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In this work we state a version of the double extension for homogeneous quadratic Lie super algebras that includes even and odd cases. We prove that any indecomposable, non-simple and homogeneous quadratic Lie super algebra is obtained by means of this type of double extension. We also show that with this construction we can recover previously studied cases as well as some other which can not be recover with former versions of double extensions.

We aim to investigate how the local environment influences the star formation history (SFH) of galaxies residing in various large-scale environments. We categorise a sample of 9384 galaxies into the three primary large scale structures (voids, walls \& filaments, and clusters) and further classify them based on their local environment (as either "singlets" or group members), through a search of companion galaxies within sky-projected distances $\Delta r_p < 0.45$ Mpc and velocity differences $\Delta v < 160$ $\text{km s}^{-1}$. Subsequently, we explore these subsamples through SFH data from previous works. Throughout the study, galaxies are divided into long-timescale SFH galaxies (LT-SFH), which assemble their mass steadily along cosmic time, and short-timescale SFH galaxies (ST-SFH), which form their stars early. We then compare characteristic mass assembly look-back times. The distributions of mass assembly look-back times in ST-SFH galaxies are statistically different for singlets and groups. These differences are only found in LT-SFH galaxies when studying these distributions in stellar mass bins. Our results indicate that the large-scale environment is related to a delay in mass assembly of up to $\sim$2 Gyr, while this delay is $<$1 Gyr in the case of local environment. The effect of both kinds of environment is more significant in less massive galaxies, and in LT-SFHs. Our results are consistent with galaxies in groups assembling their stellar mass earlier than singlets, especially in voids and lower mass galaxies. Local environment plays a relevant role in stellar mass assembly times, although we find that large-scale structures also cause a delay in mass assembly, more so in the case of cluster galaxies., Comment: 7 pages, 4 figures. Accepted for publication in A&A

We state criteria for a nilpotent Lie algebra $\g$ to admit an invariant metric. We use that $\g$ possesses two canonical abelian ideals $\ide(\g) \subset \mathfrak{J}(\g)$ to decompose the underlying vector space of $\g$ and then we state sufficient conditions for $\g$ to admit an invariant metric. The properties of the ideal $\mathfrak{J}(\g)$ allows to prove that if a current Lie algebra $\g \otimes \Sa$ admits an invariant metric, then there must be an invariant and non-degenerate bilinear map from $\Sa \times \Sa$ into the space of centroids of $\g/\mathfrak{J}(\g)$. We also prove that in any nilpotent Lie algebra $\g$ there exists a non-zero, symmetric and invariant bilinear form. This bilinear form allows to reconstruct $\g$ by means of an algebra with unit. We prove that this algebra is simple if and only if the bilinear form is an invariant metric on $\g$.

In this work we give an inductive way to construct quadratic Hom-Lie algebras with twist maps in the centroid. We focus on those Hom-Lie algebras that are not Lie algebras. We prove that the twist map of a Hom-Lie algebra of this type must be nilpotent and the Hom-Lie algebra has trivial center. We also prove that there exists a maximal ideal containing the kernel and the image of the twist map. Then we state an inductive way to construct this type of Hom-Lie algebras -- similar to the double extension procedure for Lie algebras -- and prove that any indecomposable quadratic Hom-Lie algebra with nilpotent twist map in the centroid, which is not a Lie algebra, can be constructed using this type of double extension.

The WHT Enhanced Area Velocity Explorer (WEAVE) is a new, massively multiplexing spectrograph. This new instrument will be exploited to obtain high S/N spectra of $\sim$25000 galaxies at intermediate redshifts for the WEAVE Stellar Population Survey (WEAVE-StePS). We test machine learning methods for retrieving the key physical parameters of galaxies from WEAVE-StePS-like spectra using both photometric and spectroscopic information at various S/Ns and redshifts. We simulated $\sim$105000 galaxy spectra assuming SFH with an exponentially declining star formation rate, covering a wide range of ages, stellar metallicities, sSFRs, and dust extinctions. We then evaluated the ability of the random forest and KNN algorithms to correctly predict such parameters assuming no measurement errors. We checked how much the predictive ability deteriorates for different S/Ns and redshifts, finding that both algorithms still accurately estimate the ages and metallicities with low bias. The dispersion varies from 0.08-0.16 dex for ages and 0.11-0.25 dex for metallicity, depending on the redshift and S/N. For dust attenuation, we find a similarly low bias and dispersion. For the sSFR, we find a very good constraining power for star-forming galaxies, log sSFR$\gtrsim$ -11, where the bias is $\sim$ 0.01 dex and the dispersion is $\sim$ 0.10 dex. For more quiescent galaxies, with log sSFR$\lesssim$ -11, we find a higher bias, 0.61-0.86 dex, and a higher dispersion, $\sim$ 0.4 dex, for different S/Ns and redshifts. Generally, we find that the RF outperforms the KNN. Finally, the retrieved sSFR was used to successfully classify galaxies as part of the blue cloud, green valley, or red sequence. We demonstrate that machine learning algorithms can accurately estimate the physical parameters of simulated galaxies even at relatively low S/N=10 per angstrom spectra with available ancillary photometric information., Comment: 19 pages, 10 + 2 figures, 4 tables, accepted in A&A

Contrastive learning (CL) has emerged as a potent tool for building meaningful latent representations of galaxy properties across a broad spectrum of wavelengths, ranging from optical and infrared to radio frequencies. These representations facilitate a variety of downstream tasks, including galaxy classification, similarity searches, and parameter estimation, which is why they are often referred to as foundation models. In this study, we employ CL on the latest extended DR from CALIFA survey, which encompasses 895 galaxies with enhanced spatial resolution. We demonstrate that CL can be applied to IFU surveys, even with small training sets, to meaningful embedding where galaxies are well-separated based on their physical properties. We discover that the strongest correlations in the embedding space are observed with the EW of Ha morphology, stellar metallicity, age, stellar surface mass density, the [NII]/Ha ratio, and stellar mass, in descending order of correlation strength. Additionally, we illustrate the feasibility of unsupervised separation of galaxy populations along the SFMS, successfully identifying the BC and the RS in a two-cluster scenario, and the GV population in a three-cluster scenario. Our findings indicate that galaxy luminosity profiles have minimal impact on the construction of the embedding space, suggesting that morphology and spectral features play a more significant role in distinguishing between galaxy populations. Moreover, we explore the use of CL for detecting variations in galaxy population distributions across different environments, including voids, clusters, filaments and walls. Nonetheless, we acknowledge the limitations of the CL and our specific training set in detecting subtle differences in galaxy properties, such as the presence of an AGN or other minor scale variations that exceed the scope of primary parameters like stellar mass or morphology., Comment: 21 pages, 12 figures

