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GRANDMA is a world-wide collaboration with the primary scientific goal of studying gravitational-wave sources, discovering their electromagnetic counterparts and characterizing their emission. GRANDMA involves astronomers, astrophysicists, gravitational-wave physicists, and theorists. GRANDMA is now a truly global network of telescopes, with (so far) 30 telescopes in both hemispheres. It incorporates a citizen science programme (Kilonova-Catcher) which constitutes an opportunity to spread the interest in time-domain astronomy. The telescope network is an heterogeneous set of already-existing observing facilities that operate coordinated as a single observatory. Within the network there are wide-field imagers that can observe large areas of the sky to search for optical counterparts, narrow-field instruments that do targeted searches within a predefined list of host-galaxy candidates, and larger telescopes that are devoted to characterization and follow-up of the identified counterparts. Here we present an overview of GRANDMA after the third observing run of the LIGO/VIRGO gravitational-wave observatories in $2019-2020$ and its ongoing preparation for the forthcoming fourth observational campaign (O4). Additionally, we review the potential of GRANDMA for the discovery and follow-up of other types of astronomical transients., Comment: Submitted to the Proceedings of the SPIE, Astronomical Telescopes and Instrumentation 2022

We present here the Calar Alto Schmidt-Lemaitre Telescope (CASTLE) concept, a technology demonstrator for curved detectors, that will be installed at the Calar Alto Observatory (Spain). This telescope has a wide field of view (2.36x1.56 deg^2) and a design, optimised to generate a Point Spread Function with very low level wings and reduced ghost features, which makes it considerably less susceptible to several systematic effects usually affecting similar systems. These characteristics are particularly suited to study the low surface brightness Universe. CASTLE will be able to reach surface brightness orders of magnitude fainter than the sky background level and observe the extremely extended and faint features around galaxies such as tidal features, stellar halos, intra-cluster light, etc. CASTLE will also be used to search and detect astrophysical transients such as gamma ray bursts (GRB), gravitational wave optical counterparts, neutrino counterparts, etc. This will increase the number of precisely localized GRBs from 20% to 60% (in the case of Fermi/GMB GRBs)., Comment: 29 pages, 11 figures

The Nearby Supernova Factory has made spectrophotometric observations of Type Ia supernovae since $2004$. This work presents an interim version of the data produced, including $210$ supernovae observed between $2004$ and $2013$., Comment: 5 pages

We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $\Gamma$ be a finite index subgroup of $\mathrm{SL}(2,\mathbb{Z})$ with an action on a complex simple Lie algebra $\mathfrak g$, which can be extended to $\mathrm{SL}(2,\mathbb{C})$. We show that the Lie algebra of the corresponding $\mathfrak{g}$-valued modular forms is isomorphic to the extension of $\mathfrak{g}$ over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups $\Gamma(N), \, N\leq 6$ are considered in more details in relation to the classical results of Klein and Fricke and the celebrated Markov Diophantine equation. We finish with a brief discussion of the extensions and representations of these Lie algebras., Comment: A revised and substantially extended version

Type Ia Supernovae (SNe Ia) are widely used to measure the expansion of the Universe. Improving distance measurements of SNe Ia is one technique to better constrain the acceleration of expansion and determine its physical nature. This document develops a new SNe Ia spectral energy distribution (SED) model, called the SUpernova Generator And Reconstructor (SUGAR), which improves the spectral description of SNe Ia, and consequently could improve the distance measurements. This model is constructed from SNe Ia spectral properties and spectrophotometric data from The Nearby Supernova Factory collaboration. In a first step, a PCA-like method is used on spectral features measured at maximum light, which allows us to extract the intrinsic properties of SNe Ia. Next, the intrinsic properties are used to extract the average extinction curve. Third, an interpolation using Gaussian Processes facilitates using data taken at different epochs during the lifetime of a SN Ia and then projecting the data on a fixed time grid. Finally, the three steps are combined to build the SED model as a function of time and wavelength. This is the SUGAR model. The main advancement in SUGAR is the addition of two additional parameters to characterize SNe Ia variability. The first is tied to the properties of SNe Ia ejecta velocity, the second is correlated with their calcium lines. The addition of these parameters, as well as the high quality the Nearby Supernova Factory data, makes SUGAR an accurate and efficient model for describing the spectra of normal SNe Ia as they brighten and fade. The performance of this model makes it an excellent SED model for experiments like ZTF, LSST or WFIRST., Comment: 25 pages, 27 figures

We present a new approach to determine the rational solutions of the higher order Painleve equations associated to periodic dressing chain systems. We obtain new sets of solutions, giving determinantal representations indexed by specific Maya diagrams in the odd case or universal characters in the even case.

