Search

We report the measurement of the helicity asymmetry $E$ for the $p\pi^0$ and $n\pi^+$ final states using, for the first time, an elliptically polarized photon beam in combination with a longitudinally polarized target at the Crystal Ball experiment at MAMI. The results agree very well with data that were taken with a circularly polarized photon beam, showing that it is possible to simultaneously measure polarization observables that require linearly (e.g.~$G$) and circularly polarized photons (e.g.~$E$) and a longitudinally polarized target. The new data cover a photon energy range 270 - 1400 MeV for the $p\pi^0$ final state (230 - 842 MeV for the $n\pi^+$ final state) and the full range of pion polar angles, $\theta$, providing the most precise measurement of the observable $E$. A moment analysis gives a clear observation of the $p\eta$ cusp in the $p\pi^0$ final state.

A new data set for the helicity-dependent differential cross section of the single-meson photoproduction reaction $\gamma p \to p \pi^{0}$ was obtained for the photon energy interval 150-400 MeV. The experiment was performed at the A2 tagged photon facility of the Mainz Microtron MAMI using a circularly polarized photon beam and a longitudinally polarized proton target. The reaction products were detected with the large acceptance Crystal Ball/TAPS calorimeter covering 97\% of the full solid angle. These new results, obtained with a fine energy and polar angle binning, greatly increase both the existing quantity and quality of the data available for this observable. A moment analysis, based on a finite expansion in Legendre polynomials, was applied to these data by using a bootstrap-based fitting method to correctly account for their systematic uncertainties. From the resulting decomposition of the differential cross sections, the $E2/M1$ ratio for the $N\to \Delta(1232)$ transition was determined to be $[-2.38 \pm 0.16{\hbox{ (stat.+sys.)}} \pm 0.10 {\hbox{ (model)}}] \%$. Combining this value with previous results also allowed us to evaluate the most precise available estimate of the $E2/M1$ ratio to be used for all further reference and model comparisons., Comment: 16 pages, 14 figures

Photoproduction of $\pi^0\pi^{\pm}$-pairs from quasifree nucleons bound in the deuteron has been investigated to study the helicity dependence of this reaction. Measurements with a liquid deuterium target were used to extract the unpolarized cross sections for reactions on protons and neutrons. A deuterated, longitudinally polarized solid-butanol target, together with a circularly polarized photon beam, determined the double polarization observable $E$. From these results the spin-dependent cross sections $\sigma_{1/2}$ and $\sigma_{3/2}$, corresponding to the anti-parallel and parallel spin configurations of the beam photon and target nucleon, have been derived. The measurements were performed at the Mainz MAMI accelerator with tagged, circularly-polarized photon beams produced via bremsstrahlung from longitudinally polarized electron beams. The reaction products were detected with an almost $4\pi$ solid-angle covering calorimeter composed of the Crystal Ball and TAPS detectors, supplemented by plastic scintillation detectors for charged particle identification. The results are sensitive to sequential decays of nucleon resonances via intermediate states and also to the decay of nucleon resonances by emission of charged $\rho$ mesons, and are compared to recent model results., Comment: 9 pages, 6 figures

We report a first measurement of the double-polarisation observable, $C_{x'}$, in $\pi^+$ photoproduction off the proton. The $C_{x'}$ double-polarisation observable represents the transfer of polarisation from a circularly polarised photon beam to the recoiling neutron. The MAMI circularly polarised photon beam impinged on a liquid deuterium target cell, with reaction products detected in the Crystal Ball calorimeter. Ancillary apparatus surrounding the target provided tracking, particle identification and determination of recoil nucleon polarisation. The $C_{x'}$ observable is determined for photon energies 800-1400 MeV, providing new constraints on models aiming to elucidate the spectrum and properties of nucleon resonances. This is the first determination of any polarisation observable from the beam-recoil group of observables for this reaction, providing a valuable constraint and systematic check of the current solutions of partial wave analysis based theoretical models.