We have learnt in the last decades that the majority of galaxies belong to high density regions interconnected in a sponge-like fashion. This large-scale structure is characterised by clusters, filaments, walls, where most galaxies concentrate, but also under-dense regions, called voids. The void regions and the galaxies within represent an ideal place for the study of galaxy formation and evolution as they are largely unaffected by the complex physical processes that transform galaxies in high-density environments. These void galaxies can hold the key as well to answer current challenges to the $\Lambda$CDM paradigm. The Calar Alto Void Integral-field Treasury surveY (CAVITY) is a Legacy project approved by the Calar Alto Observatory to obtain spatially resolved spectroscopic information of $\sim300$ void galaxies in the Local Universe (0.005 < z < 0.050) covering from -17.0 to -21.5 in $\rm r$ band absolute magnitude. It officially started in January 2021 and has been awarded 110 useful dark observing nights at the 3.5 m telescope using the PMAS spectrograph. Complementary follow-up projects including deep optical imaging, integrated, as well as resolved CO data, and integrated HI spectra, have joint the PMAS observations and naturally complete the scientific aim of characterising galaxies in cosmic voids. The extension data has been denominated CAVITY+. The data will be available to the whole community in different data releases, the first of which is planned for July 2024, and it will provide the community with PMAS data cubes for around 100 void galaxies through a user friendly, and well documented, database platform. We present here the survey, sample selection, data reduction, quality control schemes, science goals, and some examples of the scientific power of the CAVITY and CAVITY+ data., Comment: 19 pages, 11 figures. Accepted for publication in A&A - Replacement after A&A minor language edition

In this work we state a result that relates the cohomology groups of a Lie algebra $\mathfrak{g}$ and a current Lie algebra $\mathfrak{g} \otimes \mathcal{S}$, by means of a short exact sequence -- similar to the universal coefficients theorem for modules -- where $\mathcal{S}$ is a finite dimensional, commutative and associative algebra with unit over a field $\mathbb{F}$. Although this result can be applied to any Lie algebra, we determine the cohomology group of $\mathfrak{g} \otimes \mathcal{S}$, where $\mathfrak{g}$ is a semisimple Lie algebra.

Galaxy clusters are an essential tool to understand and constrain the cosmological parameters of our Universe. Thanks to its multi-band design, J-PAS offers a unique group and cluster detection window using precise photometric redshifts and sufficient depths. We produce galaxy cluster catalogues from the miniJPAS, which is a pathfinder survey for the wider J-PAS survey, using the PZWav algorithm. Relying only on photometric information, we provide optical mass tracers for the identified clusters, including richness, optical luminosity, and stellar mass. By reanalysing the Chandra mosaic of the AEGIS field, alongside the overlapping XMM-Newton observations, we produce an X-ray catalogue. The analysis reveals the possible presence of structures with masses of 4$\times 10^{13}$ M$_\odot$ at redshift 0.75, highlighting the depth of the survey. Comparing results with those from two other cluster catalogues, provided by AMICO and VT, we find $43$ common clusters with cluster centre offsets of 100$\pm$60 kpc and redshift differences below 0.001. We provide a comparison of the cluster catalogues with a catalogue of massive galaxies and report on the significance of cluster selection. In general, we are able to recover approximately 75$\%$ of the galaxies with $M^{\star} >$2$\times 10^{11}$ M$_\odot$. This study emphasises the potential of the J-PAS survey and the employed techniques down to the group scales., Comment: 15 pages, 11 figures, 5 tables. Submitted to A&A in December 19, 2023

We aim at developing a robust methodology for constraining the luminosity and stellar mass functions (LMFs) of galaxies by solely using data from multi-filter surveys and testing the potential of these techniques for determining the evolution of the miniJPAS LMFs up to $z\sim0.7$. Stellar mass and $B$-band luminosity for each of the miniJPAS galaxies are constrained using an updated version of the SED-fitting code MUFFIT, whose values are based on composite stellar population models and the probability distribution functions of the miniJPAS photometric redshifts. Galaxies are classified through the stellar mass versus rest-frame colour diagram corrected for extinction. Different stellar mass and luminosity completeness limits are set and parametrised as a function of redshift, for setting limits in our flux-limited sample ($r_\mathrm{SDSS}<22$). The miniJPAS LMFs are parametrised according to Schechter-like functions via a novel maximum likelihood method accounting for uncertainties, degeneracies, probabilities, completeness, and priors. Overall, our results point to a smooth evolution with redshift ($0.0510.7$)., Comment: 31 pages, 15 figures. Submitted to A&A

We present a new method for obtaining photometric redshifts (photo-z) for sources observed by multiple photometric surveys using a combination (conflation) of the redshift probability distributions (PDZs) obtained independently from each survey. The conflation of the PDZs has several advantages over the usual method of modelling all the photometry together, including modularity, speed, and accuracy of the results. Using a sample of galaxies with narrow-band photometry in 56 bands from J-PAS and deeper grizy photometry from the Hyper-SuprimeCam Subaru Strategic program (HSC-SSP), we show that PDZ conflation significantly improves photo-z accuracy compared to fitting all the photometry or using a weighted average of point estimates. The improvement over J-PAS alone is particularly strong for i>22 sources, which have low signal-to-noise ratio in the J-PAS bands. For the entire i<22.5 sample, we obtain a 64% (45%) increase in the number of sources with redshift errors |Dz|<0.003, a factor 3.3 (1.9) decrease in the normalised median absolute deviation of the errors (sigma_NMAD), and a factor 3.2 (1.3) decrease in the outlier rate compared to J-PAS (HSC-SSP) alone. The photo-z accuracy gains from combining the PDZs of J-PAS with a deeper broadband survey such as HSC-SSP are equivalent to increasing the depth of J-PAS observations by ~1.2--1.5 magnitudes. These results demonstrate the potential of PDZ conflation and highlight the importance of including the full PDZs in photo-z catalogues., Comment: 15 pages, 16 figures, 1 table. Accepted for publication in A&A

