Kévin Guerch

Justin Dekany

J R Dennison
email: jr.dennison@usu.edu

Justin Christensen

Thierry Paulmier
email: thierry.paulmier@onera.fr

Sophie Guillemet-Fritsch

Pascal Lenormand

Cathodoluminescence studies of defects in coated boron nitride

Keywords: aluminum oxide, boron nitride, cathodoluminescence, charging, coating, defects, electron irradiation

Introduction

Technical ceramic materials are subjected to extreme condi tions in plasma environments, such as those encountered by satellites in space [I], devices for plasma vapour deposition [START_REF] Helmersson | Ionized physical vapor deposition (IPVD): a review of technology and applications Thin Solid Films[END_REF], high power applications [START_REF] Prokofiev | Pyrolytical boron nitride as a window material for high power microwave electron devices 8th[END_REF], or nuclear radiation [START_REF] Treadaway | Luminescence and absorption of electron-irradiated common optical glasses, sapphire, and quartz[END_REF]. To survive, these ceramic insulators must have exceptional elec trical and thermal properties. Boron nitride (BN) is used in particular because it combines good electrical insulation and high thermal conductivity [START_REF] Haubner | Boron nitrides-properties, synthesis and applications High Performance Non-Oxide Ceram II[END_REF]. However, partially shielded environments such as those found in spacecraft interiors irra diated by electrons with preferentially higher energies can lead to charge trapping in the ceramic's bulk. With the aim to limit charging of BN under electron irradiation, studies of a coating and a thermal treatment of this substrate were per formed. Specifically, BN subslrates were coated with alumina (Al 2 O 3 ) coatings (using plasma vapour deposition) with some coated BN subsequently subjected to thermal annealing under vacuum.

This study focuses on the nature, density and occupation of the localised defects (traps), which control electrical and optical behaviour in these systems. These traps are generated from physical (e.g. vacancies, interstitial, or grain boundaries) and chemical (e.g. substitutional or interstitial impurities) defects in the crystallographic phase of materials. Trap depths (i.e. the binding energies of trapped charges) depend on the nature of the defects. Simple band theory of disordered materials, based on the energy and spatial distributions of these deep trap (DT) defect states, is used to describe the dark conductivity and radiation induced conductivity (RIC) [START_REF] Fowler | Radiation-induced conductivity in the solid state, and some applications[END_REF] and the electron beam-induced optical emission (cathodoluminescence) spectra.

On one hand, we compare the nature and densities of defects of BN substrates, alumina-coated BN (BN/Al 2 O 3 ), and annealed Al 2 O 3 -coated BN (an-BN/Al 2 O 3 ) through studies of the energies and intensities of spectral features of emitted light. On the other hand, we investigate the significant influence of thermal annealing and extended radiation doses (aging) experienced by the materials on charge trapping processes, trap creation and annihilation, electron release kinetics, and charge mobility. The sample temperature during irradiation tests has a substantial influence on electron transport and optical spectra through the electron transitions in the band gap of an-BN/Al 2 O 3 . These, in turn, are related to the charging/discharge processes and especially to the sample surface deterioration as a function of time and thus of dose.

Experimentation

The ultrahigh vacuum electron emission test (EET) chamber at Utah State University [START_REF] Dennison | Characterization of electrical materials properties related to spacecraft charging Radiation Capabilities for the Europa Jupiter System Missions Instrument Workshop[END_REF][START_REF] Jensen | Modeling the defect density of states of disordered SiO 2 through cathodoluminescence[END_REF] was used to investigate the cathodoluminescence (CL) of the ceramic samples. Figure 1 provides a schematic overview of instrumentation used during the tests described herein. This versatile facility allows the characterization of spacecraft materials in different representative space conditions. Ultrahigh vacuum (~1.10 -6 Pa) is achieved through use of mechanical, turbomolecular and ion pumps. For most tests, samples were mounted on a carousel held at room temperature which has ten sample holder faces and one location for a Faraday cup used to monitor incident beam profiles and fluxes. Low temperature tests from ~100 K up to ~300 K were made using a cryogenic sample holder [START_REF] Dekany | Ultrahigh vacuum cryostat system for extended lowtemperature space environment testing[END_REF], using a closed-cycle He refrigerator installed on a flange of the chamber. A high-energy electron gun (Kimball, Model EGPS-21B) provided an electron beam with incident energies of 5 keV to 30 keV [START_REF] Haubner | Boron nitrides-properties, synthesis and applications High Performance Non-Oxide Ceram II[END_REF][START_REF] Wilson | Electron energy-dependent charging effects of multilayered dielectric materials[END_REF][START_REF] Reimer | Scanning Electron Microscopy: Physics of Image Formation and Microanalysis[END_REF][START_REF] Silly | Luminescence properties of hexagonal boron nitride: cathodoluminescence and photoluminescence spectroscopy measurements[END_REF][START_REF] Zhang | Cathodoluminescence of cubic boron nitride films deposited by chemical vapor deposition[END_REF] and 30 keV ±0.1% during tests). Stable and reproducible incident electron fluxes of 1 nA • cm -2 to 500 nA • cm -2 (1, 10, 30, 100, 500 nA • cm -2 ±3% during tests) with a beam diameter of ~1 cm 2 FWHM were used [START_REF] Wilson | Electron energy-dependent charging effects of multilayered dielectric materials[END_REF].

The visible/near infrared (Vis/NIR) emission from samples under electron irradiation was measured with two cameras and a spectrometer [START_REF] Dekany | Ultrahigh vacuum cryostat system for extended lowtemperature space environment testing[END_REF]. Colours and intensity of glow were captured with a single-lens reflex (SLR) CCD camera (Cannon, EOS Rebel XT DS126071; ~400 nm-700 nm in RGB format, capture time of 1 s/frame, with an average spectral response of ~4 • 10 9 counts/(W cm -2 • sr • µm)). The low intensity of light was monitored with a Vis/NIR image-intensified video camera (Xybion, ISG-780-U-3; ~400 nm-900 nm, 30 frames s -1 , with an average spectral response of ~4 • 10 10 counts/(W cm -2 • sr • µm)). The absolute sensitivity of the cameras were calibrated against National Institute of Standards and Technology (NIST) traceable sources. A UV/Vis fibre optic spectrometer (Stellarnet, 13LK-C-SR; ~350 nm-1080 nm with 0.5 nm or 2 meV resolution) was used to measure the luminescence emission spectra [START_REF] Jensen | Properties of cathodoluminescence for cryogenic applications of SiO 2 -based space observatory optics and coatings[END_REF].

The materials studied are industrial samples (~1 × 1 × 10 mm) of pyrolytic hexagonal boron nitride (BN), alumina-coated boron nitride (BN/Al 2 O 3 ), and annealed alumina-coated boron nitride (an-BN/Al 2 O 3 ). Commercial BN was coated with alumina (~0.3 µm thickness) using an industrial PVD-RF method. The industrial thermal annealing treatment was carried out under vacuum. Surface roughness was equal to ~0.5 µm prior to coating [START_REF] Guerch | Etude des propriétés physiques et électriques de matériaux céramiques utilisés en application spatiale[END_REF]. Before irradiation tests, samples were cleaned with methanol and underwent a 48 h vacuum bakeout at ~375 K and <1.10 -3 Pa in order to eliminate adsorbed water and volatile contaminates. Subsequently, they were installed in EET chamber at low pressure (<1.10 -6 Pa) for a duration higher than 30 hours to allow outgassing prior to irradiation tests. Samples for room temperature tests were mounted on grounded Cu sample holders on the EET sample carousel. Several samples of an-BN/Al 2 O 3 were mounted side-by-side on a similar grounded Cu sample holder attached to the cryostat in order to increase the exposed sample surface area for temperature-dependent tests.

Accurate prediction of luminescence intensity from multilayered materials is difficult, especially for materials with different densities, thicknesses, atomic weights, backscatter coefficients and scattering cross-sections [START_REF] Brillson | Applications of depth-resolved cathodoluminescence spectroscopy[END_REF]. Production of cathodoluminescence results from the deposition of incident electron energy and creation of electron-hole pairs in the material. The energy and depth dependence can be modelled through complex Monte Carlo simulations such as CASINO [14], or can be characterized by a single parameter such as the electron penetration depth or the range [START_REF] Reimer | Scanning Electron Microscopy: Physics of Image Formation and Microanalysis[END_REF] in the continuous slow down approximation (CSDA) [START_REF] Wilson | Approximation of range in materials as a function of incident electron energy[END_REF][START_REF]International Commission on Radiation Units and Measurements 1984 Stopping Powers for Electrons and Positrons ICRU Report[END_REF]. Table 1 compares values of the penetration depth (depth at peak energy loss) from CASINO simulations to range in the CSDA for bulk Al 2 O 3 and BN over the experimental energies. These two measures for energy deposition are consistent with one another, based on a relation developed by Bentabet [18,[START_REF] Bentabet | Range and stopping power energy relationships for 0.5-30 keV electron beams slowing down in solids: analytical model Mod[END_REF]. In a spherical geometric model for a semi-infinite target, Bentabet shows the mean penetration depth ( ) [START_REF] Silly | Luminescence properties of hexagonal boron nitride: cathodoluminescence and photoluminescence spectroscopy measurements[END_REF]. The asymptotic limits of η at high energies were measured for bulk microcrystalline Al 2 O 3 and h-BN to be 0.2 and 0.15, respectively.

( / )( ) η = - - σ σ Z R
A simple model for luminescence production in the Al 2 O 3 coating and BN substrate appropriate for the course data presented here follows from a calculation of the electron range (CSDA), R(E inc ), as a function of incident energy, E inc (figure 2(a)) assuming bulk densities ρ m of 3.45 g • cm -3 for BN and 3.97 g • cm -3 for Al 2 O 3 [START_REF] Wilson | Approximation of range in materials as a function of incident electron energy[END_REF]. The range of hexagonal sp 2 -coordinated BN has been approximated using an estimated effective number of valence electrons N V eff = 2.742.

N V eff is predicted from a simple formula-involving band gap energy (~6 eV for hexagonal BN [START_REF] Silly | Luminescence properties of hexagonal boron nitride: cathodoluminescence and photoluminescence spectroscopy measurements[END_REF][START_REF] Kubota | Deep ultraviolet light-emitting hexagonal boron nitride synthesized at atmospheric pressure[END_REF]) plus the average atomic weight and atomic number-which was determined by fitting NIST range data in the CSDA [START_REF]International Commission on Radiation Units and Measurements 1984 Stopping Powers for Electrons and Positrons ICRU Report[END_REF] and NIST inelastic mean free path data [START_REF] Powell | National Institute of Standards and Technology[END_REF] for a wide range of materials [START_REF] Starley | Predictive formula for electron penetration depth of diverse materials over large energy ranges[END_REF]. Figure 2(a) also shows the energy-dependent dose rate

( ) ( )/[ ( ) ] ρ = D E J E q R E ˙inc i nc inc e
inc m for a 500 nA • cm -2 nonpenetrating beam in the CSDA [START_REF] Jensen | Modeling the defect density of states of disordered SiO 2 through cathodoluminescence[END_REF]. Figure 2(b) shows the mean power density deposited by a beam of area A and range R(E inc )-which is given as

( )/ ( )/ = P E A J E q inc inc
e -in the ~300 nm thick alumina coating and the underlying BN substrate of BN/Al 2 O 3 and an-BN/Al 2 O 3 samples for a 500 nA • cm -2 beam. Note that at the lowest incident energy, the electron beam just penetrates the estimated thickness of the Al 2 O 3 coating. Since electron range increases with beam energy, the dose imparted to the fully penetrated Al 2 O 3 coating (or similarly, to a thin surface layer of BN) decreases with increasing incident energy from ~75% at 5 keV to ~4% at 30 keV (figure 2(b)). Use of the penetration depth for this analysis would shift the power deposition curve to higher energies by about 35%; however, penetration depth and range scale with density, so values in figure 2 are overestimated for less dense coatings and rough surfaces.