Type Ia supernova cosmology depends on the ability to fit and standardize observations of supernova magnitudes with an empirical model. We present here a series of new models of Type Ia Supernova spectral time series that capture a greater amount of supernova diversity than possible with the models that are currently customary. These are entitled SuperNova Empirical MOdels (\textsc{SNEMO}\footnote{https://snfactory.lbl.gov/snemo}). The models are constructed using spectrophotometric time series from $172$ individual supernovae from the Nearby Supernova Factory, comprising more than $2000$ spectra. Using the available observations, Gaussian Processes are used to predict a full spectral time series for each supernova. A matrix is constructed from the spectral time series of all the supernovae, and Expectation Maximization Factor Analysis is used to calculate the principal components of the data. K-fold cross-validation then determines the selection of model parameters and accounts for color variation in the data. Based on this process, the final models are trained on supernovae that have been dereddened using the Fitzpatrick and Massa extinction relation. Three final models are presented here: \textsc{SNEMO2}, a two-component model for comparison with current Type~Ia models; \textsc{SNEMO7}, a seven component model chosen for standardizing supernova magnitudes which results in a total dispersion of $0.100$~mag for a validation set of supernovae, of which $0.087$~mag is unexplained (a total dispersion of $0.113$~mag with unexplained dispersion of $0.097$~mag is found for the total set of training and validation supernovae); and \textsc{SNEMO15}, a comprehensive $15$ component model that maximizes the amount of spectral time series behavior captured., Comment: 51 page, 19 figures, accepted in ApJ

The exclusive reaction $\gamma p \to p K^+ K^-$ was studied in the photon energy range $3.0 - 3.8 \mbox{ GeV}$ and momentum transfer range $0.6<-t<1.3 \mbox{ GeV}^2$. Data were collected with the CLAS detector at the Thomas Jefferson National Accelerator Facility. In this kinematic range the integrated luminosity was approximately 20 pb$^{-1}$. The reaction was isolated by detecting the $K^+$ and the proton in CLAS, and reconstructing the $K^-$ via the missing-mass technique. Moments of the di-kaon decay angular distributions were extracted from the experimental data. Besides the dominant contribution of the $\phi$ meson in the $P$-wave, evidence for $S-P$ interference was found. The differential production cross sections $d\sigma/dt$ for individual waves in the mass range of the $\phi$ resonance were extracted and compared to predictions of a Regge-inspired model. This is the first time the $t$-dependent cross section of the $S$-wave contribution to the elastic $K^+K^-$ photoproduction has been measured., Comment: arXiv admin note: text overlap with arXiv:0907.1021

As part of an on-going effort to identify, understand and correct for astrophysics biases in the standardization of Type Ia supernovae (SNIa) for cosmology, we have statistically classified a large sample of nearby SNeIa into those located in predominantly younger or older environments. This classification is based on the specific star formation rate measured within a projected distance of 1kpc from each SN location (LsSFR). This is an important refinement compared to using the local star formation rate directly as it provides a normalization for relative numbers of available SN progenitors and is more robust against extinction by dust. We find that the SNeIa in predominantly younger environments are DY=0.163\pm0.029 mag (5.7 sigma) fainter than those in predominantly older environments after conventional light-curve standardization. This is the strongest standardized SN Ia brightness systematic connected to host-galaxy environment measured to date. The well-established step in standardized brightnesses between SNeIa in hosts with lower or higher total stellar masses is smaller at DM=0.119\pm0.032 mag (4.5 sigma), for the same set of SNeIa. When fit simultaneously, the environment age offset remains very significant, with DY=0.129\pm0.032 mag (4.0 sigma), while the global stellar mass step is reduced to DM=0.064\pm0.029 mag (2.2 sigma). Thus, approximately 70% of the variance from the stellar mass step is due to an underlying dependence on environment-based progenitor age. Standardization using only the SNeIa in younger environments reduces the total dispersion from 0.142\pm0.008 mag to 0.120\pm0.010 mag. We show that as environment ages evolve with redshift a strong bias on measurement of the dark energy equation of state parameters can develop. Fortunately, data to measure and correct for this effect is likely to be available for many next-generation experiments. [abstract shorten], Comment: Published in Astronomy and Astrophysics Rigault et al. 2020, A&A, 644, A176 | arxiv Rigault et al. 2018