We present a minimal Landau theory of plastic-to-crystal phase transitions in which the key components are a multipole-moment order parameter that describes the orientational ordering of the constituent molecules, coupling between such order parameter and elastic strains, and thermal expansion. We illustrate the theory with the simplest non-trivial model in which the orientational ordering is described by a quadrupole moment, and use such model to calculate barocaloric effects in plastic crystals that are driven by hydrostatic pressure. The model captures characteristic features of plastic-to-crystal phase transitions, namely, large changes in volume and entropy at the transition, as well as the linear dependence of the transition temperature with pressure. We identify temperature regions in the barocaloric response associated with the individual plastic and crystal phases, and those involving the phase transition. Our model is in overall agreement with previous experiments in powdered samples of fullerite C$_{60}$, and predicts peak isothermal entropy changes of $\sim90 \,{\rm J K^{-1} kg^{-1}}$ and peak adiabatic temperature changes of $\sim35 \,{\rm K}$ under $0.60\,$GPa at $265\,$K in fullerite single crystals., Comment: 24 pages, 5 figures

Samples of galaxy clusters allow us to better understand the physics at play in galaxy formation and to constrain cosmological models once their mass, position (for clustering studies) and redshift are known. In this context, large optical data sets play a crucial role. We investigate the capabilities of the Javalambre-Physics of the Accelerating Universe Astrophysical Survey (J-PAS) in detecting and characterizing galaxy groups and clusters. We analyze the data of the miniJPAS survey, obtained with the JPAS-Pathfinder camera and covering $1$ deg$^2$ centered on the AEGIS field to the same depths and with the same 54 narrow band plus 2 broader band near-UV and near-IR filters anticipated for the full J-PAS survey. We use the Adaptive Matched Identifier of Clustered Objects (AMICO) to detect and characterize groups and clusters of galaxies down to $S/N=2.5$ in the redshift range $0.05

Quadratic Hom-Lie algebras with equivariant twist maps are studied. They are completely characterized in terms of a maximal proper ideal that contains the kernel of the twist map and a complementary subspace to it that is either 1-dimensional, or has the structure of a simple Lie algebra. It is shown how the analogue of the double extension construction works well for quadratic Hom-Lie algebras with equivariant twist maps and prove that any indecomposable and quadratic Hom-Lie algebra with equivariant and nilpotent twist map can be identified with such a double extension., Comment: Theorem A, page 19, is wrong. The only simple Lie algebra for which the statement of the theorem is valid is the simple Lie algebra of rank one

This paper is part of large effort within the J-PAS collaboration that aims to classify point-like sources in miniJPAS, which were observed in 60 optical bands over $\sim$ 1 deg$^2$ in the AEGIS field. We developed two algorithms based on artificial neural networks (ANN) to classify objects into four categories: stars, galaxies, quasars at low redshift ($z < 2.1)$, and quasars at high redshift ($z \geq 2.1$). As inputs, we used miniJPAS fluxes for one of the classifiers (ANN$_1$) and colours for the other (ANN$_2$). The ANNs were trained and tested using mock data in the first place. We studied the effect of augmenting the training set by creating hybrid objects, which combines fluxes from stars, galaxies, and quasars. Nevertheless, the augmentation processing did not improve the score of the ANN. We also evaluated the performance of the classifiers in a small subset of the SDSS DR12Q superset observed by miniJPAS. In the mock test set, the f1-score for quasars at high redshift with the ANN$_1$ (ANN$_2$) are $0.99$ ($0.99$), $0.93$ ($0.92$), and $0.63$ ($0.57$) for $17 < r \leq 20$, $20 < r \leq 22.5$, and $22.5 < r \leq 23.6$, respectively, where $r$ is the J-PAS rSDSS band. In the case of low-redshift quasars, galaxies, and stars, we reached $0.97$ ($0.97$), $0.82$ ($0.79$), and $0.61$ ($0.58$); $0.94$ ($0.94$), $0.90$ ($0.89$), and $0.81$ ($0.80$); and $1.0$ ($1.0$), $0.96$ ($0.94$), and $0.70$ ($0.52$) in the same r bins. In the SDSS DR12Q superset miniJPAS sample, the weighted f1-score reaches 0.87 (0.88) for objects that are mostly within $20 < r \leq 22.5$. Finally, we estimate the number of point-like sources that are quasars, galaxies, and stars in miniJPAS.

The upcoming new generation of optical spectrographs on four-meter-class telescopes will provide valuable opportunities for forthcoming galaxy surveys through their huge multiplexing capabilities, excellent spectral resolution, and unprecedented wavelength coverage. WEAVE is a new wide-field spectroscopic facility mounted on the 4.2 m William Herschel Telescope in La Palma. WEAVE-StePS is one of the five extragalactic surveys that will use WEAVE during its first five years of operations. It will observe galaxies using WEAVE MOS (~950 fibres across a field of view of ~3 deg2 on the sky) in low-resolution mode (R~5000, spanning the wavelength range 3660-9590 AA). WEAVE-StePS will obtain high-quality spectra (S/N ~ 10 per AA at R~5000) for a magnitude-limited (I_AB = 20.5) sample of ~25,000 galaxies, the majority selected at z>=0.3. The survey goal is to provide precise spectral measurements in the crucial interval that bridges the gap between LEGA-C and SDSS data. The wide area coverage of ~25 deg2 will enable us to observe galaxies in a variety of environments. The ancillary data available in each observed field (including X-ray coverage, multi-narrow-band photometry and spectroscopic redshift information) will provide an environmental characterisation for each observed galaxy. This paper presents the science case of WEAVE-StePS, the fields to be observed, the parent catalogues used to define the target sample, and the observing strategy chosen after a forecast of the expected performance of the instrument for our typical targets. WEAVE-StePS will go back further in cosmic time than SDSS, extending its reach to encompass more than ~6 Gyr, nearly half of the age of the Universe. The spectral and redshift range covered by WEAVE-StePS will open a new observational window by continuously tracing the evolutionary path of galaxies in the largely unexplored intermediate-redshift range., Comment: 15 pages, 9 figures, A&A in press