Results and discussion

Results for BN and BN/Al 2 O 3

Emission spectra as a function of photon energy and wavelength are shown in figure 3 for an uncoated BN sample irradiated with 5 keV and 30 keV electrons. The red curves represent the emission spectra raw data; the small intensity of luminescence from BN substrates results in a low signal-to-noise ratio (SNR). These spectra were fitted with least-squares methods for up to four Gaussian peaks with peak central energy, width, and amplitude as fitting parameters. The black curves show the sum of the individual fitted peak profiles and component curves are shown below.

The intensities of BN/Al 2 O 3 spectra have consistently even lower SNR than those of BN. Hence, for clarity sake, the composite fit emission spectra of BN/Al 2 O 3 at various incident electron energies are shown in figure 4(b), while only raw data are shown for BN. Since the observed features were broad, such smoothing is acceptable. Gaussian peak profiles were found to produce better fits than Lorentzian or Voight functions, as would be expected for long-lived deep level trap states (see below) with very little lifetime broadening [START_REF] Jensen | Modeling the defect density of states of disordered SiO 2 through cathodoluminescence[END_REF]. Gaussian widths are a reasonable measure of the DT energy density of states widths, since measured widths are much larger than the 2 meV instrumental resolution.

For the BN substrate (figure 3), two broad peaks were observed at ~2.4 eV (~516 nm) and ~3.0 eV (~416 nm). Cathodoluminescence spectral features of BN have been difficult to identify and associate with specific defect structures due to the different band gap energies (~4.5 eV to ~6.5 eV) [START_REF] Silly | Luminescence properties of hexagonal boron nitride: cathodoluminescence and photoluminescence spectroscopy measurements[END_REF] and structures associated with cubic (sp 3 -bonded) c-BN [START_REF] Berzina | Photoluminescence excitation spectroscopy in boron nitride nanotubes compared to microcrystalline h-BN and c-BN[END_REF][START_REF] Zhang | Cathodoluminescence of cubic boron nitride films deposited by chemical vapor deposition[END_REF], hexagonal (sp 2 -bonded) h-BN [START_REF] Silly | Luminescence properties of hexagonal boron nitride: cathodoluminescence and photoluminescence spectroscopy measurements[END_REF][START_REF] Berzina | Photoluminescence excitation spectroscopy in boron nitride nanotubes compared to microcrystalline h-BN and c-BN[END_REF][START_REF] Kim | Photoluminescence of hexagonal boron nitride (h-BN) film[END_REF], boron nitride nanotubes [START_REF] Berzina | Photoluminescence excitation spectroscopy in boron nitride nanotubes compared to microcrystalline h-BN and c-BN[END_REF], and disordered BN such as pyrolytic pBN [START_REF] Berzina | Photoluminescence excitation spectroscopy in boron nitride nanotubes compared to microcrystalline h-BN and c-BN[END_REF]. Broad peaks at 2.48 eV (attributed to multivacancy complexes of B and N vacancies in c-BN) and 3.12 eV (of unknown origin) have been reported in CL spectra of higher purity c-BN close to our observed peaks at ~2.4 eV and ~3.0 eV. However, sharper peaks of similar intensities at 2.81 eV, 3.01 eV and 3.20 eV attributed to c-BN and a broad intense peak at ~3.55 eV attributed to h-BN seen in [START_REF] Zhang | Cathodoluminescence of cubic boron nitride films deposited by chemical vapor deposition[END_REF] were not observed in our spectra.

The intensities of BN/Al 2 O 3 spectra are lower than those of BN. The integrated peak intensities at given incident energies of both of the two observed broad ~2.40 eV (figure 5(a)) and ~2.98 eV (figure 5(b)) peaks for the BN/Al 2 O 3 spectra are consistently ~50% smaller than corresponding peaks for BN. However, the shapes of the ~2.40 eV and ~2.98 eV peaksindeed the shape of the full spectra from 1.5 eV to 4.5 eV-are

Results for an-BN/Al 2 O 3

Emission spectra of an-BN/Al 2 O 3 for several incident electron energies are dramatically different than BN and BN/Al 2 O 3 spectra. Overall, CL intensity from an-BN/Al 2 O 3 is substantially higher, with higher SNR; consequently, the curves shown in figure 4(c) are raw data rather than fitted curves.

First, we consider the broad peaks at ~2.40 eV and ~2.98 eV which we previously attributed to the BN substrate. The CL intensities of these two peaks at 20 keV when the peak intensity is highest are ~4-6 times more intense than the corresponding BN peaks and 8-12 times more intense than those These increased enhancements for more penetrating higher energies suggest that they originate preferentially from deeper BN material. It was suggested in a companion study-which included measurements of Raman spectroscopy, XPS, electrical conductivity, and electron-induced surface charging and relaxation-that the annealing treatment of an-BN/Al 2 O 3 generated a higher density of physical and chemical defects especially near the surface [START_REF] Guerch | Electrical properties of annealed and coated boron nitride under electron-beam irradiation[END_REF]. Thermal annealing could allow either chemical or physical defects inherent in the surface layers to diffuse further into the bulk. An alternate cause of the observed increased emissions from BN could be due to reduced surface charging from the incident electron beam.

The reduced charging could result from increased total electron emission [START_REF] Hoffmann | Measurement methods of electron emission over a full range of sample charging[END_REF], as well as from increased conductivity which increased charge dissipation. Measurements showed the bulk conductivity of an-BN/Al 2 O 3 was 15 times that of BN and 40 times that of unannealed BN/Al 2 O 3 [START_REF] Guerch | Electrical properties of annealed and coated boron nitride under electron-beam irradiation[END_REF]. Negative surface potential of up to 3500 V was measured for BN irradiated with a 11 keV beam; this was reduced to ~1700 V for BN/ Al 2 O 3 and ~200 V for an-BN/Al 2 O 3 [START_REF] Guerch | Electrical properties of annealed and coated boron nitride under electron-beam irradiation[END_REF]. Enhanced negative surface charging reduces the landing energies of incident electrons, thereby reducing the absorbed power and the penetration range and hence reducing the CL intensity [START_REF] Dennison | Characterization of electrical materials properties related to spacecraft charging Radiation Capabilities for the Europa Jupiter System Missions Instrument Workshop[END_REF].

There are some small differences in the shape of these broad peaks at ~2.40 eV and ~2.98 eV. The ~2.40 eV and ~2.98 eV peak energies of an-BN/Al 2 O 3 at incident energies >10 keV are about 1.5% higher than those for BN and BN/Al 2 O 3 spectra and increase modestly at lower energies instead of decreasing as for BN and BN/Al 2 O 3 peaks (figure 5(d)). The widths of these two peaks are similar for all three materials, with some differences below 15 keV (figure 5(f)). The relative density of deeper traps to shallower trapsas reflected in the ratio of peak intensities (I 2.98 /I 2.40 ) (figure 5(d))-are nearly identical for all three samples above 10 keV, although the ratio does decrease for an-BN/Al 2 O 3 below 15 keV. The energy dependences of the normalized integrated intensities exhibit very similar energy dependencies for all three materials (figure 5(c)). Altogether, these differences in spectral shape suggest the added intensity in the broad an-BN/Al 2 O 3 spectra is not from the Al 2 O 3 coating, though we cannot rule out some contributions due to broad CL peaks in similar energy ranges of 3.5 eV-5.0 eV sometimes observed for Al 2 O 3 materials [START_REF] Jensen | Defects density of states model of cathodoluminescent intensity and spectra of disordered SiO 2[END_REF][START_REF] Garcia-Guinea | Luminescence of α-Al2O3 and α-AlOOH natural mixtures[END_REF].

Second, we consider additional sharp features in figure 4(c). These include a strong sharp emission peak at ~1.785 eV with numerous small sharp satellite peaks visible in figure 6, plus a sharp absorption peak at ~2.93 eV. Annealing induces both chemical and structural changes in the Al 2 O 3 , leading to CL which can be clearly attributed to the annealed Al 2 O 3 surface layer. Chemical characterization of an-BN/Al 2 O 3 with Raman [START_REF] Guerch | Electrical properties of annealed and coated boron nitride under electron-beam irradiation[END_REF] and XPS [START_REF] Guerch | Etude des propriétés physiques et électriques de matériaux céramiques utilisés en application spatiale[END_REF] methods found evidence for (Cr 3+ ) impurities and chemical bonds of compounds Al(2p) and O(1s). O vacancies are known to be produced in Al 2 O 3 in a reducing (vacuum) environment [START_REF] Garcia-Guinea | Luminescence of α-Al2O3 and α-AlOOH natural mixtures[END_REF] and by electron irradiation [START_REF] Kristianpoller | Radiation effects in pure and doped Al 2 O 3 crystals[END_REF]; these have been associated with luminescence attributed to one electron (F + emission) or two electron (F emission) trapping processes [START_REF] Garcia-Guinea | Luminescence of α-Al2O3 and α-AlOOH natural mixtures[END_REF].

The most prominent feature attributed to the alumina layer in figure 4(c) is a sharp and intense low energy peak at ~1.785 eV. This peak is called the 'ruby peak' because it is characteristic of ruby (crystalline Al 2 O 3 (sapphire) of the mineral family of corundum with Cr impurities). This peak corresponds to the no-phonon line caused by localized chemical defects (colour centres) related to Cr 3+ impurities [START_REF] Treadaway | Luminescence and absorption of electron-irradiated common optical glasses, sapphire, and quartz[END_REF][START_REF] Guerch | Electrical properties of annealed and coated boron nitride under electron-beam irradiation[END_REF][START_REF] Garcia-Guinea | Luminescence of α-Al2O3 and α-AlOOH natural mixtures[END_REF][START_REF] Powell | Fluorescence studies of energy transfer between single and pair Cr 3+ systems in Al 2 O 3[END_REF][START_REF] Green | Broad-band radioluminescence in electron-irradiated ionic compounds[END_REF]. Very low concentrations of <5 ppm of Cr 3+ can cause intense emissions [START_REF] Jonnard | F + and F centers in α-Al 2 O 3 by electron-induced x-ray emission spectroscopy and cathodoluminescence[END_REF].

Around this intense no-phonon peak at ~1.785 eV, symmetric peaks were observed at ~1.758 eV and ~1.835 eV (figure 6) corresponding to vibrational (phonon-) assisted electron transitions, similar to those seen in previous studies [START_REF] Treadaway | Luminescence and absorption of electron-irradiated common optical glasses, sapphire, and quartz[END_REF][START_REF] Guerch | Electrical properties of annealed and coated boron nitride under electron-beam irradiation[END_REF][START_REF] Powell | Fluorescence studies of energy transfer between single and pair Cr 3+ systems in Al 2 O 3[END_REF]. Indeed six or more smaller satellite peaks are evident in figure 6, evenly spaced at 14 ± 1 meV intervals, at energies both above and below the 1.785 eV peak. This incremental energy is the phonon energy. Additional satellite peaks are observed at ~1.737 eV and 1.853 eV; peaks at these energies have been previously attributed to point defects of Fe 2+ and Ti 4+ in Al 2 O 3 [START_REF] Garcia-Guinea | Luminescence of α-Al2O3 and α-AlOOH natural mixtures[END_REF].

A narrow absorption peak is observed at ~2.93 eV, superimposed on the emission from the ~2.98 eV BN peak. Charge carriers are excited from deep traps (DT) to shallow traps (ST) in the annealed alumina coating by absorption of photons coming from the luminescence of BN substrate. Carriers excited to these ST states can then be thermally excited into conducting states, thereby activating charge transport in the Al 2 O 3 layer. This absorption band is near a band observed at ~2.89 eV from radiation induced defects [START_REF] Springis | Red luminescence of color centres in sapphire[END_REF].