Type Ia Supernovae (SNe Ia) are widely used to measure the expansion of the Universe. To perform such measurements the luminosity and cosmological redshift ($z$) of the SNe Ia have to be determined. The uncertainty on $z$ includes an unknown peculiar velocity, which can be very large for SNe Ia in the virialized cores of massive clusters. We determine which SNe Ia exploded in galaxy clusters. We then study how the correction for peculiar velocities of host galaxies inside the clusters improves the Hubble residuals. Using 145 SNe Ia from the Nearby Supernova Factory we found 11 candidates for membership in clusters. To estimate the redshift of a cluster we applied the bi-weight technique. Then, we use the galaxy cluster redshift instead of the host galaxy redshift to construct the Hubble diagram. For SNe Ia inside galaxy clusters the dispersion around the Hubble diagram when peculiar velocities are taken into account is smaller in comparison with a case without peculiar velocity correction, with a $wRMS=0.130\pm0.038$ mag instead of $wRMS=0.137\pm0.036$ mag. The significance of this improvement is 3.58 $\sigma$. If we remove the very nearby Virgo cluster member SN2006X ($z<0.01$) from the analysis, the significance decreases to 1.34 $\sigma$. The peculiar velocity correction is found to be highest for the SNe Ia hosted by blue spiral galaxies, with high local specific star formation rate and smaller stellar mass, seemingly counter to what might be expected given the heavy concentration of old, massive elliptical galaxies in clusters. As expected, the Hubble residuals of SNe Ia associated with massive galaxy clusters improve when the cluster redshift is taken as the cosmological redshift of the SN. This fact has to be taken into account in future cosmological analyses in order to achieve higher accuracy for cosmological redshift measurements. Here we provide an approach to do so., Comment: 13 pages, 11 figures, 2 tables, accepted for publication in A&A

Context. Observations of Type Ia supernovae (SNe Ia) can be used to derive accurate cosmological distances through empirical standardization techniques. Despite this success neither the progenitors of SNe Ia nor the explosion process are fully understood. The U-band region has been less well observed for nearby SNe, due to technical challenges, but is the most readily accessible band for high-redshift SNe. Aims. Using spectrophotometry from the Nearby Supernova Factory, we study the origin and extent of U-band spectroscopic variations in SNe Ia and explore consequences for their standardization and the potential for providing new insights into the explosion process. Methods. We divide the U-band spectrum into four wavelength regions {\lambda}(uNi), {\lambda}(uTi), {\lambda}(uSi) and {\lambda}(uCa). Two of these span the Ca H&K {\lambda}{\lambda} 3934, 3969 complex. We employ spectral synthesis using SYNAPPS to associate the two bluer regions with Ni/Co and Ti. Results. (1) The flux of the uTi feature is an extremely sensitive temperature/luminosity indicator, standardizing the SN peak luminosity to 0.116 $\pm$ 0.011 mag RMS. A traditional SALT2.4 fit on the same sample yields a 0.135 mag RMS. Standardization using uTi also reduces the difference in corrected magnitude between SNe originating from different host galaxy environments. (2) Early U-band spectra can be used to probe the Ni+Co distribution in the ejecta, thus offering a rare window into the source of lightcurve power. (3) The uCa flux further improves standardization, yielding a 0.086 $\pm$ 0.010 mag RMS without the need to include an additional intrinsic dispersion to reach {\chi}$^2$/dof $\sim$ 1. This reduction in RMS is partially driven by an improved standardization of Shallow Silicon and 91T-like SNe., Comment: 19 pages, 17 figures, accepted for publication in A&A