A first measurement of the polarisation transfer from a circularly-polarised photon to the final state neutron ($C^n_{x'}$) in deuterium photodisintegration has been carried out. This quantity is determined over the photon energy range 370~--~700~MeV and for neutron centre-of-mass breakup angles $\sim45-120^{\circ}$. The polarisation of the final state neutrons was determined by an ancillary large-acceptance nucleon polarimeter, surrounding a cryogenic liquid deuterium target within the Crystal Ball detector at MAMI. The polarimeter characterised $(n,p)$ charge exchange of the ejected neutrons to determine their polarisation. The new $C^n_{x'}$ data are also compared to a theoretical model based on nucleonic and nucleon resonance degrees of freedom constrained by the current world-database of deuterium photodisintegration measurements. Structures in $C^n_{x'}$ observed in the region of the $d^*(2380)$ could not be explained by conventional models of deuteron photodisintegration.

The helicity-dependent single $\pi^{0}$ photoproduction cross section on the deuteron and the angular dependence of the double polarisation observable $E$ for the quasi-free single $\pi^0$ production off the proton and the neutron have been measured for the first time from the threshold region up to the photon energy 1.4 GeV. The experiment was performed at the tagged photon facility of the MAMI accelerator and used a circularly polarised photon beam and longitudinally polarised deuteron target. The reaction products were detected using the large acceptance Crystal Ball/TAPS calorimeter, which covered 97% of the full solid angle. Comparing the cross section from the deuteron with the sum of free nucleon cross sections provides a quantitative estimate of the effects of the nuclear medium on pion production. In contrast, comparison of $E$ helicity asymmetry data from quasi-free protons off deuterium with data from a free proton target indicates that nuclear effects do not significantly affect this observable. As a consequence, it is deduced that the helicity asymmetry $E$ on a free neutron can be reliably extracted from measurements on a deuteron in quasi-free kinematics., Comment: 22 pages, 22 figures; version accepted for publication on Eur. Phys. J. A

A precise measurement of the differential cross-sections $d\sigma/d\Omega$ and the linearly polarized photon beam asymmetry $\Sigma_3$ for Compton scattering on the proton below pion threshold has been performed with a tagged photon beam and almost $4\pi$ detector at the Mainz Microtron. The incident photons were produced by the recently upgraded Glasgow-Mainz photon tagging facility and impinged on a cryogenic liquid hydrogen target, with the scattered photons detected in the Crystal Ball/TAPS set-up. Using the highest statistics Compton scattering data ever measured on the proton along with two effective field theories (both covariant baryon and heavy-baryon) and one fixed-$t$ dispersion relation model, constraining the fits with the Baldin sum rule, we have obtained the proton electric and magnetic polarizabilities with unprecedented precision: \begin{align*} &{}\alpha_{E1} = 10.99 \pm 0.16 \pm 0.47 \pm 0.17 \pm 0.34 &{}\beta_{M1} = 3.14 \pm 0.21 \pm 0.24 \pm 0.20 \pm 0.35 \end{align*} in units of $10^{-4}$\,fm$^3$ where the errors are statistical, systematic, spin polarizability dependent and model dependent.

The double-polarization observable $E$ and helicity-dependent cross sections $\sigma_{1/2}$, $\sigma_{3/2}$ have been measured for the photoproduction of $\pi^0$ pairs off quasi-free protons and neutrons at the Mainz MAMI accelerator with the Crystal Ball/TAPS setup. A circularly polarized photon beam was produced by bremsstrahlung from longitudinally polarized electrons and impinged on a longitudinally polarized deuterated butanol target. The reaction products were detected with an almost $4\pi$ covering calorimeter. The results reveal for the first time the helicity- and isospin-dependent structure of the $\gamma N\rightarrow N\pi^0\pi^0$ reaction. They are compared to predictions from reaction models in view of nucleon resonance contributions and also to a refit of one model that predicted results for the proton and for the neutron target. The comparison of the prediction and the refit demonstrate the large impact of the new data., Comment: Submitted to Phys. Rev. Lett