All these chemical defects generating low energy electron transitions are activated by thermal annealing of the alumina coating. Consequently, the annealing process must generate several kinds of relatively low energy localized defects with substantial densities within the Al 2 O 3 band gap. These low energy trap states can potentially act to increase conductivity in the alumina layer in several ways. Absorption of higher energy photons from the BN luminescence could excite electrons from the low energy Al 2 O 3 chemical defect states directly into the CB or into ST states which are then thermally excited into the CB. Further, more mobile charge carriers in the annealed sample would reach recombination centres more easily thereby enhancing electron-hole recombination. This in turn may act to limit charging in the annealed alumina layer as compared to the unannealed alumina layer or the BN surface layer, thereby increasing the incident electrons flux reaching the BN and enhancing BN CL intensities in the an-BN/Al 2 O 3 samples.

The glow intensity of an-BN/Al 2 O 3 after the end of irradiation is shown in figure 7. During this relaxation phase after the beam is turned off, delayed luminescence (phosphorescence) is observed with an exponential decrease in intensity with two discernible decay constants of 0.81 ± 0.01 s and 5.4 ± 0.1 s. This delayed luminescence is characteristic of recombination processes of charges carriers after irradiation that involves detrapping of longer-lifetime trap states at the beginning of the relaxation phase.

Influence of temperature

A thorough study of an-BN/Al 2 O 3 at various temperatures was performed in order to get more information about electron transitions within the band gap of the annealed coating. Figure 8 shows the temperature evolution of the strong CL 'ruby' peak (related to Cr 3+ defects) from an-BN/Al 2 O 3 exposed to an electron beam with energy of 5 keV and flux of 500 nA • cm -2 . This low incident energy was chosen for three main reasons: (i) at this energy with a electron penetration range is ~0.25 µm (figure 2(a)), most electrons were deposited in the ~0.3 µm thick alumina coating, thereby optimising the CL intensity from the annealed Al 2 O 3 layer; (ii) to limit charging of the ceramic sample, thereby avoiding a decrease in the local incident energy which would affect CL measurements as a function of time and (iii) lower incident energies above the second crossover energies such as 5 keV have total electron yields near unity that cause less charging.

The overall CL intensity increases when the temperature decreases and the luminescence hue is more magenta than violet especially at the center of specimen (figure 8). That is due to a larger increase of the 'ruby' peak intensity than that of the higher energy broad peaks attributed to the BN substrate. Powell [START_REF] Powell | Fluorescence studies of energy transfer between single and pair Cr 3+ systems in Al 2 O 3[END_REF][START_REF] Powell | The interaction of chronium ions in ruby crystals[END_REF] and Green [START_REF] Green | Broad-band radioluminescence in electron-irradiated ionic compounds[END_REF] performed detailed studies of the temperature dependence of the 'ruby' peak (~1.785 eV). Green showed that the conversion efficiency (which is directly proportional to CL intensity) is linearly proportional to the optical phonon density which increases exponentially at lower temperatures as an Arrhenius function; his theoretical prediction of a ~20 increase in CL intensity from room temperature to 100 K is larger than the four-fold increase shown in figure 8(c).

This increase in the intensity of 'ruby' peak is due to a relative increase of electron transitions to the more shallow (low energy) DT states (emission from Cr 3+ -center: transition 2 E → A 4 2 [START_REF] Ghamnia | Luminescent centres F and F + in α-alumina detected by cathodoluminescence technique[END_REF]). Indeed, when T decreases the deeper traps (blue and violet BN traps) become more filled because charges are decreasingly thermally excited into ST or CB states, limiting the number of these traps that electrons can decay into and emit photons. By contrast, the occupied density of the shallower DT traps (red an-BN/Al 2 O 3 traps) remains lower due to thermal excitations into the ST and CB and thus electron transitions from ST into the red traps and the accompanying CL remain high. The same sort of increase in CL with decreasing temperature was observed for four DT states in SiO 2 . The behavior of these SiO 2 peaks was attributed to the thermal dependence of the occupancy of the deep traps, as explained by a model for CL for highly disordered insulating materials based on band models of the localized disordered trap states found in the forbidden band gap region of insulating materials. CL intensity increases with decreasing occupancy and increased number of available low energy states to decay into [START_REF] Jensen | Modeling the defect density of states of disordered SiO 2 through cathodoluminescence[END_REF][START_REF] Jensen | Defects density of states model of cathodoluminescent intensity and spectra of disordered SiO 2[END_REF].

Figure 8(c) shows a small (~22 meV from 100 K to 290 K) red shift of the 'ruby' peak at higher temperatures. This red shift of the 'ruby' peak was similar to small decreases in peak energy observed for three of the four SiO 2 peaks; the peak shifts can be explained as due to energy of the emitted photons increasing as the mean occupation level of the DT decreases; this too is related to the occupancy of the DT states [START_REF] Jensen | Defects density of states model of cathodoluminescent intensity and spectra of disordered SiO 2[END_REF].

Influence of long irradiation time (dose)

An annealed coated sample was irradiated for several hours under high electron flux in order to study the evolution of defects and their occupation in an-BN/Al 2 O 3 as a function of the ionizing dose. For the same reasons noted in section 3.3, an incident energy of 5 keV was used to irradiate the sample.

Figure 9(a) shows the temporal evolution of CL intensity under prolonged irradiation, with a noticeable diminution in the red colour of the sample at lower temperatures. The general intensity, brightness and hue of CL of the an-BN/Al2O3 sample approach those of the BN substrate over irradiation time. However, some regions of the an-BN/Al 2 O 3 sample becomes brown (dark spots on the picture in figure 9(a) for 1177 min exposure). On occasion, the sample would abruptly increase in intensity after prolonged beam exposure; this and the asymptotic behaviour suggest that the intensity variations were related to sample charging and a concomitant decrease in power deposited in the sample. This is consistent with the temporal evolution of the surface potential of an-BN/Al 2 O 3 , which after 300 min exposed to a 750 nA • cm -2 20 keV beam at room temperature reached approximately -3700 V [START_REF] Guerch | Etude des propriétés physiques et électriques de matériaux céramiques utilisés en application spatiale[END_REF][START_REF] Guerch | Electrical properties of annealed and coated boron nitride under electron-beam irradiation[END_REF].

More specifically, the CL intensity of the an-Al 2 O 3 'ruby' peak decreases ~50% after 1177 min (figure 9(b)). This decrease could be a result of the surface potential build up described above. Alternately, the decrease could result from CL saturation as the occupancy of final 'ruby' trap increases with charge accumulation thereby diminishing the CL transition rate [START_REF] Jensen | Defects density of states model of cathodoluminescent intensity and spectra of disordered SiO 2[END_REF]. The CL decrease could also result from chemical or structural evolution of the Al 2 O 3 surface from defect modification, contamination or aging [START_REF] Guerch | Etude des propriétés physiques et électriques de matériaux céramiques utilisés en application spatiale[END_REF]. Further detailsrelated to CL-of the effects of electron irradiation on the electrical aging from an-BN/Al 2 O 3 samples are discussed in [START_REF] Guerch | Etude des propriétés physiques et électriques de matériaux céramiques utilisés en application spatiale[END_REF][START_REF] Guerch | Electrical properties of annealed and coated boron nitride under electron-beam irradiation[END_REF].

Conclusion

A thorough investigation of defects states in samples of boron nitride (BN), alumina coated BN (BN/Al 2 O 3 ), and annealed BN with Al 2 O 3 coating (an-BN/Al 2 O 3 ) was carried out under electron bombardment through cathodoluminescence (CL) measurements. The natures and densities of defects and their occupation were studied as a function of incident electron energy and flux, sample temperature, and dose. Evidence suggests that much of the broad signature emission attributed to BN was produced in the rough near surface regions of the samples and its CL intensity was diminished by surface charging. The Al 2 O 3 coating deposited through physical vapour deposition does not have substantial CL-active defects, which could impact the surface conductivity of Al 2 O 3 coating. However, the vacuum thermal annealing treatment performed on BN/Al 2 O 3 has a very significant effect on CL emissions. Indeed, this thermal treatment generates many sharp features in the CL spectra ascribed to specific types of defects in the sample and especially in the Al 2 O 3 coating. This increase of the nature and density of Al 2 O 3 defects states, especially Cr 3+ defects, seems to favour electron transitions and charge mobility under electron irradiation. The influence of temperature on electron transitions from the conduction band (CB) and shallow trap (ST) states to red trap levels (upper band of DT) is significant. In fact, these states become predominant when the temper ature decreases and charges in the deeper bands remain trapped. Nonetheless, the CL decreases as a function of time under high electron flux. The ionizing dose deposited in the Al 2 O 3 coating gradually degrades this material which affects electron transitions.

Figure 1 .

 1 Figure 1. Diagram of instrumentation used to characterise samples under electron bombardment in EET chamber.

Figure 3 .

 3 Figure 3. Emission spectra of BN at room temperature with a 500 nA • cm -2 incident beam at (a) 5 keV and (b) 30 keV. Lower graphs show the component curves in the fits.

Figure 2 .

 2 Figure 2. (a) Range (solid curves, left axis) and dose for non-penetrating beam (dashed curves, right axis) as a function of incident energy. (b) Power deposited (left axis) in 0.3 µm Al 2 O 3 coating (solid curve) and BN substrate (dashed curve) as a function of incident energy for a 500 nA • cm -2 beam. Percentage of incident electron power absorbed in Al 2 O 3 coating (right axis, dash-dotted curve). The 5 keV to 30 keV energy span of the acquired data is indicated by vertical dashed red lines.

Figure 4 .

 4 Figure 4. Emission spectra of (a) BN, (b) BN/Al 2 O 3 and (c) an-BN/Al 2 O 3 acquired at 500 nA • cm -2 beam current, 10, 20 and 30 keV beam energies, and room temperature.

Figure 5 .

 5 Figure 5. BN cathodoluminescence peak fitting parameters as a function of incident electron energy for ~2.4 eV (filled) and ~2.98 eV (open) peaks at 500 nA • cm -2 beam current for the three samples BN (circle), BN/Al 2 O 3 (square) and an-BN/Al 2 O 3 (triangle). Vertical dashed black lines indicate the boundary between low energy peak behavior and high energy peak behavior between 10 keV and 15 keV. (a) and (b) Integrated peak intensities, for ~2.4 eV and ~2.98 eV peaks, respectively. (c) Integrated peak intensities normalized to largest intensity for each peak of each sample. The horizontal dashed red line indicates mean low energy behavior. The sloped dashed red line is a linear fit to decreasing intensity above 15 keV, with a slope of -76%/10 keV. (d) Ratio of intensities of ~2.98 eV peak to ~2.4 eV peak (I 2.98 /I 2.40 ), with 0.41 mean ratio at high energies (horizontal dashed red line). (e) Percent shift in peak energy normalized to the mean low energy unannealed peak positions at 2.388 ± 0.005 eV (solid) and 2.995 ± 0.005 eV (open), respectively. The an-BN/Al 2 O 3 (triangle) peaks have a mean 1.3% shift at high energies. (f) Peak width (FWHM) normalized to the mean high energy peak width of 0.49 ± 0.04 eV.

Figure 6 .

 6 Figure 6. Emission spectra of an-BN/Al 2 O 3 near the 1.785 eV ruby peak acquired at 500 nA • cm -2 beam current, 10,15, 20, 25 and 30 keV beam energies, and room temperature. Larger satellite peaks are observed at ~1.737 eV, ~1.758 eV, ~1.835 eV and ~1.853 eV are indicated by vertical dashed lines.

Figure 7 .

 7 Figure 7. Delayed luminescence of an-BN/Al 2 O 3 . The electron beam (30 keV at 500 nA • cm -2 ) incident on a room temperature sample is turned off at 737 s, as denoted by the vertical dashed line. The CL intensity measured with the Xybion CCD camera (green solid line) has had the background (black curve) subtracted; the camera is saturated when the beam is on. Red lines are exponential decay fits with decay constants of 0.81 ± 0.01 s and 5.4 ± 0.1 s.

Figure 8 .