Studies indicate strong evidence of a scaling relation in the local Universe between the supermassive black hole mass ($M_\rm{BH}$) and the stellar mass of their host galaxies ($M_\star$). They even show similar histories across cosmic times of their differential terms: star formation rate (SFR) and black hole accretion rate (BHAR). However, a clear picture of this coevolution is far from being understood. We select an X-ray sample of active galactic nuclei (AGN) up to $z=2.5$ in the miniJPAS footprint. Their X-ray to infrared spectral energy distributions (SEDs) have been modeled with CIGALE, constraining the emission to 68 bands. For a final sample of 308 galaxies, we derive their physical properties (e.g., $M_\star$, $\rm{SFR}$, $\rm{SFH}$, and $L_\rm{AGN}$). We also fit their optical spectra for a subsample of 113 sources to estimate the $M_\rm{BH}$. We calculate the BHAR depending on two radiative efficiency regimes. We find that the Eddington ratios ($\lambda$) and its popular proxy ($L_\rm{X}$/$M_\star$) have 0.6 dex of difference, and a KS-test indicates that they come from different distributions. Our sources exhibit a considerable scatter on the $M_\rm{BH}$-$M_\star$ relation, which can explain the difference between $\lambda$ and its proxy. We also model three evolution scenarios to recover the integral properties at $z=0$. Using the SFR and BHAR, we show a notable diminution in the scattering between $M_\rm{BH}$-$M_\star$. For the last scenario, we consider the SFH and a simple energy budget for the AGN accretion, obtaining a relation similar to the local Universe. Our study covers $\sim 1$ deg$^2$ in the sky and is sensitive to biases in luminosity. Nevertheless, we show that, for bright sources, the link between SFR and BHAR, and their decoupling based on an energy limit is the key that leads to the local $M_\rm{BH}$-$M_\star$ scaling relation., Comment: Accepted for publication in A&A

The J-PAS survey will observe ~1/3 of the northern sky with a set of 56 narrow-band filters using the dedicated 2.55 m JST telescope at the Javalambre Astrophysical Observatory. Prior to the installation of the main camera, in order to demonstrate the scientific potential of J-PAS, two small surveys were performed with the single-CCD Pathfinder camera: miniJPAS (~1 deg2 along the Extended Groth Strip), and J-NEP (~0.3 deg2 around the JWST North Ecliptic Pole Time Domain Field), including all 56 J-PAS filters as well as u, g, r, and i. J-NEP is ~0.5-1.0 magnitudes deeper than miniJPAS, providing photometry for 24,618 r-band detected sources and photometric redshifts (photo-z) for the 6,662 sources with r<23. In this paper we describe the photometry and photo-z of J-NEP and demonstrate a new method for the removal of systematic offsets in the photometry based on the median colours of galaxies, dubbed "galaxy locus recalibration". This method does not require spectroscopic observations except in a few reference pointings and, unlike previous methods, is applicable to the whole J-PAS survey. We use a spectroscopic sample of 787 galaxies to test the photo-z performance for J-NEP and in comparison to miniJPAS. We find that the deeper J-NEP observations result in a factor ~1.5-2 decrease in sigma_NMAD (a robust estimate of the standard deviation of the photo-z error) and the outlier rate relative to miniJPAS for r>21.5 sources, but no improvement in brighter ones. We find the same relation between sigma_NMAD and odds in J-NEP and miniJPAS, suggesting sigma_NMAD can be predicted for any set of J-PAS sources from their odds distribution alone, with no need for additional spectroscopy to calibrate the relation. We explore the causes for photo-z outliers and find that colour-space degeneracy at low S/N, photometry artifacts, source blending, and exotic spectra are the most important factors., Comment: 16 pages, 25 figures, accepted for publication in Astronomy and Astrophysics

Hom-Lie algebras having non-invertible twist maps in their centroids are studied. Central extensions of Hom-Lie algebras having these properties are obtained and shown how the same properties are preserved. Conditions are given so that the produced central extension has an invariant metric with respect to its Hom-Lie product making its twist map self-adjoint when the original Hom-Lie algebra has such a metric. This work is focused on algebras with these properties and following Benayadi and Makhloufwe call them quadratic Hom-Lie algebras. It is shown how a quadratic Hom-Lie algebra gives rise to a quadratic Lie algebra and that the Lie algebra associated to the given Hom-Lie central extension is a Lie algebra central extension of it. It is also shown that if the Hom-Lie product is not a Lie product, there exists a non-abelian algebra, which is in general non-associative too, the commutator of whose product is precisely the Hom-Lie product of the Hom-Lie central extension. Moreover, the algebra whose commutator realizes this Hom-Lie product is shown to be simple if the associated Lie algebra is nilpotent. Non-trivial examples are provided.

The importance of photometric galaxy redshift estimation is rapidly increasing with the development of specialised powerful observational facilities. We develop a new photometric redshift estimation workflow TOPz to provide reliable and efficient redshift estimations for the upcoming large-scale survey J-PAS which will observe 8500 deg2 of the northern sky through 54 narrow-band filters. TOPz relies on template-based photo-z estimation with some added J-PAS specific features and possibilities. We present TOPz performance on data from the miniJPAS survey, a precursor to the J-PAS survey with an identical filter system. First, we generated spectral templates based on the miniJPAS sources using the synthetic galaxy spectrum generation software CIGALE. Then we applied corrections to the input photometry by minimising systematic offsets from the template flux in each filter. To assess the accuracy of the redshift estimation, we used spectroscopic redshifts from the DEEP2, DEEP3, and SDSS surveys, available for 1989 miniJPAS galaxies with r < 22 magAB. We also tested how the choice and number of input templates, photo-z priors, and photometric corrections affect the TOPz redshift accuracy. The general performance of the combination of miniJPAS data and the TOPz workflow fulfills the expectations for J-PAS redshift accuracy. Similarly to previous estimates, we find that 38.6% of galaxies with r < 22 mag reach the J-PAS redshift accuracy goal of dz/(1 + z) < 0.003. Limiting the number of spectra in the template set improves the redshift accuracy up to 5%, especially for fainter, noise-dominated sources. Further improvements will be possible once the actual J-PAS data become available., Comment: 20 pages, 24 figures

In this work we state conditions for a current Lie algebra $\g \otimes \mathcal{S}$ to admit an invariant metric, where $\g$ is a quadratic Lie algebra and $\mathcal{S}$ is an associative and commutative algebra with unit. We also consider the reciprocal: if $\g \otimes \mathcal{S}$ admits an invariant metric, we state necessary and sufficient conditions for $\g$ to admit an invariant metric. In particular, we show that if $\g$ is an indecomposable quadratic Lie algebra, then $\g \otimes \mathcal{S}$ admits an invariant metric if and only if $\mathcal{S}$ also admits an invariant, symmetric and non-degenerate bilinear form. In addition, we prove a theorem similar to the double extension for $\g \otimes \mathcal{S}$, where $\g$ is an indecomposable, nilpotent and quadratic Lie algebra.