We report a measurement of the spin polarisation of the recoiling neutron in deuterium photodisintegration, utilising a new large acceptance polarimeter within the Crystal Ball at MAMI. The measured photon energy range of 300~--~700~MeV provides the first measurement of recoil neutron polarisation at photon energies where the quark substructure of the deuteron plays a role, thereby providing important new constraints on photodisintegration mechanisms. A very high neutron polarisation in a narrow structure centred around $E_{\gamma}\sim$~570~MeV is observed, which is inconsistent with current theoretical predictions employing nucleon resonance degrees of freedom. A Legendre polynomial decomposition suggests this behaviour could be related to the excitation of the $d^*(2380)$ hexaquark.

The $\gamma n \to \pi^0 n$ differential cross section evaluated for 27 energy bins span the photon-energy range 290-813 MeV (W = 1.195-1.553 GeV) and the pion c.m. polar production angles, ranging from 18 deg to 162 deg, making use of model-dependent nuclear corrections to extract pi0 production data on the neutron from measurements on the deuteron target. Additionally, the total photoabsorption cross section was measured. The tagged photon beam produced by the 883-MeV electron beam of the Mainz Microtron MAMI was used for the 0-meson production. Our accumulation of 3.6 x 10^6 $\gamma n \to \pi^0 n$ events allowed a detailed study of the reaction dynamics. Our data are in reasonable agreement with previous A2 measurements and extend them to lower energies. The data are compared to predictions of previous SAID, MAID, and BnGa partial-wave analyses and to the latest SAID fit MA19 that included our data. Selected photon decay amplitudes $N^* \to \gamma n$ at the resonance poles are determined for the first time., Comment: 16 pages, 12 figures, 3 tables

This work measured $d\sigma/d\Omega$ for neutral kaon photoproduction reactions from threshold up to a c.m.\ energy of 1855MeV, focussing specifically on the $\gamma p\rightarrow K^0\Sigma^+$, $\gamma n\rightarrow K^0\Lambda$, and $\gamma n\rightarrow K^0 \Sigma^0$ reactions. Our results for $\gamma n\rightarrow K^0 \Sigma^0$ are the first-ever measurements for that reaction. These data will provide insight into the properties of $N^*$ resonances and, in particular, will lead to an improved knowledge about those states that couple only weakly to the $\pi N$ channel. Integrated cross sections were extracted by fitting the differential cross sections for each reaction as a series of Legendre polynomials and our results are compared with prior experimental results and theoretical predictions., Comment: 16 pages, 13 figures, 7 tables

We report the first large-acceptance measurement of the beam-spin asymmetry for deuteron photodisintegration ($\vec{\gamma} d\to pn$) in the photon energy range $400

Photoproduction of mesons off quasi-free nucleons bound in the deuteron allows to study the electromagnetic excitation spectrum of the neutron and the isospin structure of the excitation of nucleon resonances. The database for such reactions is much more sparse than for free proton targets. Single $\pi^0$ photoproduction off quasi-free nucleons from the deuteron was experimentally studied. Nuclear effects were investigated by a comparison of the results for free protons and quasi-free protons and used as a correction for the quasi-free neutron data. The experiment was performed at the tagged photon beam of the Mainz MAMI accelerator for photon energies between 0.45~GeV and 1.4~GeV, using an almost $4\pi$ electromagnetic calorimeter composed of the Crystal Ball and TAPS detectors. A complete kinematic reconstruction of the final state removed the effects of Fermi motion. Reaction model predictions and PWA for $\gamma n\rightarrow n\pi^{0}$, based on fits to data for the other isospin channels, disagreed between themselves and no model provided a good description of the new data. The results demonstrate clearly the importance of a measurement of the fully neutral final state for the isospin decomposition of the cross section. Model refits, for example from the Bonn-Gatchina analysis, show that the new and the previous data for the other three isospin channels can be simultaneously described when the contributions of several partial waves are modified. The results are also relevant for the suppression of the higher resonance bumps in total photoabsorption on nuclei, which are not well understood., Comment: excepted for publication in Phys. Rev. C

