What a classical r-matrix really is

The notion of classical $r$-matrix is re-examined, and a definition suitable to differential (-difference) Lie algebras, -- where the standard definitions are shown to be deficient, -- is proposed, the notion of an ${\mathcal O}$-operator. This notion has all the natural properties one would expect form it, but lacks those which are artifacts of finite-dimensional isomorpisms such as not true in differential generality relation \mbox{End}\, (V) \simeq V^* \otimes V for a vector space $V$. Examples considered include a quadratic Poisson bracket on the dual space to a Lie algebra; generalized symplectic-quadratic models of such brackets (aka Clebsch representations); and Drinfel'd's 2-cocycle interpretation of nondegenate classical $r$-matrices

Jordanian Twist Quantization of D=4 Lorentz and Poincare Algebras and D=3 Contraction Limit

We describe in detail two-parameter nonstandard quantum deformation of D=4 Lorentz algebra $\mathfrak{o}(3,1)$, linked with Jordanian deformation of $\mathfrak{sl} (2;\mathbb{C})$. Using twist quantization technique we obtain the explicit formulae for the deformed coproducts and antipodes. Further extending the considered deformation to the D=4 Poincar\'{e} algebra we obtain a new Hopf-algebraic deformation of four-dimensional relativistic symmetries with dimensionless deformation parameter. Finally, we interpret $\mathfrak{o}(3,1)$ as the D=3 de-Sitter algebra and calculate the contraction limit $R\to\infty$ ($R$ -- de-Sitter radius) providing explicit Hopf algebra structure for the quantum deformation of the D=3 Poincar\'{e} algebra (with masslike deformation parameters), which is the two-parameter light-cone $\kappa$-deformation of the D=3 Poincar\'{e} symmetry.Comment: 13 pages, no figure

Braided algebras and the kappa-deformed oscillators

Recently there were presented several proposals how to formulate the binary relations describing kappa-deformed oscillator algebras. In this paper we shall consider multilinear products of kappa-deformed oscillators consistent with the axioms of braided algebras. In general case the braided triple products are quasi-associative and satisfy the hexagon condition depending on the coassociator $Phi \in A\otimes A\otimes A$. We shall consider only the products of kappa-oscillators consistent with co-associative braided algebra, with Phi =1. We shall consider three explicite examples of binary kappa-deformed oscillator algebra relations and describe briefly their multilinear coassociative extensions satisfying the postulates of braided algebras. The third example, describing kappa-deformed oscillators in group manifold approach to kappa-deformed fourmomenta, is a new result.Comment: v2, 13 pages; Proc. of 2-nd Corfu School on Quantum Gravity and Quantum Geometry, September 2009, Corfu; Gen. Rel. Grav. (2011),special Proceedings issue; version in pres

Once again about quantum deformations of D=4 Lorentz algebra: twistings of q-deformation

This paper together with the previous one (arXiv:hep-th/0604146) presents the detailed description of all quantum deformations of D=4 Lorentz algebra as Hopf algebra in terms of complex and real generators. We describe here in detail two quantum deformations of the D=4 Lorentz algebra o(3,1) obtained by twisting of the standard q-deformation U_{q}(o(3,1)). For the first twisted q-deformation an Abelian twist depending on Cartan generators of o(3,1) is used. The second example of twisting provides a quantum deformation of Cremmer-Gervais type for the Lorentz algebra. For completeness we describe also twisting of the Lorentz algebra by standard Jordanian twist. By twist quantization techniques we obtain for these deformations new explicit formulae for the deformed coproducts and antipodes of the o(3,1)-generators.Comment: 17 page

Compton scattering beyond the impulse approximation

We treat the non-relativistic Compton scattering process in which an incoming photon scatters from an N-electron many-body state to yield an outgoing photon and a recoil electron, without invoking the commonly used frameworks of either the impulse approximation (IA) or the independent particle model (IPM). An expression for the associated triple differential scattering cross section is obtained in terms of Dyson orbitals, which give the overlap amplitudes between the N-electron initial state and the (N-1) electron singly ionized quantum states of the target. We show how in the high energy transfer regime, one can recover from our general formalism the standard IA based formula for the cross section which involves the ground state electron momentum density (EMD) of the initial state. Our formalism will permit the analysis and interpretation of electronic transitions in correlated electron systems via inelastic x-ray scattering (IXS) spectroscopy beyond the constraints of the IA and the IPM.Comment: 7 pages, 1 figur

Theoretical study of the valence ionization energies and electron affinities of linear C2n+1 (n=1–6) clusters

The valence level hole spectral functions of linear C2n+1 (n = 1-6) clusters are calculated by the ab initio third order algebraic diagrammatic construction [ADC(3)] Green function method and the outer-valence Green function (OVGF) method using an extended basis set. The vertical electron affinities of linear C2n+1 (n= 1-6) clusters are also evaluated by the same methods. With an increase of the number of carbon atoms, the KT energy levels become more closely spaced and start to form quasi-continua. The original spectral strength of the main line becomes distributed over several lines of comparable intensity. With an increase of the number of carbon atoms, the one-electron (or even quasi-particle) picture of the ionization breaks down because of the interaction between the initial single hole level and the final two-hole-one-particle levels. The spectral intensity of the first four ionization levels remains fairly constant independent of the number of carbon atoms. The agreement of the affinities of C2n+1 (n = 1-6) with experiment is in general very good. Two anionic states are found to be bound for C9, C11 and C13

Electron impact ionization cross sections of beryllium and beryllium hydrides

We report calculated electron impact ionization cross sections (EICSs) for beryllium (Be) and some of its hydrides from the ionization threshold to 1 keV using the Deutsch-Märk (DM) and the Binary-Encounter-Bethe (BEB) formalisms. The positions of the maxima of the DM and BEB cross sections are very close in each case while the DM cross section values at the maxima are consistently higher. Our calculations for Be are in qualitative agreement with results from earlier calculations (convergent close-coupling, R matrix, distorted-wave and plane-wave Born approximation) in the low energy region. For the various beryllium hydrides, we know of no other available data. The maximum cross section values for the various compounds range from 4.0 × 10-16 to 9.4 × 10-16 cm2 at energies of 44 to 56 eV for the DM cross sections and 3.0 × 10-16 to 5.4 × 10-16 cm2 at energies of 40.5 to 60 eV for the BEB cross sections