 8 Figure 8. Evolution of cathodoluminescence of an-BN/Al 2 O 3 as a function of temperature under electron bombardment at 5 keV and 500 nA • cm -2 . (a) and (b) Intensity evolution captured through SLR CCD camera photographs. (c) Temperature variation of 'ruby' peak at ~1.785 eV.

Figure 9 .

 9 Figure 9. Evolution of cathodoluminescence of an-BN/Al 2 O 3 as a function of irradiation time under electron bombardment of 5 keV at 500 nA • cm -2 and at 110 K: (a) evolution of overall intensity captured through SLR CCD camera photographs. (b) Emission CL spectra of 'ruby' peak at ~1.79 eV.

Table 1 .

 1 Penetration depth and range of Al 2 O 3 and h-BN.

			Bulk Al 2 O 3			Bulk h-BN
	E (keV) Z 12 (µm) R (µm) Z 12 /R	Z 12 (µm) R (µm) Z 12 /R
	5	0.2	0.36	0.56	0.5	0.44	1.1
	10	0.7	1.1	0.63	1.1	1.4	0.77
	15	1.4	2.1	0.65	1.9	2.8	0.67
	20	2.4	3.4	0.71	3.0	4.6	0.66
	25	3.6	4.9	0.74	4.3	6.6	0.65
	30	5.0	6.5	0.77	5.9	9.0	0.66

Acknowledgment

The authors would like to thank A Jensen and A Souvall of Mat erials Physics Group for their technical support. The authors gratefully acknowledge the laboratories and companies which contributed to this international collaboration, including the USU Physics Department, ONERA Toulouse, CIRIMAT and THALES.


Innocent Ipek Karacan
email: ipek.karacan@student.uts.edu.aufrance

Jacob Macha
email: bimacha@udsm.ac.tz

Gina Choi

Sophie Cazalbou
email: csophie.cazalbou@univ-tlse3.fr

Besim Ben

B Ben-Nissan

Antibiotic Containing Poly Lactic Acid/Hydroxyapatite Biocomposite Coatings for Dental Implant Applications

Keywords: PLA thin film composites, hydroxyapatite, hard coral, drug release studies, coating dental implants

published or not. The documents may come

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Introduction

Despite the considerable evolution in biomaterials, the bone repair processes in dental and orthopedic implantology are too often compromised due to post-operative complications. In general, when the implant-related infections are observed, the worst scenario is to replace with the new implant which is very painful and sometimes complicated. Another solution is to apply different conventional antibiotic therapies which are not always successful treatments [1,2]. This is the reason a large number of researches currently focuses on the use of new multifunctional bioactive coated implants for in situ drug delivery to prevent post-operative complications or treat certain bone diseases such as osteomyelitis which can avoid treatments. Especially, using drug release composites for implant coating are effective on the prevention and treating of these infections and diseases [3,4].

Marine invertebrates are one of the potential and utility sources which can be converted to calcium phosphate bioceramics as a drug carrier in the drug delivery systems. For instance, foraminifera whose unique structure of exoskeleton has interconnected porous network are very attractive marine species [1,5]. In order to provide a wide range for drug delivery systems, biodegradable and biocompatible thin film composites are generally used. Biodegradable thin film composites are easily applicable materials in clinical application because they allow controlling drug dissolution rate [6].

The main aim of this investigation was to apply Gentamicin containing converted coralline microspheres embedded in PLA matrix for sustained and efficient drug release coatings on titanium dental implants. doi:10.4028/www.scientific.net/

Experimental

Materials

Coral samples were obtained from the Great Barrier Reef, QLD, Australia. All chemicals were obtained from Sigma Aldrich, Australia.

Methods

In this research microspheres of coral were converted into HAp using hydrothermal technique and incorporated within biodegradable thin poly-lactic acid (PLA) to form thin film composites. Treatment of the coral and the conversion methods were covered in our previous publications [8].

Hydrothermal Synthesis of Calcium Phosphate

The coral samples were crushed then cleaned with 2% (v/v) NaClO, and then ground within an aluminium oxide ball mill (46 rpm, 2 h), sieved with 100 μm sieve, then cleaned again with 2% (v/v) NaClO and then dried at 100 o C for 2 hours before use. Cleaned coral were converted to hydroxyapatite (HAp) by hydrothermal conversion method which was described in Ben-Nissan procedures [7]. Briefly, the coral powder was mixed with the required amount of diammonium hydrogen phosphate (NH 4 ) 2 HPO 4 which is calculated according to Ca/P ratio of HAp to obtain HAp. This chemical reaction was occurred under high temperature and pressure. Therefore, cleaned coral powder is converted with (NH 4 ) 2 HPO 4 in a Parr reactor at 250 o C and 8.0MPa pressure for three hours.

Production of PLA composites and Drug Loading

The solvent casting technique which was identified in [4] was used to prepare PLA thin films and their composites which were denited as PLA with gentamicin (PLA-GM), PLA with HAp (PLA-HAp) and PLA with GM loaded HAp (PLA-HAp-GM), respectively. The drug loading and composites preparation were illustrated in Fig. 1 [8]. This method allows us to preserve the nano and meso-porous structure as appropriate drug carrier systems for sustained release of antibiotics (gentamicin (GM)) on titanium dental implants. The characterization of PLA composites were carried out by SEM (ZEISS Supra55VP, Zeiss, Germany) and phase analysis of the products were carried out by X-ray powder diffraction using D8 ADVANCE (BRUKER), radiation Cu K(alpha)1+2 = 0.15418 nm, no monochromator, from 20 to 80° (2 theta), steps of 0.02°, 2 seconds/step, divergence slit 0.3°, Sollers slit 2.5°, detector linear: LynxEye (2.73°). Dissolution studies were carried out with phosphate buffer saline (PBS) for gentamicin under the constant speed of 100 rpm in a temperature controlled water bath shaker at 37 o C. Gentamicin concentration was measured by Cary 100 UV-Vis Spectrophotometer (Agilent Technologies, Victoria, Australia, Cary Series UV-Vis Spectrophotometer) at 332 nm.

Coating of Titanium Plate with PLA composites

PLA thin film composite coatings were carried out on the titanium (Ti) plate and dental implants by spin and dip coating respectively. The spin coating method was given in previous publications [9], however, the method was modified in this study. The coating was carried out at 3000, 5000 and 10000 rpm speeds on titanium flat plates using Headway rotary spinner (Headway, USA). A range of coatings was tried: PLA film alone, PLA-HAp, PLA-GM and PLA-HAp-GM. Films on titanium dental implants were carried out and characterized.

Results and Discussion

PLA thin film composites which are PLA-HAp, PLA-GM, and PLA-HAp-GM, respectively were produced successfully.

The release kinetics of drugs was studied in order to evaluate the effectiveness of the drug loaded HA particles on the drug release behavior. Fig. 2 The Gentamicin release rate changes during 8 weeks in PBS solution.

A number of different stages were observed during the gentamicin release in PBS during 8 weeks as shown in Fig. 2. The first stage is an initial burst of surface bound drug release which can be referred as the direct dissolution of drug from the film PLA surface, the rest of the stages dependent on the particle size and amount and their dissociation rates. It is thought that the order of dissolution progresses according to a specific dissolution under the experimental environmental condition but it might involve: (i) surface drug release from the PLA coating, (ii) PLA matrix dissolution, and release of the drug from the matrix (iii) drug release from the particle surface and particle dissolution and break down further drug release, (iv) combination of all at once after break down of the particles. As stated above the drug release rates depends on the number of material and environmental factors, but it was observed that the initial stage was very fast for PLA-GM composite than PLA-HAp-GM composite. Knowing the particle size and composition effect on the dissolution this is not surprising. Fig. 3 The SEM images of PLA composites during dissolution study at the start of the testing and after 3 weeks period.

Drug particles were entrapped within the nano and meso porous structure of coralline HAp, so HAp provided to control of gentamicin release rate in PLA composite according to its dissolution rate. It has been shown that PLA used in this study starts to degrade after one to two weeks. There is no strong affinity between drug and HAp only secondary forces and the effect of interconnected pores. Also PLA control the release by its degradation rate through its molecular weight which can be adjusted to give a specific dissolution rate, so the higher the molecular weight, the lower the degradation rate. The release mechanism is by diffusion; however, degradation of PLA and dissociation of HAp particle triggers the dissolution and influence the diffusion rates of the drug. Because of the complex structure or the drug release study, the dissolution data were analyzed statistically, and they were fitted by Korsmeyer-Peppas method with a mathematical expression of F = kt n in this study. According to results of the statistical calculations, the drug might be released by diffusion in PLA-GM (n< 0.5) while PLA-HAp-GM (n> 1) had mixed transport mechanism. Therefore, the drug release mechanism of PLA-HAp-GM can be considered more complex than PLA-GM.

During the dissolution studies, the release of GM highlights a burst followed by a sustained release period extending for three weeks. The morphological changes of PLA-GM, PLA-HAp-GM and PLA-HAp composites and the degradation of the PLA film in samples were observed during 3 weeks by SEM at the Fig. 3. End of the 3 rd week, the PLA film degradation was seen clearly in PLA-HAp films. The progression of film degradation for PLA-GM and PLA-HAp-GM was also shown in Fig. 3. By week 8 most of the drug encapsulated within the polymer matrix is expected to be released. The film degradation and correlate are well with drug dissolution. All drug releasing and kinetic studies described in detail at the research paper of Macha et al. [START_REF] Macha | Kinetics and the theoretical aspects of drug dissolution from PLA/ HAp thin films[END_REF]. According to results of this research, when the drugs are associated with HAp particles the release is substantially delayed on dental implants under experimental conditions. It can be postulated that we have succeeded to control the release rate of GM in the PBS solution by the micro-porous structure of coralline HAp, so the devices could be used for a relative slow drug release applications and the release rate could be tailored so as to increase the release time. Indeed, the release of the drug from the coated film not only depends on the polymeric matrix but by the accessibility of the drug positioned within the porous network of the particles and by interactions within the immediate physiologic environment. Morphological analysis of coating titanium flat plates with the composites was studied by SEM. Fig. 4-A andB show the surface morphology of Ti plates coated with HAp-PLA while the morphology of Ti plates coated with GM-HAp-PLA was seen in Fig. 4-C andD. Finally, the porous structure of PLA film which was coated on Ti plate was obtained at low and high magnifications in Fig. 4-E andF, respectively. HAp particles in Fig. 4-B and GM loaded HAp particles in Fig. 4-D on PLA film was illustrated at high magnification, so there is no chemical interaction between polymer and bioceramic material-drug particles. The results suggest that GM loaded HAp particles have more smooth morphology than the HAp particles according to SEM images. Figure 5 shows FTIR spectra of HAp and PLA together with their thin film composites on Ti plates. The peaks of 10% HAp coated samples were matched with the peaks of coralline HAp in Fig. 5-A and B. The FT-IR spectra of Ti coated with GM-HAp-PLA in Fig. 5-D which was produced 10% GM loaded HAp coincides with the spectra of PLA thin film which was illustrated in Fig. 5-C. In this study, the FT-IR analysis suggested that HAp and GM drug particles did not interact chemically with PLA when mixed together and used as a coating material on Ti plate, so PLA matrix only acts as a barrier during the drug releasing process.

Conclusions

The results from this study demonstrated that dental implants could be coated with biodegradable polymer such poly lactic acid that contains antibiotic loaded HAp microspheres that can be produced and used to reduce or in certain instances even eliminate bacterial infection. Further characterization on in vitro and in vivo studies is needed to confirm this more accurately.

Fig. 1

 1 Fig. 1 Schematic illustration of the drug loading and dissolution procedure [8].

Fig. 4

 4 Fig. 4 The SEM images of coated Ti plates with low (A) and high (B) magnification of HAp-PLA thin film, the black arrows refer to HAp particles, and low (C) and high (D) magnification of GM-HAp-PLA thin film, the white arrows refer to GM loaded HAp particles, and low (E) and high (F) magnification of PLA film.

Fig. 5

 5 Fig.5 The Spectra of FT-IR for PLA thin film, Coralline HAp, HAp-PLA coated on Ti plate, and GM-HAp-PLA coated on Ti plates.