The aim of the paper is twofold. First, we introduce analogs of (partial) derivatives on certain Noncommutative algebras, including some enveloping algebras and their "braided counterparts", namely, the so-called modified Reflection Equation algebras. By using these derivatives we construct an analog of the de Rham complex on these algebras. Second, we discuss deformation property of some quantum algebras and show that contrary to a commonly held view, in the so-called q-Witt algebra there is no analog of the PBW theorem. In this connection, we discuss different forms of the Jacobi condition related to quadratic-linear algebras., Comment: Some typos in formulae were corrected and minor chhanges and comments were added

We introduce the notion of a braided algebra and study some examples of these. In particular, R-symmetric and R-skew-symmetric algebras of a linear space V equipped with a skew-invertible Hecke symmetry R are braided algebras. We prove the "mountain property" for the numerators and denominators of their Poincare-Hilbert series (which are always rational functions). Also, we further develop a differential calculus on modified Reflection Equation algebras. Thus, we exhibit a new form of the Leibniz rule for partial derivatives on such algebras related to involutive symmetries R. In particular, we present this rule for the algebra U(gl(m)). The case of the algebra U(gl(2)) and its compact form U(u(2)) (which can be treated as a deformation of the Minkowski space algebra) is considered in detail. On the algebra U(u(2)) we introduce the notion of the quantum radius, which is a deformation of the usual radius, and compute the action of rotationally invariant operators and in particular of the Laplace operator. This enables us to define analogs of the Laplace-Beltrami operators corresponding to certain Schwarzschild-type metrics and to compute their actions on the algebra U(u(2)) and its central extension. Some "physical" consequences of our considerations are presented., Comment: LaTeX file, 24 pages

On any Reflection Equation algebra corresponding to a skew-invertible Hecke symmetry (i.e. a special type solution of the Quantum Yang-Baxter Equation) we define analogs of the partial derivatives. Together with elements of the initial Reflection Equation algebra they generate a "braided analog" of the Weyl algebra. When $q\to 1$, the braided Weyl algebra corresponding to the Quantum Group $U_q(sl(2))$ goes to the Weyl algebra defined on the algebra $\Sym((u(2))$ or that $U(u(2))$ depending on the way of passing to the limit. Thus, we define partial derivatives on the algebra $U(u(2))$, find their "eigenfunctions", and introduce an analog of the Laplace operator on this algebra. Also, we define the "radial part" of this operator, express it in terms of "quantum eigenvalues", and sketch an analog of the de Rham complex on the algebra $U(u(2))$. Eventual applications of our approach are discussed., Comment: LaTex, 18 pages. Accepted in Journal of Geometry and Physics

We define the braided differential algebras which can be interpreted as quantization of the differential operator algebra defined on some algebraic varieties supplied with the action of the group GL(m). The algebra is generated by right invariant or coajoint vector fields. Our main example is gl^*(m) and coadjoint orbits in it. The Heisenberg double on the quantum group Fun_q(GL(m)is a particular case of the suggested construction., Comment: Corrected and extended version. An illustrative example is added into the Section 5. LaTex file, 24 pp. Accepted for publication in "Journal of Geometry and Physics"

In the algebra Sym(gl(m)) we consider Poisson pencils generated by the linear Poisson-Lie bracket {,}_{gl(m)} and that corresponding to the so-called Reflection Equation Algebra. Each bracket of such a pencil has the Poisson center coinciding with that of the bracket {,}_{gl(m)}. Consequently, any bracket from this pencil can be restricted to a generic GL(m)-orbit O in gl(m)*. Quantization of such a restricted bracket can be done in the frameworks of braided affine geometry. In the paper we consider these Poisson structures, their super-analogs as well as their quantum (braided) counterparts. Also, we exhibit some detailed examples.