The scientific yield of current and future optical surveys is increasingly limited by systematic uncertainties in the flux calibration. This is the case for Type Ia supernova (SN Ia) cosmology programs, where an improved calibration directly translates into improved cosmological constraints. Current methodology rests on models of stars. Here we aim to obtain flux calibration that is traceable to state-of-the-art detector-based calibration. We present the SNIFS Calibration Apparatus (SCALA), a color (relative) flux calibration system developed for the SuperNova Integral Field Spectrograph (SNIFS), operating at the University of Hawaii 2.2 m (UH 88) telescope. By comparing the color trend of the illumination generated by SCALA during two commissioning runs, and to previous laboratory measurements, we show that we can determine the light emitted by SCALA with a long-term repeatability better than 1%. We describe the calibration procedure necessary to control for system aging. We present measurements of the SNIFS throughput as estimated by SCALA observations. The SCALA calibration unit is now fully deployed at the UH\,88 telescope, and with it color-calibration between 4000 {\AA} and 9000 {\AA} is stable at the percent level over a one-year baseline., Comment: 17 pages, 15 figures, accepted for publication by A&A in 8 of August 2017

Correction of Type Ia Supernova brightnesses for extinction by dust has proven to be a vexing problem. Here we study the dust foreground to the highly reddened SN 2012cu, which is projected onto a dust lane in the galaxy NGC 4772. The analysis is based on multi-epoch, spectrophotometric observations spanning 3,300 - 9,200 {\AA}, obtained by the Nearby Supernova Factory. Phase-matched comparison of the spectroscopically twinned SN 2012cu and SN 2011fe across 10 epochs results in the best-fit color excess of (E(B-V), RMS) = (1.00, 0.03) and total-to-selective extinction ratio of (RV , RMS) = (2.95, 0.08) toward SN 2012cu within its host galaxy. We further identify several diffuse interstellar bands, and compare the 5780 {\AA} band with the dust-to-band ratio for the Milky Way. Overall, we find the foreground dust-extinction properties for SN 2012cu to be consistent with those of the Milky Way. Furthermore we find no evidence for significant time variation in any of these extinction tracers. We also compare the dust extinction curve models of Cardelli et al. (1989), O'Donnell (1994), and Fitzpatrick (1999), and find the predictions of Fitzpatrick (1999) fit SN 2012cu the best. Finally, the distance to NGC4772, the host of SN 2012cu, at a redshift of z = 0.0035, often assigned to the Virgo Southern Extension, is determined to be 16.6$\pm$1.1 Mpc. We compare this result with distance measurements in the literature., Comment: 48 pages, 13 figures. Accepted for publication in The Astrophysical Journal. The spectral time series data presented in this article can be found at http://snfactory.lbl.gov/snf/data/

We show that the relatively small but coincident excesses observed around 2 TeV in the ATLAS Run 1 and Run 2 hadronic diboson searches --- when a cut on the number of tracks in the fat jets is not applied --- and the null results of all remaining high-mass diboson searches are compatible with the decay of a triboson resonance $R$ into $WZ$ plus an extra particle $X$. These decays can take place via new neutral ($Y^0$) or charged ($Y^\pm$) particles, namely $R \to Y^0 \, W$, with $Y^0 \to Z X$, or $R \to Y^\pm Z$, with $Y^\pm \to W X$. An obvious candidate for such intermediate particle is a neutral one $Y^0$, given a $3.9\sigma$ excess found at 650 GeV by the CMS Collaboration in searches for intermediate mass diboson resonances decaying to $ZV$, with $V=W,Z$. We discuss discovery strategies for triboson resonances with small modifications of existing hadronic searches., Comment: LaTeX 32 pages, 38 figures. v2: Figs. 6, 9 and 11 modified. v3: added VH search in hadronic final state. v4: final published version

We introduce a method for identifying "twin" Type Ia supernovae, and using them to improve distance measurements. This novel approach to Type Ia supernova standardization is made possible by spectrophotometric time series observations from the Nearby Supernova Factory (SNfactory). We begin with a well-measured set of supernovae, find pairs whose spectra match well across the entire optical window, and then test whether this leads to a smaller dispersion in their absolute brightnesses. This analysis is completed in a blinded fashion, ensuring that decisions made in implementing the method do not inadvertently bias the result. We find that pairs of supernovae with more closely matched spectra indeed have reduced brightness dispersion. We are able to standardize this initial set of SNfactory supernovae to 0.083 +/- 0.012 magnitudes, implying a dispersion of 0.072 +/- 0.010 magnitudes in the absence of peculiar velocities. We estimate that with larger numbers of comparison SNe, e.g, using the final SNfactory spectrophotometric dataset as a reference, this method will be capable of standardizing high-redshift supernovae to within 0.06-0.07 magnitudes. These results imply that at least 3/4 of the variance in Hubble residuals in current supernova cosmology analyses is due to previously unaccounted-for astrophysical differences among the supernovae, Comment: 37 pages, 9 figures, 5 tables. Accepted for publication in ApJ. Fixed typo in arXiv abstract