The miniJPAS is a 1 deg$^2$ survey that uses the Javalambre-Physics of the Accelerating Universe Astrophysical Survey (J-PAS) filter system (54 narrow-band filters) with the Pathfinder camera. We study mJPC2470-1771, the most massive cluster detected in miniJPAS. We study the stellar population properties of the members, their star formation rates (SFR), star formation histories (SFH), the emission line galaxy (ELG) population, their spatial distribution, and the effect of the environment on them, showing the power of J-PAS to study the role of environment in galaxy evolution. We use a spectral energy distribution (SED) fitting code to derive the stellar population properties of the galaxy members: stellar mass, extinction, metallicity, colours, ages, SFH (a delayed-$\tau$ model), and SFRs. Artificial Neural Networks are used for the identification of the ELG population through the detection of H$\alpha$, [NII], H$\beta$, and [OIII] nebular emission. We use the WHAN and BPT diagrams to separate them into star-forming galaxies and AGNs. We find that the fraction of red galaxies increases with the cluster-centric radius. We select 49 ELG, 65.3\% of the them are probably star forming galaxies, and they are dominated by blue galaxies. 24% are likely to host an AGN (Seyfert or LINER galaxies). The rest are difficult to classify and are most likely composite galaxies. Our results are compatible with an scenario where galaxy members were formed roughly at the same epoch, but blue galaxies have had more recent star formation episodes, and they are quenching from inside-out of the cluster centre. The spatial distribution of red galaxies and their properties suggest that they were quenched prior to the cluster accretion or an earlier cluster accretion epoch. AGN feedback and/or mass might also be intervening in the quenching of these galaxies., Comment: 24 pages, 17 figures, 4 tables, accepted in Astronomy & Astrophysics

The miniJPAS survey has observed $\sim 1$ deg$^2$ on the AEGIS field with 60 bands (spectral resolution of $R \sim 60$) in order to demonstrate the capabilities of the Javalambre-Physics of the Accelerating Universe Astrophysical Survey (J-PAS) that will map $\sim 8000$ deg$^2$ of the northern sky in the next years. This paper shows the power of J-PAS to detect low mass groups and characterise their galaxy populations up to $z \sim 1$. We use the spectral energy distribution fitting code BaySeAGal to derive the stellar population properties of the galaxy members in 80 groups at $z \leq 0.8$ previously detected by the AMICO code, as well as for a galaxy field sample retrieved from the whole miniJPAS sample. We identify blue, red, quiescent, and transition galaxy populations through their rest-frame (extinction corrected) colour, stellar mass ($M_\star$) and specific star formation rate. We measure their abundance as a function of $M_\star$ and environment. We find: (i) The fraction of red and quiescent galaxies in groups increases with $M_\star$ and it is always higher in groups than in the field. (ii) The quenched fraction excess (QFE) in groups strongly increases with $M_\star$, (from a few percent to higher than 60% in the mass range $10^{10} - 3 \times 10 ^{11}$ $M_\odot$. (iii) The abundance excess of transition galaxies in groups shows a modest dependence with $M_\star$ (iv) The fading time scale is very short ($<1.5$ Gyr), indicating that the star formation declines very rapidly in groups. (v) The evolution of the galaxy quenching rate in groups shows a modest but significant evolution since $z\sim0.8$, compatible with an evolution with constant $QFE=0.4$, previously measured for satellites in the nearby Universe, and consistent with a scenario where the low-mass star-forming galaxies in clusters at $z= 1-1.4$ are environmentally quenched., Comment: 21 pages, 20 figures, accepted for publication in Astronomy & Astrophysics

It has been suggested that the gravitational potential can have a significant role in suppressing the star formation in the nearby galaxies. To establish observational constrains on this scenario, we investigate the connection between the dynamics, through the circular velocity curves (CVCs) as a proxy of the inner gravitational potential, and star formation quenching in 215 non-active galaxies across Hubble sequence from the Calar Alto Legacy Integral Field Area (CALIFA) survey. Our results show that galaxies with similar CVCs tend to have a certain star-formation quenching pattern. To explore these findings in more details, we construct kpc-resolved relations of the equivalent width of the H$\alpha$ ($W_{{\rm H}\alpha}$) versus the amplitude ($V_c$) and shape ($\beta= d\ln V_c/ d\ln R$) of the circular velocity at given radius. We find that the $W_{{\rm H}\alpha}-V_c$ is a declining relationship, where the retired regions of the galaxies (the ones with $W_{{\rm H}\alpha}$ values below 3 \r{A}) tend to have higher $V_c$. Differently, $W_{{\rm H}\alpha}-\beta$ is a bi-modal relationship, characterised by two peaks: concentration of the star forming regions at a positive $\beta$ (rising CVC) and another one of the retired regions with a negative $\beta$ (declining CVC). Our results show that both the amplitude of the CVC, driven by the mass of the galaxies, and the shape of the CVC, reflecting the internal structure of the galaxies, play an important role in galaxy's quenching history., Comment: 16 pages, 12 figures, Accepted for publication in A&A

The Javalambre-Physics of the Accelerating Universe Astrophysical Survey (J-PAS) is expected to map thousands of square degrees of the northern sky with 56 narrowband filters in the upcoming years. This will make J-PAS a very competitive and unbiased emission line survey compared to spectroscopic or narrowband surveys with fewer filters. The miniJPAS survey covered 1 deg$^2$, and it used the same photometric system as J-PAS, but the observations were carried out with the pathfinder J-PAS camera. In this work, we identify and characterize the sample of emission line galaxies (ELGs) from miniJPAS with a redshift lower than $0.35$. Using a method based on artificial neural networks, we detect the ELG population and measure the equivalent width and flux of the $H\alpha$, $H\beta$, [OIII], and [NII] emission lines. We explore the ionization mechanism using the diagrams [OIII]/H$\beta$ versus [NII]/H$\alpha$ (BPT) and EW(H$\alpha$) versus [NII]/H$\alpha$ (WHAN). We identify 1787 ELGs ($83$%) from the parent sample (2154 galaxies) in the AEGIS field. For the galaxies with reliable EW values that can be placed in the WHAN diagram (2000 galaxies in total), we obtained that $72.8 \pm 0.4$%, $17.7 \pm 0.4$% , and $9.4 \pm 0.2$% are star-forming (SF), active galactic nucleus (Seyfert), and quiescent galaxies, respectively. Based on the flux of $H\alpha$ we find that the star formation main sequence is described as $\log$ SFR $[M_\mathrm{\odot} \mathrm{yr}^{-1}] = 0.90^{+ 0.02}_{-0.02} \log M_{\star} [M_\mathrm{\odot}] -8.85^{+ 0.19}_{-0.20}$ and has an intrinsic scatter of $0.20^{+ 0.01}_{-0.01}$. The cosmic evolution of the SFR density ($\rho_{\text{SFR}}$) is derived at three redshift bins: $0 < z \leq 0.15$, $0.15 < z \leq 0.25$, and $0.25 < z \leq 0.35$, which agrees with previous results that were based on measurements of the $H\alpha$ emission line., Comment: 22 pages, 19 figures