We introduce some braided varieties -- braided orbits -- by considering quotients of the so-called Reflection Equation Algebras associated with Hecke symmetries (i.e. special type solutions of the quantum Yang-Baxter equation). Such a braided variety is called regular if there exists a projective module on it, which is a counterpart of the cotangent bundle on a generic orbit O in gl(m)* in the framework of the Serre approach. We give a criterium of regularity of a braided orbit in terms of roots of the Cayley-Hamilton identity valid for the generating matrix of the Reflection Equation Algebra in question. By specializing our general construction we get super-orbits in gl(m|n)* and a criterium of their regularity.

We introduce an analog of the Maxwell operator on a q-Minkowski space algebra (treated as a particular case of the so-called Reflection Equation Algebra) and on certain of its quotients. We treat the space of "quantum differential forms" as a projective module in the spirit of the Serre-Swan approach. Also, we use "braided tangent vector fields" which are q-analogs of Poisson vector fields associated to the Lie bracket sl(2).

We discuss the consistency of the axioms which the definition of quantum Lie algebras is usually based on., Comment: Talk, given by D.Gurevich at the Satellite Conference "From Lie algebras to quantum groups", Coimbra (Portugal), Juin 2006. To be published in the Proceedings of the Conference. Some essential typos in macrocommands were corrected

Let R: V x V -> V x V be a Hecke type solution of the quantum Yang-Baxter equation (a Hecke symmetry). Then, the Hilbert-Poincre' series of the associated R-exterior algebra of the space V is a ratio of two polynomials of degree m (numerator) and n (denominator). Assuming R to be skew-invertible, we define a rigid quasitensor category SW(V) of vector spaces, generated by the space V and its dual V*, and compute certain numerical characteristics of its objects. Besides, we introduce a braided bialgebra structure in the modified Reflection Equation Algebra, associated with R, and equip objects of the category SW(V) with an action of this algebra. In the case related to the quantum group U_q(sl(m)), we consider the Poisson counterpart of the modified Reflection Equation Algebra and compute the semiclassical term of the pairing, defined via the categorical (or quantum) trace., Comment: Some misprints were corrected

The classical Cayley-Hamilton identities are generalized to quantum matrix algebras of the GL(m|n) type., Comment: To be published in Algebra i Analiz, vol. 17, 2005. A reference to a grant number is added

To some braiding R of Hecke type (a Hecke symmetry) we put into correspondence an associative algebra called the modified Reflection Equation Algebra (mREA). We construct a series of matrices L_(m), m=1,2,... with entries belonging to mREA such that each of them satisfies a version of the Cayley-Hamilton identity with central coefficients. We also consider some quotients of the mREA which are called the non-commutative orbits. For each of these orbits we construct a large family of projective modules. In this family we introduce an algebraic structure which is close to that of $K^0(\Fl(\C^n))$. The algebraic structure respects an equivalence relation motivated by a "quantum" trace compatible with the initial Hecke symmetry R. For a subclass of non-commutative orbits we compute the spectrum of central elements of the mREA Tr_R(L_(m)^k), k\in {\Bbb N}.

To some Hecke symmetries (i.e. Yang-Baxter braidings of Hecke type) we assign algebras called braided non-commutative spheres. For any such algebra, we introduce and compute a q-analog of the Chern-Connes index. Unlike the standard Chern-Connes index, ours is based on the so-called categorical trace specific for a braided category in which the algebra in question is represented., Comment: Essentially revised version. Latex file, 22 pages, no figures