The non-topological, stationary and propagating, soliton solutions of the classical continuous Heisenberg ferromagnet equation are investigated. A general, rigorous formulation of the Inverse Scattering Transform for this equation is presented, under less restrictive conditions than the Schwartz class hypotheses and naturally incorporating the non-topological character of the solutions. Such formulation is based on a new triangular representation for the Jost solutions, which in turn allows an immediate computation of the asymptotic behaviour of the scattering data for large values of the spectral parameter, consistently improving on the existing theory. A new, general, explicit multi-soliton solution formula, amenable to computer algebra, is obtained by means of the matrix triplet method, producing all the soliton solutions (including breather-like and multipoles), and allowing their classification and description.

Previously we used the Nearby Supernova Factory sample to show that SNe~Ia having locally star-forming environments are dimmer than SNe~Ia having locally passive environments.Here we use the \constitution\ sample together with host galaxy data from \GALEX\ to independently confirm that result. The effect is seen using both the SALT2 and MLCS2k2 lightcurve fitting and standardization methods, with brightness differences of $0.094 \pm 0.037\ \mathrm{mag}$ for SALT2 and $0.155 \pm 0.041\ \mathrm{mag}$ for MLCS2k2 with $R_V=2.5$. When combined with our previous measurement the effect is $0.094 \pm 0.025\ \mathrm{mag}$ for SALT2. If the ratio of these local SN~Ia environments changes with redshift or sample selection, this can lead to a bias in cosmological measurements. We explore this issue further, using as an example the direct measurement of $H_0$. \GALEX{} observations show that the SNe~Ia having standardized absolute magnitudes calibrated via the Cepheid period--luminosity relation using {\textit{HST}} originate in predominately star-forming environments, whereas only ~50% of the Hubble-flow comparison sample have locally star-forming environments. As a consequence, the $H_0$ measurement using SNe~Ia is currently overestimated. Correcting for this bias, we find a value of $H_0^{corr}=70.6\pm 2.6\ \mathrm{km\ s^{-1}\ Mpc^{-1}}$ when using the LMC distance, Milky Way parallaxes and the NGC~4258 megamaser as the Cepheid zeropoint, and $68.8\pm 3.3\ \mathrm{km\ s^{-1}\ Mpc^{-1}}$ when only using NGC~4258. Our correction brings the direct measurement of $H_0$ within $\sim 1\,\sigma$ of recent indirect measurements based on the CMB power spectrum., Comment: 3 Figures ; Submitted to ApJ: Oct.~30, 2014 -- Accepted: Dec.~17, 2014

We develop a new framework for use in exploring Type Ia Supernova (SN Ia) spectra. Combining Principal Component Analysis (PCA) and Partial Least Square analysis (PLS) we are able to establish correlations between the Principal Components (PCs) and spectroscopic/photometric SNe Ia features. The technique was applied to ~120 supernova and ~800 spectra from the Nearby Supernova Factory. The ability of PCA to group together SNe Ia with similar spectral features, already explored in previous studies, is greatly enhanced by two important modifications: (1) the initial data matrix is built using derivatives of spectra over the wavelength, which increases the weight of weak lines and discards extinction, and (2) we extract time evolution information through the use of entire spectral sequences concatenated in each line of the input data matrix. These allow us to define a stable PC parameter space which can be used to characterize synthetic SN Ia spectra by means of real SN features. Using PLS, we demonstrate that the information from important previously known spectral indicators (namely the pseudo-equivalent width (pEW) of Si II 5972 / Si II 6355 and the line velocity of S II 5640 / Si II 6355) at a given epoch, is contained within the PC space and can be determined through a linear combination of the most important PCs. We also show that the PC space encompasses photometric features like B or V magnitudes, B-V color and SALT2 parameters c and x1. The observed colors and magnitudes, that are heavily affected by extinction, cannot be reconstructed using this technique alone. All the above mentioned applications allowed us to construct a metric space for comparing synthetic SN Ia spectra with observations., Comment: 22 pages, 26 figures, 3 tables, accepted for publication in MNRAS