MiniJPAS is a ~1 deg^2 imaging survey of the AEGIS field in 60 bands, performed to demonstrate the scientific potential of the upcoming JPAS survey. Full coverage of the 3800-9100 \AA range with 54 narrow and 6 broad optical filters allow for extremely accurate photo-z, which applied over 1000s of deg^2 will enable new applications of the photo-z technique such as measurement of baryonic acoustic oscillations. In this paper we describe the method used to obtain the photo-z included in the publicly available miniJPAS catalogue, and characterise the photo-z performance. We build 100 \AA resolution photo-spectra from the PSF-corrected forced-aperture photometry. Systematic offsets in the photometry are corrected by applying magnitude shifts obtained through iterative fitting with stellar population synthesis models. We compute photo-z with a customised version of LePhare, using a set of templates optimised for the J-PAS filter-set. We analyse the accuracy of miniJPAS photo-z and their dependence on multiple quantities using a subsample of 5,266 galaxies with spectroscopic redshifts from SDSS and DEEP, that we find to be representative of the whole r<23 miniJPAS sample. Formal uncertainties for the photo-z that are calculated with the \delta\chi^2 method underestimate the actual redshift errors. The odds parameter has the stronger correlation with |Dz|, and accurately reproduces the probability of a redshift outlier (|Dz|>0.03) irrespective of the magnitude, redshift, or spectral type of the sources. We show that the two main summary statistics characterising the photo-z accuracy for a population of galaxies (snmad and \eta) can be predicted by the distribution of odds in such population, and use this to estimate them for the whole miniJPAS sample. At r<23 there are 17,500 galaxies/deg^2 with valid photo-z estimates, of which 4,200 are expected to have |Dz|<0.003 (abridged)., Comment: 25 pages, 32 figures, 5 tables, accepted for publication in A&A

We analyze LHC data in order to constrain the parameter space of new spin-2 particles universally coupled to the energy-momentum tensor. These new hypothetical particles are the so-called hidden gravitons, whose phenomenology at low energies is determined by two parameters: its mass and its dimensional coupling constant. Hidden gravitons arise in many different extensions of the Standard Model of particles and interactions and General Relativity. Their phenomenology has been studied mainly in relation to modifications of gravity and astrophysical signatures. In this work, we extend the constraints for heavy hidden gravitons, with masses larger than $1$ GeV, by taking into account events collected by ATLAS and CMS in the WW channel, Drell-Yan processes, and the diphoton channel from proton-proton collisions at $\sqrt{s}=8$ TeV., Comment: 9 pages, 5 figures

J-PAS will soon start imaging 8000 deg2 of the northern sky with its unique set of 56 filters (R $\sim$ 60). Before, we observed 1 deg2 on the AEGIS field with an interim camera with all the J-PAS filters. With this data (miniJPAS), we aim at proving the scientific potential of J-PAS to identify and characterize the galaxy populations with the goal of performing galaxy evolution studies across cosmic time. Several SED-fitting codes are used to constrain the stellar population properties of a complete flux-limited sample (rSDSS <= 22.5 AB) of miniJPAS galaxies that extends up to z = 1. We find consistent results on the galaxy properties derived from the different codes, independently of the galaxy spectral-type or redshift. For galaxies with SNR>=10, we estimate that the J-PAS photometric system allows to derive stellar population properties with a precision that is equivalent to that obtained with spectroscopic surveys of similar SNR. By using the dust-corrected (u-r) colour-mass diagram, a powerful proxy to characterize galaxy populations, we find that the fraction of red and blue galaxies evolves with cosmic time, with red galaxies being $\sim$ 38% and $\sim$ 18% of the whole population at z = 0.1 and z = 0.5, respectively. At all redshifts, the more massive galaxies belong to the red sequence and these galaxies are typically older and more metal rich than their counterparts in the blue cloud. Our results confirm that with J-PAS data we will be able to analyze large samples of galaxies up to z $\sim$ 1, with galaxy stellar masses above of log(M$_*$/M$_{\odot}$) $\sim$ 8.9, 9.5, and 9.9 at z = 0.3, 0.5, and 0.7, respectively. The SFH of a complete sub-sample of galaxies selected at z $\sim$ 0.1 with log(M$_*$/M$_{\odot}$) > 8.3 constrain the cosmic evolution of the star formation rate density up to z $\sim$ 3 in good agreement with results from cosmological surveys., Comment: 27 pages, 23 figures, 3 tables, accepted for publication in Astronomy & Astrophysics

The mechanisms that bring galaxies to strongly reduce their star formation activity (star-formation quenching) is still poorly understood. To better study galaxy evolution, we propose a classification based on the maps of the ionised hydrogen distribution, traced by kpc-resolved, equivalent width of H$\alpha$ maps, and the nuclear activity of the galaxies using information from the BPT diagnostic diagrams. Using these tools, we group a sample of 238 galaxies from the CALIFA survey in six quenching stages (QS): objects dominated by recent star formation; systems that present a quiescent-nuclear-ring structure in their centre; galaxies that are centrally-quiescent; galaxies with no clear pattern in their ionisation gas distribution - mixed; systems that posses only a few star-forming regions - nearly-retired, or galaxies that are completely quiescent - fully-retired. Regarding their nuclear activity, we further divide the galaxies into two groups - active systems that host a weak or strong AGN in their centre, and non-active objects. Galaxies grouped into quenching stage classes occupy specific locations on the star-formation-rate versus stellar mass diagram. The "Blue cloud" is populated by the star-forming and the quiescent-nuclear-ring galaxies, the "Green valley" is populated by centrally-quiescent and mixed systems, "Red sequence" by the nearly- and fully-retired objects. Generally, galaxies that host a weak or strong AGN show properties, comparable to the non-active counterparts at the same quenching stages, except for the AGN-hosting star-forming systems. The degree of the star-formation quenching increases along the present emission-line pattern sequence from star-forming to fully-retired. The proposed emission-line classes reinforce the "inside-out" quenching scenario, which foresees that the suppression of the star-formation begins from the central regions of the galaxies., Comment: 24 pages, 11 figures, 2 tables, accepted for publication in A&A