Noncommutative (NC) sphere is introduced as a quotient of the enveloping algebra of the Lie algebra su(2). Using the Cayley-Hamilton identities we introduce projective modules which are analogues of line bundles on the usual sphere (we call them quantum line bundles) and define a multiplicative structure in their family. Also, we compute a pairing between certain quantum line bundles and finite dimensional representations of the NC sphere in the spirit of the NC index theorem. A new approach to constructing the differential calculus on a NC sphere is suggested. The approach makes use of the projective modules in question and gives rise to a NC de Rham complex being a deformation of the classical one., Comment: LaTeX file, 15 pp, no figures. Some clarifying remarks are added at the beginning of section 2 and into section 5

With any even Hecke symmetry R (that is a Hecke type solution of the Yang-Baxter equation) we associate a quasitensor category. We formulate a condition on R implying that the constructed category is rigid and its commutativity isomorphisms R_{U,V} are natural. We show that this condition leads to rescaling of the initial Hecke symmetry. We suggest a new way of introducing traces as properly normalized categorical morphisms End(V) --> K and deduce the corresponding normalization from categorical dimensions., Comment: Source: Revised version, a more attention is given to the problem of trace definition and its proper normalization in braided categories with Hecke type braidings. Minor corrections in Introduction. LaTex file, all macros included, no figures

Let $k(S^2_q)$ be the "coordinate ring" of a quantum sphere. We introduce the cotangent module on the quantum sphere as a one-sided $k(S^2_q)$-module and show that there is no Yang-Baxter type operator converting it into a $k(S^2_q)$-bimodule which would be a flatly deformed object w.r.t. its classical counterpart. This implies non-flatness of any covariant differential calculus on the quantum sphere making use of the Leibniz rule. Also, we introduce the cotangent and tangent modules on generic quantum orbits and discuss some related problems of "braided geometry"., Comment: 13 pages, Latex

Quantum sphere is introduced as a quotient of the so-called Reflection Equation Algebra. This enables us to construct some line bundles on it by means of the Cayley-Hamilton identity whose a quantum version was discovered in \cite{PS}, \cite{GPS}. A new way to introduce some elements of "braided geometry" on the quantum sphere is discussed., Comment: LaTex file, 15 pages, minor correction in Acknowledgement

As was shown in \cite{GPS} the matrix $L=|| l_i^j||$ whose entries $l_i^j$ are generators of the so-called reflection equation algebra is subject to some polynomial identity looking like the Cayley-Hamilton identity for a numerical matrix. Here a similar statement is presented for a matrix whose entries are generators of a filtered algebra being a "non-commutative analogue" of the reflection equation algebra. In an appropriate limit we get a similar statement for the matrix formed by the generators of the algebra $U(gl(n))$. This property is used to introduce the notion of line bundles over quantum orbits in the spirit of the Serre-Swan approach. The quantum orbits in question are presented explicitly as some quotients of one of the mentioned above algebras both in the quasiclassical case (i.e. that related to the quantum group $U_q(sl(n))$) and a non-quasiclassical one (i.e. that arising from a Hecke symmetry with non-standard Poincar\'e series of the corresponding symmetric and skewsymmetric algebras)., Comment: LaTex file, 21 pages, minor correction in Aknowledgement

To a vector space V equipped with a non-quasiclassical involutary solution of the quantum Yang-Baxter equation and a partition $\lambda$, we associate a vector space $\Vl$ and compute its dimension. The functor $V\mapsto \Vl$ is an analogue of the well-known Schur functor. The category generated by the objects $\Vl$ is called the Schur-Weyl category. We suggest a way to construct some related twisted varieties looking like orbits of semisimple elements in sl(n)^*. We consider in detail a particular case of such "twisted orbits", namely the twisted non-quasiclassical hyperboloid and we define the twisted Casimir operator on it. In this case, we obtain a formula looking like the Weyl formula, and describing the asymptotic behavior of the function $N(\la)=\{\sharp \la_i\leq\la\}$, where $\la_i$ are the eigenvalues of this operator., Comment: Latex file, 35 pages