Acknowledgement

The authors would like to thank the Australian Academy of Science and the Horizon 2020 European Commission Grants for this collaborative research.


Jacques Alev
email: jacques.alev@univ-reims.fr

¸ois Dumas
email: francois.dumas@uca.fr

ENVELOPING SKEWFIELDS OF THE NILPOTENT POSITIVE PART AND THE BOREL SUBSUPERALGEBRA OF osp(1, 2n)

Keywords: Mathematics Subject Classification. Primary 17B35; Secondary 16S30, 16S85, 16K40 Simple Lie superalgebra, enveloping algebra, Gelfand-Kirillov hypothesis, Weyl algebra, skewfield

We study an analogue of the Gelfand-Kirillov property for some Lie superalgebras. More precisely, we consider in the classical simple orthosymplectic Lie superalgebra osp(1, 2n) of dimension n 2 + 3n the positive nilpotent subsuperalgebra n + of dimension n 2 + n and the solvable Borel subsuperalgebra b + of dimension n 2 + 2n. We prove that the enveloping algebras U(n + ) and U(b + ) are rationally equivalent to some super-analogues of Weyl algebras over polynomial superalgebras.

Introduction

In the classical context of the seminal paper [START_REF] Gelfand | Sur les corps liés aux algèbres enveloppantes des algèbres de Lie[END_REF], an algebraic finitedimensional complex Lie algebra g is said to satisfy the Gelfand-Kirillov property when its enveloping algebra U(g) is rationaly equivalent to a Weyl algebra A n (K) over a commutative purely transcendental field extension K of C. It was proved that this property is satisfied when g is nilpotent ( [START_REF] Gelfand | Sur les corps liés aux algèbres enveloppantes des algèbres de Lie[END_REF]), or more generally solvable ( [START_REF] Borho | Primideale in Einhüllenden auflösbarer Lie-Algebren[END_REF], [START_REF] Joseph | Proof of the Gelfand-Kirillov conjecture for solvable Lie algebras[END_REF], [START_REF] Mcconnell | Representations of solvable Lie algebras and the Gelfand-Kirillov conjecture[END_REF]). This profound problem gave rise to an abundant literature with many developments within Lie theory itself (see references in [START_REF] Alev | The Gelfand-Kirillov conjecture for Lie algebras of dimension at most eight[END_REF] or [START_REF] Premet | Modular Lie algebras and the Gelfand-Kirillov conjecture[END_REF]) or for variants concerning quantum groups, invariant algebras or other classes of interesting noncommutatives algebras.

The study of an analogue of the Gelfand-Kirillov property for Lie superalgebras was explored in [START_REF] Musson | On the Goldie quotient ring of the enveloping algebra of a classical simple Lie superalgebra[END_REF] and [START_REF] Alev | On enveloping skew fields of some Lie superalgebras[END_REF]. In this context, it is relevant to introduce for any integer n ≥ 1 the noncommutative polynomial superalgebra O(C n ) and an analogue A n (C) of the Weyl algebra. By definition O(C n ) is the algebra generated over C by n indeterminates y 1 , . . . , y n with relations y i y j + y j y i = 0 for all 1 ≤ i = j ≤ n. Then A n (C) is the algebra generated over O(C n ) by n other indeterminates x 1 , . . . , x n with relations x i y j + y j x i = x i x j + x j x i = 0 for all 1 ≤ i = j ≤ n and x i y i -y i x i = 1 for any 1 ≤ i ≤ n. These algebras are noetherian domains and then they admit a skewfield of fractions. Otherwise the only case in the classification of classical simple finite dimensional complex Lie superalgebras g where the enveloping algebra U(g) is a domain is when g is an orthosymplectic Lie superalgebra osp(1, 2n) (see [START_REF] Aubry | Zero divisors in enveloping algebras of graded Lie algebras[END_REF], [START_REF] Gorelik | The annihilation theorem for the Lie superalgebra osp(1,2 )[END_REF]). This is the case we consider here.

For any integer n ≥ 1, the superalgebra g = osp(1, 2n) is of dimension 2n 2 + 3n. In its Z 2 -graded decomposition g = g 0 ⊕ g 1 , the even part g 0 is isomorphic to the symplectic Lie algebra sp(2n) of dimension 2n 2 + n, and the odd part g 1 is a vector space of dimension 2n. There is a natural definition of the nilpotent positive part n + = osp + (1, 2n), which is a nilpotent subsuperalgebra of g of dimension n 2 + n. In its Z 2 -graded decomposition n + = n + 0 ⊕ n + 1 , the even part n + 0 sp + (2n) is just the nilpotent positive part of dimension n 2 in the ordinary triangular decomposition of the Lie algebra g 0 sp(2n), and the odd part n + 1 is a subspace of dimension n in g 1 . Denoting by h the Cartan Lie subalgebra of g 0 , the Borel subsuperalgebra b + of g is defined as the subsuperalgebra generated by n + and h. We have

b + = n + ⊕h = b + 0 ⊕n + 1
, where the even part b + 0 = n + 0 ⊕h is the positive Borel Lie subalgebra of g 0 sp(2n). In particular, b + 0 and b + are of dimensions n 2 + n and n 2 + 2n respectively. The enveloping algebras U(n + ) and U(b + ) can be described as iterated skew polynomial algebras; then they are noetherian domains and therefore we can consider their skewfields of fractions. We prove in the paper: Main theorem. For any n ≥ 2, the enveloping algebras of the subsuperalgebras n + and b + of osp(1, 2n) satisfy the following isomorphisms: The method we use to prove this theorem allows to recover as a corollary the classical Gelfand-Kirillov property for the even parts, with already known descriptions of Frac U(n + 0 ) as a Weyl skewfield D n(n-1)/2 (K) over a commutative field K which is a purely transcendental extension of dimension n over C, and of Frac U(b + 0 ) as a Weyl skewfield D n(n+1)/2 (C) over C. The particular case n = 2 of the main theorem was previously proved in Proposition 3.2 and Theorem 3.4 of [START_REF] Alev | On enveloping skew fields of some Lie superalgebras[END_REF]. The natural and probably difficult question of an extension of the main theorem to the enveloping algebra of the Lie superalgebra osp(1, 2n) itself must be appreciated by recalling that even for the even part the answer is unknown, the case of sp(2n) being one of the situations where the original Gelfand-Kirillov conjecture remains open (see [START_REF] Premet | Modular Lie algebras and the Gelfand-Kirillov conjecture[END_REF]).

(i) Frac U(n + ) Frac A n(n-1)/2 (C)⊗ O(C n ) ,
The paper is in three parts. The first one introduces the analogues of Weyl algebras for polynomial superalgebras and some canonical form of their skewfields of fractions. The second part describes the enveloping algebras U(n + ) and U(b + ) as iterated skew polynomial algebras. The last section contains the proof of the main theorem, based on suitable rational changes of generators reducing step by step the commutation relations. 1.1.1. Notations. It is usual in quantum group theory to denote by O Λ (C n ) the algebra of polynomials in n indeterminates y 1 , . . . , y n with coefficients in C and noncommutative multiplication twisted by relations y i y j = λ ij y j y i for any 1 ≤ i, j ≤ n, where Λ = (λ ij ) is a n × n multiplicatively skew-symmetric matrix with entries in C. In the particular cases where all λ ij = 1, or λ ij = -1 for all i = j, we use respectively the two following notations:

(i) O(C n ) is the commutative algebra of polynomials in n indeterminates y 1 , y 2 , . . . , y n with coefficients in C.

(ii) O(C n ) is the noncommutative algebra of polynomials in n indeterminates y 1 , y 2 , . . . , y n with coefficients in C and product twisted by relations y i y j = -y j y i for any 1 ≤ i = j ≤ n.

1.1.2. Superalgebra structure. The monomials y 1

β 1 y 2 β 2 • • • y n βn whose total degree β 1 + β 2 + • • • + β n is even generate a commutative subalgebra R 0 of O(C n ). Denoting by R 1 the R 0 -submodule of O(C n ) generated by y 1 , . . . , y n , the C-submodules R 0 and R 1 satisfy O(C n ) = R 0 ⊕ R 1 and R i R j ⊆ R i+j
If n = 2p, then Frac ( O(C n )) Frac ( O(C 2 ) ⊗p ) If n = 2p + 1, then Frac ( O(C n )) Frac ( O(C 2 ) ⊗p ⊗ O(C))
Proof. Suppose that n ≥ 3 and define from generators y 1 , . . . , y n of O(C n ) with y i y j = -y j y i for all 1 ≤ i = j ≤ n the monomials: y 1 = y 1 , y 2 = y 2 and y i = y 1 y 2 y i for any 3 ≤ i ≤ n. They satisfy y 1 y 2 = -y 2 y 1 and

y 1 y i = y i y 1 , y 2 y i = y i y 2 , y i y j = -y j y i for all 3 ≤ i = j ≤ n. It is clear that the subfield of Frac O(C n ) generated by y 1 , y 2 , . . . , y n is equal to Frac O(C n ). We deduce that Frac O(C n ) Frac ( O(C 2 ) ⊗ O(C n-2
)), and the result follows by iteration. 1.2. Weyl algebras for polynomial superalgebras.

1.2.1.

Notations. The definition of quantum differential calculi on various quantum algebras with adapted de Rham complexes gives rise to suitable versions of quantum Weyl algebras. In the case of quantum spaces O Λ (C n ), these algebras A q,Λ n (introduced in [START_REF] Alev | Sur le corps de fractions de certaines algèbres quantiques[END_REF] and [START_REF] Wambst | Complexes de Koszul quantiques[END_REF] and studied in many papers) take in consideration both the quantization parameters Λ = (λ ij ) of the space and the quantization parameters q = (q 1 , . . . , q n ) of the "differential" operators on this space. We are interested here in two particular cases:

(i) For λ ij = q i = 1 for all 1 ≤ i, j ≤ n, we find the usual Weyl algebra A n (C), that is the algebra of polynomials in 2n indeterminates x 1 , . . . , x n , y 1 , . . . , y n with coefficients in C and a noncommutative product twisted by relations:

x i x j -x j x i = y i y j -y j y i = x i y j -y j x i = 0 (1 ≤ i = j ≤ n), x i y i -y i x i = 1 (1 ≤ i ≤ n). (1)
We also introduce the subalgebra U n (C) of A n (C) generated by y 1 , . . . , y n and w 1 , . . . , w n with w i = y i x i for any 1 ≤ i ≤ n. We have:

w i w j -w j w i = y i y j -y j y i = w i y j -y j w i = 0 (1 ≤ i = j ≤ n), w i y i -y i w i = y i (1 ≤ i ≤ n). (2)
(ii) For q i = 1 and λ ij = -1 for all 1 ≤ i = j ≤ n, we denote by A n (C) the algebra of polynomials in 2n indeterminates x 1 , . . . , x n , y 1 , . . . , y n with coefficients in C and a noncommutative product twisted by:

x i x j + x j x i = y i y j + y j y i = x i y j + y j x i = 0 (1 ≤ i = j ≤ n), x i y i -y i x i = 1 (1 ≤ i ≤ n). (3)
We denote by U n (C) the subalgebra of A n (C) generated by y 1 , . . . , y n and w 1 , . . . , w n with w i = y i x i for any 1 ≤ i ≤ n. We have:

     y i y j + y j y i = 0 (1 ≤ i = j ≤ n), w i w j -w j w i = w i y j -y j w i = 0 (1 ≤ i = j ≤ n), w i y i -y i w i = y i (1 ≤ i ≤ n). (4)
1.2.2. Remarks. The algebras A n (C) are the algebras denoted by S Λ n,n and studied in [START_REF] Richard | Hochschild homology and cohomology of some classical and quantum noncommutative polynomial algebras[END_REF] and [START_REF] Richard | Équivalence rationnelle d'algèbres polynomiales classiques et quantiques[END_REF] when all non diagonal entries of Λ are equal to -1. Section 11 of [START_REF] Wambst | Complexes de Koszul quantiques[END_REF] gives a slightly different presentation of these algebras as "generalized enveloping algebras"; with the notations of this article, case (i) corresponds to the values q ij = p i = q = 1, and case (ii) is for q ij = -1 and p i = q = 1. They also appear in example 2.1 of [START_REF] Giaquinto | Quantum Weyl algebras[END_REF] (taking q = 1 and p ij = -1). We refer to paragraph 1.3.3 of [START_REF] Richard | Hochschild homology and cohomology of some classical and quantum noncommutative polynomial algebras[END_REF] for more ringtheoretical references and a survey, and only mention here that, for any integer n ≥ 1, the algebra A n (C) is simple, has center C, and has the same Hochschild homology and cohomology as the usual Weyl algebra A n (C). 1.2.3. Superalgebra structure. (i) Let g 0 and g 1 be two C-vector spaces of dimension n with respective basis {z 1 , . . . , z n } and {y 1 , . . . , y n }. Let k 0 be a linear extension of g 0 of dimension 2n with basis {z 1 , . . . , z n , w 1 , . . . , w n }. Then we define a Lie superalgebra structure g = g 0 ⊕ g 1 of dimension 2n and a Lie superalgebra structure k = k 0 ⊕ g 1 of dimension 3n by setting for the values of the super-bracket on these generators:

{y i , y i } = z i , [w i , y i ] = y i , [w i , z i ] = 2z i for any 1 ≤ i ≤ n,
the brackets are zero for all other pairs of generators. By construction g 0 is a subsuperalgebra of k 0 and g = g 0 ⊕ g 1 is a subsuperalgebra of k. It is clear by PBW theorem (see Theorem 6.1.2 in [START_REF] Musson | Lie superalgebras and enveloping algebras[END_REF]) that the enveloping algebra U(k) is isomorphic to U n (C) of GK-dim 2n. Moreover the subalgebra U(k 0 ) is isomorphic to U n (C) of GK-dim 2n, and the subalgebra

U(g) is isomorphic to O(C n ) of GK-dim n.
(ii) We have for the usual Weyl algebra the classical isomorphism A n (C) A 1 (C) ⊗n . We also have A n (C) A 1 (C) ⊗n in the tensor category of superalgebras assigning the parity 1 to the generators x i and y i . (iii) All isomorphims and tensor products considered in the following are in the tensor category of associative algebras. 1.2.4. Remark. All algebras introduced in 1.2.1 can be described as iterated skew polynomial algebras over C; then they are noetherian domains, and therefore they admit a skewfield of fractions. The skewfield Frac A n (C) = Frac U n (C) is the well known Weyl skewfield D n (C), with center C. Similarly we deduce from (4) that Frac A n (C) = Frac U n (C), and it is proved in Proposition 3.3.1 of [START_REF] Richard | Équivalence rationnelle d'algèbres polynomiales classiques et quantiques[END_REF] that its center is also C. In parallel with Proposition 1.1.3, we have for these skewfields the following canonical form. 1.2.5. Theorem. For any integer n ≥ 1, we have the following isomorphisms:

(i) If n = 2p is even, then Frac A n (C) = Frac ( A 2 (C) ⊗p ). (ii) If n = 2p + 1 is odd, then Frac A n (C) = Frac ( A 2 (C) ⊗p ⊗ A 1 (C)).
Proof. We can suppose n ≥ 1. We consider the subalgebra U n (C) defined in 1.2.1 with generators w 1 , . . . , w n , y 1 , . . . y n and relations [START_REF] Aubry | Zero divisors in enveloping algebras of graded Lie algebras[END_REF]. As in the proof of Proposition 1.1.3, we introduce: y 1 = y 1 , y 2 = y 2 and y i = y 1 y 2 y i for 3 ≤ i ≤ n. Then we define:

w 1 = w 1 -(w 3 + • • • + w n ), w 2 = w 2 -(w 3 + • • • + w n ) and w i = w i for 3 ≤ i ≤ n.
Obvious calculations give:

y 1 y 2 = -y 2 y 1 , w 1 w 2 = w 2 w 1 , y 1 w 2 = w 2 y 1 , y 2 w 1 = w 1 y 2 , [w 1 , y 1 ] = y 1 , [w 2 , y 2 ] = y 2 . (5)
y 1 y j = y j y 1 , y 2 y j = y j y 2 , y 1 w j = w j y 1 , y 2 w j = w j y 2 , w 1 w j = w j w 1 , w 2 w j = w j w 2 , for 3 ≤ j ≤ n.

We calculate for = 1 or 2 and any 3 ≤ j ≤ n the commutator: [w , y j ] = [w , y 1 y 2 y j ] -n i=3 [w i , y 1 y 2 y j ] = y 1 y 2 y j -[w j , y 1 y 2 y j ] = 0 to conclude : y j w 1 = w 1 y j and y j w 2 = w 2 y j for any 3 ≤ j ≤ n.

Moreover: [w j , y j ] = y j for any 3

≤ j ≤ n, (8)
y i y j = -y j y i , w i w j = w j w i , w i y j = y j w i , for 3

≤ i = j ≤ n. (9)
We denote by V the subalgebra of U n (C) generated by y 1 , . . . , y n ,w 1 , . . . , w n , by W the subalgebra of V generated by y 1 , , y 2 , w 1 , w 2 , and by W the subalgebra of V generated by y 3 , . . . , y n , w 3 , . . . , w n . It follows from ( 5) that W U 2 (C), from ( 8) and ( 9) that W U n-2 (C), and from ( 7) and ( 9

) that V W ⊗ W . Since it is clear that Frac V = Frac U n (C), we deduce that Frac U n (C) Frac ( U 2 (C) ⊗ U n-2 (C)). We conclude that Frac A n (C) Frac ( A 2 (C) ⊗ A n-2 (C)
) and finish by induction on n.

2. The Lie superalgebra osp(1, 2n) and its enveloping algebra 2.1. Generators and relations. We refer for instance to [START_REF] Ch | A Lie superalgebraic interpretation of the para-Bose statistics[END_REF], [START_REF] Palev | The quantum superalgebra Uq[osp(1|2n)]: deformed para-Bose operators and root of unity representations[END_REF], [START_REF] Alev | On enveloping skew fields of some Lie superalgebras[END_REF]. The basefield is C. For any n ≥ 1, the odd part g 1 of g = osp(1, 2n) is a vector space of dimension 2n and the even part g 0 is isomorphic to the symplectic Lie algebra sp(2n) of dimension 2n 2 + n. As a Lie superalgebra, osp(1, 2n) is generated by the elements b

± i (1 ≤ i ≤ n) of a basis of g 1 . Then the elements {b ± j , b ± k } (1 ≤ j ≤ k ≤ n) and {b + j , b - k } (1 ≤ j, k ≤ n)
form a basis of g 0 . The brackets in osp(1, 2n) are given by the "para-Bose" relations on the generators:

[{b ξ j , b η k }, b ] = ( -ξ)δ j b η k + ( -η)δ k b ξ j (10) [{b ξ i , b η j }, {b k , b ϕ }] = ( -η)δ jk {b ξ i , b ϕ } + ( -ξ)δ ik {b η j , b ϕ } + (ϕ -η)δ j {b ξ i , b k } + (ϕ -ξ)δ i {b η j , b k }, (11)
with 1 ≤ i, j, k, ≤ n and , ϕ, ξ, η are ±. By PBW theorem, the enveloping algebra U(osp(1, 2n)) is generated by the 2n 2 + n elements:

b ± i and k i := 1 2 {b - i , b + i } for 1 ≤ i ≤ n, (12)
a ± ij := 1 2 {b ± i , b ± j } for 1 ≤ i < j ≤ n, (13)
s ij := 1 2 {b - i , b + j } and t ij := 1 2 {b + i , b - j } for 1 ≤ i < j ≤ n, (14)
of osp(1, 2n). In particular, denoting

c ± i := 1 2 {b ± i , b ± i } for 1 ≤ i ≤ n, (15)
the enveloping algebra U(sp(2n)) of the even part g 0 is the subalgebra of U(osp(1, 2n)) generated by the 2n

2 + n elements (b ± i ) 2 = c ± i , k i for 1 ≤ i ≤ n, and a ± ij , s ij , t ij for 1 ≤ i < j ≤ n.
The commutation relations in the associative algebra U(osp(1, 2n)) are deduced from ( 10) and ( 11) taking {x, y} = xy + yx if x, y ∈ g 1 , and [x, y] = xy -yx otherwise. In particular commutation relations between the generators defined by ( 12), ( 13), ( 14), [START_REF] Palev | The quantum superalgebra Uq[osp(1|2n)]: deformed para-Bose operators and root of unity representations[END_REF] are obtained by injecting them in relations [START_REF] Joseph | Proof of the Gelfand-Kirillov conjecture for solvable Lie algebras[END_REF] and [START_REF] Kac | Lie superalgebras[END_REF]. For instance:

b + i b + j + b + j b + i = 2a + ij for all 1 ≤ i < j ≤ n, (16)
[c + i , b + j ] = [c + i , c + j ] = 0 for all 1 ≤ i < j ≤ n, (17)
[a + ij , b + ] = [a + ij , c + ] = 0 for all 1 ≤ i < j ≤ n, 1 ≤ ≤ n, (18)
[a + ij , a + k ] = 0 for all 1 ≤ i < j ≤ n, 1 ≤ k < ≤ n. (19)
Similarly with ( 10) and ( 14), we deduce for all 1 ≤ i < j ≤ n and 1 ≤ k ≤ n:

[t ij , b + j ] = b + i and [t ij , b + k ] = 0 if k = j. ( 20
)
The relations involving the t ij 's are obtained from ( 11), ( 13) and ( 14), which give for 1 ≤ i < j ≤ n and 1 ≤ k < ≤ n :

[t ij , a + k ] = δ jk 1 2 {b + i , b + } + δ j 1 2 {b + i , b + k }, ( 21
) [t ij , t k ] = δ jk 1 2 {b + i , b -} -δ i 1 2 {b + k , b - j }, (22)
or in other words:

[t ij , a + k ] = 0 if j = k, j = (1 ≤ i < j ≤ n, 1 ≤ k < ≤ n), ( 23
) [t ij , a + j ] = a + i (1 ≤ i < j < ≤ n), ( 24
) [t ij , a + ij ] = c + i (1 ≤ i < j ≤ n), ( 25
) [t ij , a + kj ] = a + ik if i < k (1 ≤ i < j ≤ n, 1 ≤ k < j ≤ n, ), ( 26
) [t ij , a + kj ] = a + ki if k < i (1 ≤ i < j ≤ n, 1 ≤ k < j ≤ n, ). ( 27
) [t ij , t j ] = t i (1 ≤ i < j < ≤ n), ( 28
) [t ij , t ki ] = -t kj (1 ≤ k < i < j ≤ n), ( 29
) [t ij , t k ] = 0 if j = k, i = (1 ≤ i < j ≤ n, 1 ≤ k < ≤ n). ( 30
)
The action of the generators k i follows from ( 10) and ( 12):

[k i , k j ] = 0 (1 ≤ i, j ≤ n), (31)
[k i , b + i ] = b + i , [k i , b ± j ] = 0 (1 ≤ i = j ≤ n), (32)
then with ( 13) and ( 14), for all 1 ≤ i < j ≤ n and 1 ≤ ≤ n:

     [k i , a + ij ] = a + ij , [k j , a + ij ] = a + ij , [k , a + ij ] = 0 if = i, j.      [k i , t ij ] = t ij , [k j , t ij ] = -t ij , [k , t ij ] = 0 if = i, j. (33)
2.2. Chevalley generators and Serre relations. Following [START_REF] Palev | The quantum superalgebra Uq[osp(1|2n)]: deformed para-Bose operators and root of unity representations[END_REF] (up to a renormalization of the n-th generator by √ 2), we introduce:

t i := t i,i+1 for any 1 ≤ i ≤ n -1, and t n := b + n , (34)
which satisfy:

[t i , t j ] = 0 if |i -j| > 1 (1 ≤ i, j ≤ n), ( 35
) [t i , t i+1 ] = t i,i+2 (1 ≤ i ≤ n -2), ( 36
) [t n-1 , t n ] = b + n-1 . ( 37
)
It also follows from (20) that for any 1 ≤ i ≤ n -1 and 1 ≤ j ≤ n:

[t i , b + i+1 ] = b + i and [t i , b + j ] = 0 if j = i + 1. ( 38
)
where the derivations d ij are suitably defined to account for relations (20) and ( 23) to (30). To sum up we conclude:

A = C[a + ij ] 1≤i<j<n , (44)
B = A[b + i ; σ i , δ i ] 1≤i≤n , (45)
U + = B[t ij ; d ij ] 1≤i<j≤n . ( 46
)
Let us recall that in this presentation, A is commutative and the b + i 's commute with the elements of A.