Our goal is to estimate the star formation main sequence (SFMS) and the star formation rate density (SFRD) at z <= 0.017 (d < 75 Mpc) using the Javalambre Photometric Local Universe Survey (J-PLUS) first data release, that probes 897.4 deg2 with twelve optical bands. We extract the Halpha emission flux of 805 local galaxies from the J-PLUS filter J0660, being the continuum level estimated with the other eleven J-PLUS bands, and the dust attenuation and nitrogen contamination corrected with empirical relations. Stellar masses (M), Halpha luminosities (L), and star formation rates (SFRs) were estimated by accounting for parameters covariances. Our sample comprises 689 blue galaxies and 67 red galaxies, classified in the (u-g) vs (g-z) color-color diagram, plus 49 AGN. The SFMS is explored at log M > 8 and it is clearly defined by the blue galaxies, with the red galaxies located below them. The SFMS is described as log SFR = 0.83 log M - 8.44. We find a good agreement with previous estimations of the SFMS, especially those based on integral field spectroscopy. The Halpha luminosity function of the AGN-free sample is well described by a Schechter function with log L* = 41.34, log phi* = -2.43, and alpha = -1.25. Our measurements provide a lower characteristic luminosity than several previous studies in the literature. The derived star formation rate density at d < 75 Mpc is log rho_SFR = -2.10 +- 0.11, with red galaxies accounting for 15% of the SFRD. Our value is lower than previous estimations at similar redshift, and provides a local reference for evolutionary studies regarding the star formation history of the Universe., Comment: 26 pages. Accepted for publication in Astronomy and Astrophysics

Hom-Lie algebras defined on central extensions of a given quadratic Lie algebra that in turn admit an invariant metric, are studied. It is shown how some of these algebras are naturally equipped with other symmetric, bilinear forms that satisfy an invariant condition for their twisted multiplication maps. The twisted invariant bilinear forms so obtained resemble the Cartan-Killing forms defined on ordinary Lie algebras. This fact allows one to reproduce on the Hom-Lie algebras hereby studied, some results that are classically associated to the ordinary Cartan-Killing form.

Throughout this paper we present a new method to detect and measure emission lines in J-PAS up to $z = 0.35$. J-PAS will observe $8000$~deg$^2$ of the northern sky in the upcoming years with 56 photometric bands. The release of such amount of data brings us the opportunity to employ machine learning methods in order to overcome the difficulties associated with photometric data. We used Artificial Neural Networks (ANNs) trained and tested with synthetic J-PAS photometry from CALIFA, MaNGA, and SDSS spectra. We carry out two tasks: firstly, we cluster galaxies in two groups according to the values of the equivalent width (EW) of $H\alpha$, $H\beta$, $[NII]{\lambda 6584}$, and $ [OIII]{\lambda 5007}$ lines measured in the spectra. Then, we train an ANN to assign to each galaxy a group. We are able to classify them with the uncertainties typical of the photometric redshift measurable in J-PAS. Secondly, we utilize another ANN to determine the values of those EWs. Subsequently, we obtain the $[NII]/H\alpha$, $[OIII]/H\beta$, and \ion{O}{3}\ion{N}{2} ratios recovering the BPT diagram . We study the performance of the ANN in two training samples: one is only composed of synthetic J-PAS photo-spectra (J-spectra) from MaNGA and CALIFA (CALMa set) and the other one is composed of SDSS galaxies. We can reproduce properly the main sequence of star forming galaxies from the determination of the EWs. With the CALMa training set we reach a precision of 0.093 and 0.081 dex for the $[NII]/H\alpha$ and $[OIII]/H\beta$ ratios in the SDSS testing sample. Nevertheless, we find an underestimation of those ratios at high values in galaxies hosting an AGN. We also show the importance of the dataset used for both training and testing the model. ANNs are extremely useful to overcome the limitations previously expected concerning the detection and measurements of the emission lines in surveys like J-PAS., Comment: 19 pages, 14 figures, Accepted to A&A

Future astrophysical surveys such as J-PAS will produce very large datasets, which will require the deployment of accurate and efficient Machine Learning (ML) methods. In this work, we analyze the miniJPAS survey, which observed about 1 deg2 of the AEGIS field with 56 narrow-band filters and 4 ugri broad-band filters. We discuss the classification of miniJPAS sources into extended (galaxies) and point-like (e.g. stars) objects, a necessary step for the subsequent scientific analyses. We aim at developing an ML classifier that is complementary to traditional tools based on explicit modeling. In order to train and test our classifiers, we crossmatched the miniJPAS dataset with SDSS and HSC-SSP data. We trained and tested 6 different ML algorithms on the two crossmatched catalogs. As input for the ML algorithms we use the magnitudes from the 60 filters together with their errors, with and without the morphological parameters. We also use the mean PSF in the r detection band for each pointing. We find that the RF and ERT algorithms perform best in all scenarios. When analyzing the full magnitude range of 1521). We use our best classifiers, with and without morphology, in order to produce a value added catalog available at https://j-pas.org/datareleases ., Comment: 19 pages, 17 figures. New Appendices E & F and improved discussion. VAC available at https://j-pas.org/datareleases . Trained models available at https://github.com/J-PAS-collaboration/StarGalClass-MachineLearning . Abstract abridged to meet arXiv requirements. Version accepted for publication in A&A

The Javalambre-Physics of the Accelerating Universe Astrophysical Survey (J-PAS) will soon start to scan thousands of square degrees of the northern extragalactic sky with a unique set of $56$ optical filters from a dedicated $2.55$m telescope, JST, at the Javalambre Astrophysical Observatory. Before the arrival of the final instrument (a 1.2 Gpixels, 4.2deg$^2$ field-of-view camera), the JST was equipped with an interim camera (JPAS-Pathfinder), composed of one CCD with a 0.3deg$^2$ field-of-view and resolution of 0.23 arcsec pixel$^{-1}$. To demonstrate the scientific potential of J-PAS, with the JPAS-Pathfinder camera we carried out a survey on the AEGIS field (along the Extended Groth Strip), dubbed miniJPAS. We observed a total of $\sim 1$ deg$^2$, with the $56$ J-PAS filters, which include $54$ narrow band (NB, $\rm{FWHM} \sim 145$Angstrom) and two broader filters extending to the UV and the near-infrared, complemented by the $u,g,r,i$ SDSS broad band (BB) filters. In this paper we present the miniJPAS data set, the details of the catalogues and data access, and illustrate the scientific potential of our multi-band data. The data surpass the target depths originally planned for J-PAS, reaching $\rm{mag}_{\rm {AB}}$ between $\sim 22$ and $23.5$ for the NB filters and up to $24$ for the BB filters ($5\sigma$ in a $3$~arcsec aperture). The miniJPAS primary catalogue contains more than $64,000$ sources extracted in the $r$ detection band with forced photometry in all other bands. We estimate the catalogue to be complete up to $r=23.6$ for point-like sources and up to $r=22.7$ for extended sources. Photometric redshifts reach subpercent precision for all sources up to $r=22.5$, and a precision of $\sim 0.3$% for about half of the sample. (Abridged), Comment: The miniJPAS data and associated value added catalogues are publicly accessible via this url: http://archive.cefca.es/catalogues/minijpas-pdr201912

In this work we state conditions on a covariant right exact functor so that it commutes with direct limits. These conditions are related to the commutativity of the functor under direct limits of projective modules. We prove that if the functor commutes with direct limits of projective modules, then the functor commutes with direct limits.