Kim et al. (2013) [K13] introduced a new methodology for determining peak-brightness absolute magnitudes of type Ia supernovae from multi-band light curves. We examine the relation between their parameterization of light curves and Hubble residuals, based on photometry synthesized from the Nearby Supernova Factory spectrophotometric time series, with global host-galaxy properties. The K13 Hubble residual step with host mass is $0.013\pm 0.031$ mag for a supernova subsample with data coverage corresponding to the K13 training; at $\ll 1\sigma$, the step is not significant and lower than previous measurements. Relaxing the data coverage requirement the Hubble residual step with host mass is $0.045\pm 0.026$ mag for the larger sample; a calculation using the modes of the distributions, less sensitive to outliers, yields a step of 0.019 mag. The analysis of this article uses K13 inferred luminosities, as distinguished from previous works that use magnitude corrections as a function of SALT2 color and stretch parameters: Steps at $>2\sigma $ significance are found in SALT2 Hubble residuals in samples split by the values of their K13 $x(1)$ and $x(2)$ light-curve parameters. $x(1)$ affects the light-curve width and color around peak (similar to the $\Delta m_{15}$ and stretch parameters), and $x(2)$ affects colors, the near-UV light-curve width, and the light-curve decline 20 to 30 days after peak brightness. The novel light-curve analysis, increased parameter set, and magnitude corrections of K13 may be capturing features of SN~Ia diversity arising from progenitor stellar evolution., Comment: 17 pages, 6 figures. Accepted by Astrophysical Journal

(Abridged) We study the host galaxy regions in close proximity to Type Ia supernovae (SNe Ia) to analyze relations between the properties of SN Ia events and environments most similar to where their progenitors formed. We focus on local H\alpha\ emission as an indicator of young environments. The Nearby Supernova Factory has obtained flux-calibrated spectral timeseries for SNe Ia using integral field spectroscopy, allowing the simultaneous measurement of the SN and its immediate vicinity. For 89 SNe Ia we measure H\alpha\ emission tracing ongoing star formation within a 1 kpc radius around each SN. This constitutes the first direct study of the local environment for a large sample of SNe Ia also having accurate luminosity, color and stretch measurements. We find that SNe Ia with local H\alpha\ emission are redder by 0.036+/-0.017 mag, and that the previously-noted correlation between stretch and host mass is entirely driven by the SNe Ia coming from passive regions. Most importantly, the mean standardized brightness for SNe Ia with local H\alpha\ emission is 0.094+/-0.031 mag fainter than for those without. This offset arises from a bimodal structure in the Hubble residuals, that also explains the previously-known host-mass bias. We combine this bimodality with the cosmic star-formation rate to predict changes with redshift in the mean SN Ia brightness and the host-mass bias. This change is confirmed using high-redshift SNe Ia from the literature. These environmental dependences point to remaining systematic errors in SNe Ia standardization. The observed brightness offset is predicted to cause a significant bias in measurements of the dark energy equation of state. Recognition of these effects offers new opportunities to improve SNe Ia as cosmological probes - e.g. SNe Ia having local H\alpha\ emission are more homogeneous, having a brightness dispersion of 0.105+/-0.012 mag., Comment: accepted for publication in Section 3. Cosmology of A&A (The official date of acceptance is 30/08/2013)

In order to build a new concept of photomultiplier based on silicon technology, design and characterization of 5x5 arrays of a new generation of single photon avalanche diodes (SPAD) manufactured by ST-Microelectronics have been performed. Single photons sensitivity, dark noise and timing resolution of the SPAD- STM devices in several experimental conditions have been evaluated. Moreover, the effects arising from the multiple integration of many elements and the study of their common read-out have been deeply investigated., Comment: 23 pages, 15 figures, 2 tables

We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic Lie algebras. For automorphic Lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly. These algebras have a useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts., Comment: 21 pages, standard LaTeX2e, corrected typos, accepted for publication in CMP - Communications in Mathematical Physics

We discuss algebraic and analytic structure of rational Lax operators. With algebraic reductions of Lax equations we associate a reduction group - a group of twisted automorphisms of the corresponding infinite dimensional Lie algebra. We present a complete study of dihedral reductions for sl(2,C) Lax operators with simple poles and corresponding integrable equations. In the last section we give three examples of dihedral reductions for sl(N,C) Lax operators., Comment: 17 pages, submitted to Journal of Physics A: Mathematical and General