A new approach is suggested to quantum differential calculus on certain quantum varieties. It consists in replacing quantum de Rham complexes with differentials satisfying Leibniz rule by those which are in a sense close to Koszul complexes from \cite{G1}. We also introduce the tangent space on a quantum hyperboloid equipped with an action on the quantum function space and define the notions of quantum (pseudo)metric and quantum connection (partially defined) on it. All objects are considered from the viewpoint of flatness of quantum deformations. A problem of constructing a flatly deformed quantum gauge theory is discussed as well., Comment: 19 pages, Latex

It is known that symmetric orbits in ${\bf g}^*$ for any simple Lie algebra ${\bf g}$ are equiped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the "canonical" R-matrix. We realize quantization of this Poisson pencil on $\cp$ type orbits (i.e. orbits in $sl(n+1)^*$ whose real compact form is $ CP^n$) by means of q-deformed Verma modules., Comment: 21 pages, LaTeX, no figures

The q-Legendre polynomials can be treated as some special "functions in the quantum double cosets $U(1)\setminus SU_q(2)/U(1)$". They form a family (depending on a parameter $q$) of polynomials in one variable. We get their further generalization by introducing a two parameter family of polynomials. If the former family arises from an algebra which is in a sense "q-commutative", the latter one is related to its noncommutative counterpart. We introduce also a two parameter deformation of the invariant integral on a sphere., Comment: 14 pages Latex

Two types of Poisson pencils connected to classical R-matrices and their quantum counterparts are considered. A representation theory of the quantum algebras related to some symmetric orbits in $sl(n)^*$ is constructed. A twisted version of quantum mechanics is discussed., Comment: 23 pages Latex, to appear in Lie goups, their representations, generalizations and applications, Kluwer

We consider nonquasiclassical solutions to the quantum Yang-Baxter equation and the corresponding quantum cogroups $\Fun(SL(S))$ constructed earlier by one of the authors . We give a criterion of the existence of a dual quasitriangular structure in the algebra $\Fun(SL(S))$ and describe a large class of such objects related to the Temperley-Lieb algebra satisfying this criterion. We show also that this dual quasitriangular structure is in some sense nondegenerate., Comment: 18 pages Latex, to appear in Lie goups, their representations, generalizations and applications, Kluwer

We introduce a representation theory of q-Lie algebras defined earlier in \cite{DG1}, \cite{DG2}, formulated in terms of braided modules. We also discuss other ways to define Lie algebra-like objects related to quantum groups, in particular, those based on the so-called reflection equations. We also investigate the truncated tensor product of braided modules., Comment: 18 pp., Latex

We discuss how properties of Hecke symmetry (i.e., Hecke type R-matrix) influence the algebraic structure of the corresponding Reflection Equation (RE) algebra. Analogues of the Newton relations and Cayley-Hamilton theorem for the matrix of generators of the RE algebra related to a finite rank even Hecke symmetry are derived., Comment: 11 pages, LaTeX file, reference to preprint No. added

When a quantum hyperboloid is realized, as a three - parameter algebra $\ahqc$, in the usual manner, the following problem arises: what is a ``representation theory'' of this algebra? We construct the series of all spin representations of $\ahqc$, and we discuss a braided version of the orbit method, i.e. a correspondence between orbits in $\gggg^*$ and $\gggg$-modules. A braided trace and a braided involution are discussed as well., Comment: 18 pp., Latex

We describe two types of Poisson pencils generated by a linear bracket and a quadratic one arising from a classical R-matrix. A quantization scheme is discussed for each. The quantum algebras are represented as the enveloping algebras of ``generalized Lie algebras''., Comment: 20 pp., Latex

Given a simple Lie algebra $\gggg$, we consider the orbits in $\gggg^*$ which are of R-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-called R-matrix bracket. We call an algebra quantizing the latter bracket a quantum orbit of R-matrix type. We describe some orbits of this type explicitly and we construct a quantization of the whole Poisson pencil on these orbits in a similar way. The notions of q-deformed Lie brackets, braided coadjoint vector fields and tangent vector fields are discussed as well., Comment: 18 pp., Latex