2.3.3.

Restriction to the even part. Because of relations ( 17) and ( 18), we can consider in B the commutative subalgebra C generated over A by the elements c

+ k = (b + k ) 2 for 1 ≤ k ≤ n.
It follows from previous results that:

C = C[a + ij , c + k ] 1≤i<j<n,1≤k≤n , (47)
U(n + 0 ) = C[t ij ; d ij ] 1≤i<j≤n . (48) 2.4.
Enveloping algebra of the solvable Borel subsuperalgebra b + .

2.4.1.

Definition of b + . The Cartan Lie subalgebra h of g 0 is the Lie subalgebra with basis (k i ) 1≤i≤n , which is abelian by relation (31). The positive Borel subsuperalgebra b + is defined as the subsuperalgebra of osp(1, 2n) generated by n + and h. We have clearly:

b + = n + ⊕ h = b + 0 ⊕ n + 1
, with b + 0 := n + 0 ⊕ h, where the even part b + 0 is the positive Borel Lie subalgebra of g 0 sp(2n).

Description of U(b +

) as an iterated polynomial algebra. We denote by U + b the enveloping algebra U(b + ) of b + . The commutation relations of the k i 's with the generators of U + given by formulas (31) to (33) allow an easy description of U + b as the following iterated skew polynomial algebra:

U + b = U + [k 1 , ∆ 1 ][k 2 , ∆ 2 ] • • • [k n , ∆ n ], (49)
where the derivations ∆ i vanish on all k j 's and satisfy ∆ i (b

+ i ) = b + i and ∆ i (b + j ) = 0 for any 1 ≤ j = i ≤ n, ∆ i (a + ij ) = a + ij and ∆ i (t ij ) = t ij for 1 ≤ i < j ≤ n, ∆ i (a + ji ) = a + ji and ∆ i (t ji ) = -t ji for 1 ≤ j < i ≤ n, ∆ i (a + k ) = 0 and ∆ i (t k ) = 0 for k = i, = i.

2.4.3.

Restriction to the even part. It follows from previous results that, with notations (49) and (48):

U(b + 0 ) = U(n + 0 )[k i ; ∆ i ] 1≤i≤n = C[t ij ; d ij ] 1≤i<j≤n [k i ; ∆ i ] 1≤i≤n . (50)
3. Proof of the main theorem 3.1. Some general technical results. We will use the following easy lemmas.

3.1.1. Lemma. Let A be a commutative C-algebra, a an element of A, and R the iterated skew polynomial algebra generated over A by two generators x 1 and x 2 commuting with the elements of A and with relation x 1 x 2 + x 2 x 1 = 2a. Then the element y = x 1 x 2 -a satisfies x 1 y = -yx 1 and x 2 y = -yx 2 .

Proof. Obvious calculation.

3.1.2. Lemma. Let A be a commutative C-algebra. Let (a ij ) 1≤i<j≤n be a family of elements of A, and R the iterated skew polynomial algebra generated over A by n generators x 1 , x 2 , . . . , x n commuting with the elements of A and with relations x i x j + x j x i = 2a ij for all 1 ≤ i < j ≤ n. Then the n -1 elements y i = x 1 x i -a 1i for any 2 ≤ i ≤ n satisfy:

x 1 y i = -y i x 1 and y i y j + y j y i = 2(a 1i a 1j -a ij x 2 1 ). Proof. The first identity follows from previous lemma. For the second one, we observe firstly that:

x j y i = x j x 1 x i -x j a 1i = (-x 1 x j + 2a 1j )x i -a 1i x j = -x 1 x j x i + 2a 1j x i -a 1i x j = x 1 x i x j -2a ij x 1 + 2a 1j x i -a 1i x j = (x 1 x i -a 1i )x j -2a ij x 1 + 2a 1j x i = y i x j -2a ij x 1 + 2a 1j x i .
We use this identity and Lemma 3.1.1 to calculate:

y j y i = (x 1 x j -a 1j )y i = x 1 (y i x j -2a ij x 1 + 2a 1j x i ) -a 1j y i = -y i x 1 x j -2a ij x 2 1 + 2a 1j x 1 x i -a 1j y i = -y i (x 1 x j -a 1j ) -2a 1j y i -2a ij x 2 1 + 2a 1j x 1 x i = -y i y j -2a 1j y i -2a ij x 2 1 + 2a 1j x 1 x i = -y i y j + 2a 1j (x 1 x i -y i ) -2a ij x 2
1 . We conclude that y j y i = -y i y j + 2a 1j a 1i -2a ij x 2 1 . 3.2. Enveloping skewfield of the Lie subsuperalgebra n + .

3.2.1. Notations. Recalling the description (44), (45), (46) of U + , we introduce the skewfields of rational functions:

K := Frac A = C(a + ij ) 1≤i<j≤n , (51)
L := Frac B = K(b + i ; σ i , δ i ) 1≤i≤n , (52)
F := Frac U + = L(t ij ; d ij ) 1≤i<j≤n . (53)
3.2.2. Notations. We define the following families of elements of B:

y 1 := b + 1 , x 1,i := b + i , u 1,i,j := a + i,j for all 1 ≤ i < j ≤ n, (54)
and more generally for any 1 ≤ k ≤ n:

     u k,i,j := u k-1,k-1,i u k-1,k-1,j -u k-1,i,j y 2 k-1 for all k ≤ i < j ≤ n, x k,i := x k-1,k-1 x k-1,i -u k-1,k-1,i
for any k ≤ i ≤ n, y k := x k,k .

(55) y 1 y 2 = -y 2 y 1 , y 1 y i = y i y 1 , y 2 y i = y 2 y 1 , for any 3 ≤ i ≤ n, y 3 y 4 = -y 4 y 3 , y 3 y i = y i y 3 , for any 5 ≤ i ≤ n,

We deduce that Q Frac ( O(C 2 ) ⊗ T n-2 ) where T n-2 is the subalgebra of Q generated by the elements (y i ) 3≤i≤n . Applying the same reduction on the generators of T n-2 , we conclude by iteration and using Proposition 1.1.

3 that Q Frac ( O(C n )).
3.2.5. Lemma. For any 1 ≤ i ≤ n -1 and any 1 ≤ m ≤ n, we have:

[t i , u m,i,i+1 ] = x 2 m,i (63)
[t i , u m,k,i+1 ] = u m,k,i for m ≤ k < i ≤ n, (64)
[t i , u m,i+1, ] = u m,i, for m < i + 1 < ≤ n, (65)
[t i , u m,i+1, ] = 0 for m = i + 1 < ≤ n, (66)
[t i , u m,k, ] = 0 for k = i + 1 and = i + 1, ( 67
) [t i , x m,i+1 ] = x m,i for i + 1 > m, ( 68
) [t i , x m,i+1 ] = 0 for i + 1 = m, (69)
[t i , x m,k ] = 0 for k = i + 1. (70)
Proof. We study the action on B of the Chevalley generators t 1 , t 2 , . . . , t n-1 defined in 2.2. Let us recall that, by (38) we have:

[t i , b + i+1 ] = b + i and [t i , b + k ] = 0 if k = i + 1. (71)
Similarly by ( 23) to (27), we obtain:

[t i , a + i,i+1 ] = (b + i ) 2 , ( 72
) [t i , a + k,i+1 ] = a + k,i for 1 ≤ k < i ≤ n, (73)
[t i , a + i+1, ] = a + i, for i + 1 < ≤ n, (74)
[t i , a + k, ] = 0 for k = i + 1 and = i + 1.

Then for fixed i, we prove the assertions of the lemma by induction on m. By (54), assertion (71) is the case m = 1 of assertions (68) to (70), and (72) to (75) are the case m = 1 of assertions (63) to (67). Suppose that the eight assertions are satisfied to some rank m ≥ 1. By (55), we have:

[t i , u m+1,k, ] = [t i , u m,m,k ]u m,m, + u m,m,k [t i , u m,m, ] -[t i , u m,k, ]x 2 m,m -u m,k, [t i , x 2 m,m
]. It follows from (69) and (70) that the fourth term is zero. Then using induction assumptions (63) to (67), tedious but straightforward calculations prove that these five assertions remain valid at the rank m + 1. Similarly by (55) we have:

[t i , x m+1,k ] = [t i , x m,m ]x m,k + x m,m [t i , x m,k ] -[t i , u m,m,k ].
It follows from ( 69) and (70) that the first term is zero. Then using induction assumptions (63) to (70), it is easy to check that: [t i , x m+1,i+1 ] equals 0 if Proof. It is clear by (93) that Frac P Frac P and H Frac (W ⊗ P ). By tedious but simple calculations that we leave to the reader, we have:

y y m = -y m y , [h , y m ] = 0 and [h , y ] = y (94)
for all 1 ≤ = m ≤ n. We conclude with point (ii) of 1.2.1.

H 0 := Frac U(b + 0 ) = F 0 (k 1 ; ∆ 1 ) • • • (k n ; ∆ n ), (95)
where F 0 is described by Proposition 3.3.3.

3.5.2. Proposition. We have H 0 Frac (W ⊗P 0 ), where P 0 is the subalgebra of P generated by (y 1 ) 2 , . . . , (y n ) 2 , h 1 , . . . , h n , which is isomorphic to the enveloping algebra U n (C) defined in point (i) of 1.2.1.

Proof. By Proposition 3.3.3 and (92), the subfield of H 0 generated over F 0 by h 1 , . . . , h n is equal to H 0 . Since Frac P 0 = Q 0 by (93), the result follows from Proposition 3.3.3. 3.6. Final observation. A thorough reading of the previous proofs shows that the isomorphisms we obtain at the level of the skewfields of fractions are already realized at the level of the localization of the algebras under consideration by the multiplicative set generated by the n elements y 1 , . . . , y n . Since these elements are homogeneous, this result is consistent with the notion of skewfield of fractions in the category of superalgebras.

  and its center is a purely transcendental extension of dimension n over C.(ii) Frac U(b + ) Frac A n(n-1)/2 (C) ⊗ A n (C) , and its center is C.

1 . 1 .

 11 Analogues of Weyl algebras for polynomial superalgebras 1.Polynomial superalgebras.

  (with indices taken modulo 2). This gives rise to a structure of superalgebra on O(C n ). The algebras O(C n ) being noetherian domains, they admit skew fields of fractions. The following proposition gives an alternative form of O(C n ) in terms of planes O(C 2 ) up to rational equivalence. 1.1.3. Proposition. For any integer n ≥ 2, we have:

3. 4 . 6 . 3 . 5 .

 4635 Corollary. We have H Frac (A n(n-1)/2 (C) ⊗ A n (C)), with center C. Proof. We have already observed in Corollary 3.2.12 that W A n(n-1)/2 (C) and in Remark 1.2.4 that Frac A n (C) Frac U n (C). Then the isomorphism H Frac (A n(n-1)/2 (C) ⊗ A n (C)) follows from previous proposition. Using the usual notation D m (E) for the skewfield of fractions of a Weyl algebra A m (E) over a skewfield E, we have H D n(n-1) (E) for E = Frac A n (C). Its center satisfies Z(H) Z(D n(n-1) (E)) = Z(E), and Z(E) = C by Proposition 3.3.1 of [18]. 3.4.7. Conclusion. The second assertion of the main theorem is proved. Enveloping skewfield of the Lie subalgebra b + 0 . 3.5.1. Notations. In order to determine the skewfield of fractions of the enveloping algebra of the even part b + 0 of b + described in (50), we denote:

3. 5 . 3 .

