenerate onL and there exists a nonempty open subsetĽ ⊂L so that the restriction of (ad λ)εEnd(A) toL is invertible ∨λεĽ. Thisr-matrix is also well-defined ifL is the grade zero subalgebra of an affine Lie algebraA obtained from a twisted loop algebra based on a finite dimensional self-dual Lie algebraG. Application of evaluation homomorphisms to the twisted loop algebras yields spectral parameter dependentG ⊗G-valued dynamicalr-matrices that are generalizations of Felder’s ellipticr-matrices. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. This work was supported in part by the Hungarian National Science Fund (OTKA) under T034170.     

Features of the melting dynamics of a vortex lattice in a high-T c superconductor in the presence of pinning centers

The phase transition “triangular lattice-vortex liquid” in layered high-T _c superconductors in the presence of pinning centers is studied. A two-dimensional system of vortices simulating the superconducting layers in a high-T _c Shubnikov phase is calculated by the Monte Carlo method. It was found that in the presence of defects the melting of the vortex lattice proceeds in two stages: First, the ideal triangular lattice transforms at low temperature (≃3 K)into islands which are pinned to the pinning centers and rotate around them and then, at a higher temperature (≃8 K for T _c 584 K), the boundaries of the “islands” become smeared and the system transforms into a vortex liquid. As the pinning force increases, the temperatures of both phase transitions shift: The temperature of the point “triangular lattice-rotating lattice” decreases slightly (to ≃2 K)and the temperature of the phase transition “rotating lattice-vortex liquid” increases substantially (≃70 K). Pis’ma Zh. éksp. Teor. Fiz. 66, No. 4, 269–274 (25 August 1997)     

On the Distribution of Surface Extrema in Several One- and Two-dimensional Random Landscapes

We study here a standard next-nearest-neighbor (NNN) model of ballistic growth on one-and two-dimensional substrates focusing our analysis on the probability distribution function P(M,L) of the number M of maximal points (i.e., local “peaks”) of growing surfaces. Our analysis is based on two central results: (i) the proof (presented here) of the fact that uniform one-dimensional ballistic growth process in the steady state can be mapped onto “rise-and-descent” sequences in the ensemble of random permutation matrices; and (ii) the fact, established in Ref. [G. Oshanin and R. Voituriez, J. Phys. A: Math. Gen. 37:6221 (2004)], that different characteristics of “rise-and-descent” patterns in random permutations can be interpreted in terms of a certain continuous-space Hammersley-type process. For one-dimensional system we compute P(M,L) exactly and also present explicit results for the correlation function characterizing the enveloping surface. For surfaces grown on 2d substrates, we pursue similar approach considering the ensemble of permutation matrices with long-ranged correlations. Determining exactly the first three cumulants of the corresponding distribution function, we define it in the scaling limit using an expansion in the Edgeworth series, and show that it converges to a Gaussian function as L → ∞.     

A global version of the quantum duality principle

The “quantum duality principle” states that a quantisation of a Lie bialgebra provides also a quantisation of the dual formal Poisson group and, conversely, a quantisation of a formal Poisson group yields a quantisation of the dual Lie bialgebra as well. We extend this to a much more general result: namely, for any principal ideal domainR and for each primepεR we establish an “inner” Galois’ correspondence on the categoryHA of torsionless Hopf algebras overR, using two functors (fromHA to itself) such that the image of the first and the second is the full subcategory of those Hopf algebras which are commutative and cocommutative, modulop, respectively (i.e., they are“quantum function algebras” (=QFA) and“quantum universal enveloping algebras” (=QUEA), atp, respectively). In particular we provide a machine to get two quantum groups — a QFA and a QUEA — out of any Hopf algebraH over a fieldk: apply the functors tok[ν] ⊗k H forp=ν. A relevant example occurring in quantum electro-dynamics is studied in some detail. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001     

Effect of free electrons on growth of “cancers” in mixed electronic-ionic conductors

The reasons for the anomalously rapid (for solids) growth of “cancers” (filamentary crystals, hills, pores, and other formations) on the surface of samples of superionic conductors with mixed electronic-ionic conductivity (of the type Cu_2−x Se and Ag₂Te) are discussed. The effects are attributed to the acceleration of mobile ions and activated ions in the “core” of the superionic by free electrons in the joint “chemical” diffusion of ions and electrons in the samples. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 2, 97–100 (25 July 1996)     

The sh Lie Structure of Poisson Brackets in Field Theory

A general construction of an sh Lie algebra (L _∞-algebra) from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel'fand, Dickey and Dorfman. In this way, higher order maps are constructed which combine to form an sh Lie algebra on the graded differential algebra of horizontal forms. The same construction applies for graded brackets in field theory such as the Batalin-Fradkin-Vilkovisky bracket of the Hamiltonian BRST theory or the Batalin-Vilkovisky antibracket. Received: 5 March 1997 / Accepted: 21 May 1997     

Heat conductivity and the Lorentz number of the Sm1−x GdxS “black” phase

Measurement of the heat conductivity and electrical resistivity of two Sm_1−x Gd_xS compositions with x=0.1 and 0.14 is reported within the 80–300 K interval. An analysis of experimental data on the electronic component of heat conductivity permits a conclusion that the d subband of “heavy” carriers in the conduction band of these materials lies above the s “light”-carrier subband. Fiz. Tverd. Tela (St. Petersburg) 41, 26–29 (January 1999)     

Paraboson coherent states

It is known that the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) are equivalent to the defining (triple) relations of n pairs of paraboson operators b _i ^±. In particular, the “parabosons of order p” correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of osp(1|2n). Recently we constructed these representations V(p) giving the explicit actions of the osp(1|2n) generators. We apply these results for the n = 2 case in order to obtain “coherent state” representations of the paraboson operators.     

Direct observation of phase separation in La1−x CaxMnO3

In a study of the infrared optical absorption spectra of polycrystalline La_1−x Ca_xMnO₃ (0.1⩽x⩽0.8), a contribution from free charge carriers is observed below the Curie temperature for compositions with x⩽0.4. The existence of this contribution for samples with a high resistivity is a direct proof of the existence of “metallic” drops in an insulating matrix. The relative volume of the “metallic” phase is estimated. Pis’ma Zh. éksp. Teor. Fiz. 68, No. 1, 89–92 (10 July 1998)     

Embedding euclidean lie algebras into quantum structures

We show that it is possible to express the basis elements of the Lie algebra of the Euclidean group,E(2), as simple irrational functions of certainq deformed expressions involving the generators of the quantum algebraU _q (so(2, 1)). We consider implications of these results for the representation theory of the Lie algebra ofE(2). We briefly discess analogous results forU _q (so(2, 2)). Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

Correlations of polarizations and entangled states in the two-photon system

The correlations of the linear and circular polarizations in the system of two photons have been theoretically investigated. The polarization of a two-photon state is described by the one-photon Stokes parameters and by the components of the correlation “tensor” in the Stokes space. It is shown that in the case of two-photon decays π⁰ → 2γ, η → 2γ, K _L ⁰ → 2γ, K _S ⁰ → 2γ and the cascade process |0〉 → |1〉 + γ → |0〉 + 2γ(|0〉 and |1〉 are states with the spin 0 and 1, respectively) the final two-photon state represents a characteristic example of the entangled (nonfactorizable) state, and the correlations between the Stokes parameters in all these decays have the purely quantum character: the incoherence inequalities of the Bell type for the components of the correlation “tensor”, established previously for the case of classical “mixtures”, are violated. The general analysis of the registration procedure for two correlated photons by two one-photon detectors is performed.