 53 Conclusion. Because of the isomorphism FracU n (C) Frac A n (C) observed in Remark 1.2.4 and the canonical isomorphismA n(n+1)/2 (C) A n(n-1)/2 (C) ⊗ A n (C), the previous proposition proves that: Frac U(b + 0 ) Frac A n(n+1)/2 (C).Then we retrieve in the context of classical Gelfand Kirillov property the usual description of the enveloping skewfield of the solvable Borel subalgebra of the Lie algebra sp(2n) as the Weyl skewfield D n(n+1)/2 (C).

By straightforward calculations using (29), (35), (36), (37), the elements t 1 , . . . , t n defined in (34) satisfy the following Serre identities which already appear as (2.11) in [START_REF] Palev | The quantum superalgebra Uq[osp(1|2n)]: deformed para-Bose operators and root of unity representations[END_REF]: [START_REF] Musson | On the Goldie quotient ring of the enveloping algebra of a classical simple Lie superalgebra[END_REF], n + contains the elements a + ij for 1 ≤ i < j ≤ n. It also follows inductively from (28) and (36) that n + contains the elements t ij for 1 ≤ i < j ≤ n. Hence:

where n + 1 is the C-vector space with basis (b + i ) 1≤i≤n , and n + 0 sp + (2n) is the Lie subalgebra of g 0 sp(2n) with basis (c + i , a + ij , t ij ) 1≤i<j≤n . Then: dim n + 0 = dim sp + (2n) = n 2 and dim n + 1 = n. Let us observe that relations [START_REF] Richard | Hochschild homology and cohomology of some classical and quantum noncommutative polynomial algebras[END_REF], [START_REF] Richard | Équivalence rationnelle d'algèbres polynomiales classiques et quantiques[END_REF] and [START_REF] Wambst | Complexes de Koszul quantiques[END_REF] imply that the Lie subalgebra with basis (c + i , a + ij ) 1≤i<j≤n in n + 0 is abelian of dimension n(n + 1)/2.

Description of U(n +

) as an iterated skew polynomial algebra. We denote by U + the enveloping algebra U(n + ) of n + . By PBW theorem U + is generated as an associative algebra over C by the n 2 elements b + i , a + ij and t ij for 1 ≤ i < j ≤ n. The relations detailed in 2.1 allow us to describe U + as an iterated skew polynomial extension. We start with A := C[a + ij ] 1≤i<j≤n , which is by [START_REF] Wambst | Complexes de Koszul quantiques[END_REF] a commutative polynomial algebra in n(n -1)/2 variables. Then we set:

where σ j is the automorphism of B j-1 and δ j is the σ j -derivation of B j-1 defined by σ j (a

, in order to translate relations [START_REF] Premet | Modular Lie algebras and the Gelfand-Kirillov conjecture[END_REF] and [START_REF] Richard | Équivalence rationnelle d'algèbres polynomiales classiques et quantiques[END_REF]. At the last step we denote:

Then we add the generators t ij for 1 ≤ i < j ≤ n inductively:

It is clear that we have with notation (42):

The elements y 1 , y 2 , . . . , y n of B satisfy the relations:

(ii) For any 1 ≤ k ≤ n, the elements u k,i,j with k ≤ i < j ≤ n are in the commutative subalgebra A[y 2 1 , y 2 2 , . . . , y 2 k-1 ] of B, Proof. By iterative application of Lemma 3.1.2, we have for any 1 ≤ k ≤ n:

We deduce by induction on that, for any 1 ≤ ≤ n:

Point (i) of the lemma follows obviously from these relations and point (ii) follows by iterative definition (55) of the u k,i,j 's.

Proposition. (i)

The subalgebra of B generated over C by y 1 , . . . , y n is isomophic to

, where all entries of the symmetric matrix Λ = (λ ij ) 1≤i,j≤n are equal to 1 except

where each a + ij commutes with the elements of Q. Proof. By induction from relations (55), we prove that for any 1 ≤ k ≤ n,

where the term r k-1 lies in the subalgebra of B k-1 generated by y 1 , . . . , y k-1 and the a + ij 's for 1 ≤ i < j ≤ k. It follows that the skewfield L defined by (52) is also the skewfield of rational functions generated over K by y 1 , . . . , y n . Then the commutation relations of Lemma 3.2.3 imply point (i) and assertion (60) of point (ii). In order to prove assertion (61), we suppose that n ≥ 3 and define: y 1 = y 1 , y 2 = y 2 , y 3 = y 1 y 3 and y i = y i for any 4 ≤ i ≤ n. It is clear that the subfield of Q generated by the elements (y i ) 1≤i≤n is equal to Q. Moreover by straightforward calculations, we have: m + 1 = i + 1 and equals x m+1,i if i + 1 > m + 1, and [t i , x m+1,k ] = 0 for any k = i + 1. This finishes the proof.

3.2.6. Corollary. Each generator t ij for 1 ≤ i < j ≤ n commutes with each generator y m for 1 ≤ m ≤ n.

Proof. By (69) and (70), the Chevalley generators t 1 , t 2 , . . . , t n-1 commute with y m = x m,m for any 1 ≤ m ≤ n. By (28) and (36), each generator t ij for 1 ≤ i < j ≤ n is an iterated bracket of t 1 , t 2 , . . . , t n-1 .

3.2.7. Proposition. The skewfield of fractions of U + satisfies:

where the generators t ij commute with all elements of Q, and the derivations Proof. This is just a summary of assertions ( 53) and (60), Proposition 3.2.4 and Corollary 3.2.6.

3.2.8. Lemma. Denote by (q i,j ) 1≤i<j≤n the elements of F defined by:

Then the q i,j 's commute with each other and with all elements of Q, and the subfield of L generated over Q by the (q i,j ) 1≤i<j≤n is equal to L.

Proof. The first assertion follows from Corollary 3.2.3 and third relation in (59). To prove the second assertion, we observe directly from definition (55) that for any 1 ≤ i < j ≤ n, we have:

, where the element e i-1,j lies in the subalgebra E i-1 generated by y 2 1 , y 2 2 , . . . , y 2 i-1

and the a + k, 's for 1 ≤ k ≤ i -1. Then we can write:

This proves the rest of the lemma by induction on i.

3.2.9. Lemma. For any 1 ≤ i < j ≤ n and any 1 ≤ k < ≤ n, we have:

Proof. We proceed by induction on j -i. For j = i + 1, the identities directly follows from Lemma 3.2.5. For the general case, we deduced from (28) and (34) that t i,j+1 = [t i,j , t j,j+1 ] = [t i,j , t j ]. Then we have [t i,j+1 , q k, ] = [[t i,j , t j ], q k, ] = [t i,j , [t j , q k, ]] -[t j , [t i,j , q k, ]] for any 1 ≤ k < ≤ n, and we finish the proof by induction considering successively the different cases for the indices (i, j) and (k, ).

3.2.10. Corollary. The elements (q i,j ) 1≤i<j≤n defined in Lemma 3.2.8 and the elements (p i,j ) 1≤i<j ≤ n defined inductively by: p 1,j := t 1,j and p i,j := t i,j -i-1 m=1 q m,i p m,j (82)

Proof. Follows obviously from Lemma 3.2.9 by induction on i.

3.2.11. Proposition. We have:

Proof

n ). Proof. By Corollary 3.2.10, the generators p ij and q ij commute with all elements of Q and the subalgebra W of F they generate is isomorphic to the usual Weyl algebra A n(n-1)/2 . Then the theorem follows from Proposition 3.2.11 and assertion (61) of Proposition 3.2.4.

Conclusion.

We have proved the first assertion of the main theorem.

3.3.

Enveloping skewfield of the Lie subalgebra n + 0 . 3.3.1. Notations. In order to determine the skewfield of fractions of the enveloping algebra of the even part n + 0 of n + described in (48), we complete the notations (51), ( 52), ( 53) and (61) by defining:

3.3.2. Lemma. We have the equality L 0 = K(y 2 1 , y 2 2 , . . . , y 2 n ). Proof. Returning to notations (55) and using (59) and (58), we check by straightforward calculations that:

Using this identity and point (ii) of Lemma 3.2.3, we prove that y

, and :

for any 2 ≤ i ≤ n, where the term v i-1 lies in the subalgebra generated over A by y 2 1 , y 2 2 , . . . , y 2 i-1 . Then the result follows by induction on i.

Proof. It is clear by previous lemma that L 0 = Q 0 (a ij ) 1≤i<j≤n . Then the calculations in the proof of Lemma 3.2.8 show that the q i,j 's are elements of L 0 , and moreover that the subfield of L 0 generated over Q 0 by (q ij ) 1≤i<j≤n is equal to L 0 . Finally we deduce from (82) that the skew subfield of F 0 generated over L 0 by (p ij ) 1≤i<j≤n is equal to F 0 .

Conclusion.

The previous proposition showed that:

. Hence we retrieve in the context of the classical Gelfand-Kirillov property the usual description of the enveloping skewfield of the nilpotent Lie algebra sp + (2n) as a Weyl skewfield D n(n-1)/2 over a purely transcendental extension of dimension n over C.

3.4.

Enveloping skewfield of the Lie subsuperalgebra b + . Resuming all notations of the previous section and using (49) and (53), we set:

Our first step is to determine the action of the generators k 1 , . . . , k n of the Cartan subalgebra h on the generators y i , p ij , q ij (1 ≤ i < j ≤ n) introduced in the description of F in Proposition 3.2.11.

3.4.1. Lemma. For any 1 ≤ ≤ n, we have the following identities:

Proof. We start with the action of k on b + i and a + i,j for any 1 ≤ i < j ≤ n given by relations (32) and (33). Using the iterative definitions (54) and (55), we obtain for 1 ≤ m < the identities:

Finally for m > : [k , x m,i ] = 2 m--1 x m,i and [k , u m,i,j ] = 2 m-u m,i,j . Assertion (87) follows trivially from these relations since y m = x m,m . Moreover, for 1 ≤ i, j ≤ n, we have by definition (77) and (82):

We deduce (88) applying case by case the previous identities.

3.4.2.

Notation. An obvious observation arising from (88) is:

These relations lead us to introduce the following elements of H:

which satisfy the relations described in the following lemma.

3.4.3. Lemma. For all 1 ≤ , m ≤ n and 1 ≤ i < j ≤ n, we have:

Proof. By (92) we have:

and j = , all brackets appearing in the right hand are zero by Corollary 3.2.10 and relation (88). If i = , the equality becomes [h , q ,j ] = [k , q ,j ] + q ,j [p ,j , q ,j ] = 0. If j = , we obtain [h , q i, ] = [k , q i, ] -q i, [p i, , q i, ] = 0. The calculations are similar for p i,j . It is trivial that [(p i,j q i,j ), (p i ,j q i ,j )] = 0 for any 1 ≤ i < j ≤ n and 1 ≤ i < j ≤ n. Then [h , h m ] = 0 follows from the commutativity of the Cartan subalgebra and observation (91). By Proposition 3.2.11, the generators y 1 , . . . , y n commute with all p i,j 's and q i,j 's; then the last identity of the lemma is clear.

3.4.4. Proposition. Let W be the subalgebra of F generated over C by the n(n -1) elements p ij and q ij . Let P be the subalgebra of H generated over C by y 1 , . . . , y n and h 1 , . . . , h n . Then H Frac (W ⊗ P ). Let P be the subalgebra of H generated over C by y 1 , . . . , y n and h 1 , . . . , h n . Then P U n (C) and H Frac (W ⊗ P ).