Solvable Lie algebras having at least one Abelian descending central ideal are studied. It is shown that all such Lie algebras can be built up from canonically defined ideals. The nature of such ideals is elucidated and their construction is provided in detail. An approach to study and to classify these Lie algebras is given through the theory of extensions via appropriate cocycles and representations on which a group action is naturally defined. Also, necessary and sufficient conditions for the existence of invariant metrics on the studied extensions are given. It is shown that any solvable quadratic Lie algebra $\g$ having an Abelian descending central ideal is of the form $\g=\h\oplus\a\oplus\h^*$, where $\h^*\simeq \ide(\g)$ and $\a \oplus \h^{*}\simeq \j(\g)$ are in fact two canonically defined Abelian ideals of $\g$ satisfying $\ide(\g)^\perp=\j(\g)$. As an example, a classification of this type of quadratic Lie algebras is given assuming that $\j(\g)/\ide(\g)$ is an $r$-dimensional vector space and $\g/\j(\g)$ is the 3-dimensional Heisenberg Lie algebra., Comment: This is a revised version of the manuscript On Quadratic Lie Algebras With Non-trivial Center [arXiv:1911.05009v3]. We submitted it for publication and the reviewer found a mistake in Lemma 2.6. The problem does not appear for Lie algebras with one Abelian descending central ideal. Thus, we ended up with this manuscript in whose title we have included the additional hypothesis

[Abridged] We present the apparent stellar angular momentum of 300 galaxies across the Hubble sequence, using integral-field spectroscopic data from the CALIFA survey. Adopting the same $\lambda_\mathrm{R}$ parameter previously used to distinguish between slow and fast rotating early-type (elliptical and lenticular) galaxies, we show that spiral galaxies as expected are almost all fast rotators. Given the extent of our data, we provide relations for $\lambda_\mathrm{R}$ measured in different apertures, including conversions to long-slit 1D apertures. Our sample displays a wide range of $\lambda_\mathrm{Re}$ values, consistent with previous IFS studies. The fastest rotators are dominated by relatively massive and highly star-forming Sb galaxies, which preferentially reside in the main star-forming sequence. These galaxies reach $\lambda_\mathrm{Re}$ values of $\sim$0.85, are the largest galaxies at a given mass, and display some of the strongest stellar population gradients. Compared to the population of S0 galaxies, our findings suggest that fading may not be the dominant mechanism transforming spirals into lenticulars. Interestingly, we find that $\lambda_\mathrm{Re}$ decreases for late-type Sc and Sd spiral galaxies, with values than in occasions puts them in the slow-rotator regime. While for some of them this can be explained by their irregular morphologies and/or face-on configurations, others are edge-on systems with no signs of significant dust obscuration. The latter are typically at the low-mass end, but this does not explain their location in the classical ($V/\sigma$,$\varepsilon$) and ($\lambda_\mathrm{Re}$,$\varepsilon$) diagrams. Our initial investigations, based on dynamical models, suggest that these are dynamically hot disks, probably influenced by the observed important fraction of dark matter within R$_\mathrm{e}$., Comment: 17 pages, 7 figures, A&A accepted

In this work we study the phenomenology of the process $pp\to W^+W^-jj$ at the LHC, in the scenario of the resonant vector boson scattering subprocess $W^+W^-\to W^+W^-$ which we describe within the effective field theory framework of the Electroweak Chiral Lagrangian. We assume a strongly interacting electroweak symmetry breaking sector in which dynamically generated resonances with masses in the TeV scale appear as poles in the Electroweak Chiral Lagrangian amplitudes unitarized with the Inverse Amplitude Method. The relevant resonance here, $V_0$, is the neutral component of the triplet of vector resonances which are known to emerge dynamically at the TeV scale for specific values of the Electroweak Chiral Lagrangian parameters. With the aim of studying the production and possible observation of $V^0$ at the LHC, via the resonant $W^+W^-\to W^+W^-$ scattering, a MadGraph~5 UFO model has been developed employing a phenomenological Proca Lagrangian as a practical tool to mimic the correct $V^0$ properties that are predicted with the Inverse Amplitude Method. We choose to study the fully hadronic decay channel of the final gauge bosons $W W\to J(jj)J(jj)$ since it leads to larger event rates and because in the alternative leptonic decay channels the presence of neutrinos complicates the reconstruction of the resonance properties. In this context, the 2 boosted jets from the $W$ hadronic decays, $jj$, are detected as a single fat jet, $J$, due to their extreme collinearity. We perform a dedicated analysis of the sensitivity to these vector resonances $V^0$ with masses between 1.5 and 2.5 TeV at the LHC with $\sqrt{s}=13$ TeV and the planned high luminosity of 3000 ${\rm fb}^{-1}$, paying special attention to the study of efficient cuts to extract the resonant vector boson fusion signal from the QCD background, which clearly represents the main challenge of this search., Comment: 14 pages, 4 figures (8 plots), 4 tables

The purpose of this paper is to show that there are Hom-Lie algebra structures on $\mathfrak{sl}_2(\mathbb{F}) \oplus \mathbb{F}D$, where $D$ is a special type of generalized derivation of $\mathfrak{sl}_2(\mathbb{F})$, and $\mathbb{F}$ is an algebraically closed field of characteristic zero. It is shown that the generalized derivations $D$ of $\mathfrak{sl}_2(\mathbb{F})$ that we study in this work, satisfy the Hom-Lie Jacobi identity for the Lie bracket of $\mathfrak{sl}_2(\mathbb{F})$. We study the representation theory of Hom-Lie algebras within the appropriate category and prove that any finite dimensional representation of a Hom-Lie algebra of the form $\mathfrak{sl}_2(\mathbb{F}) \oplus \mathbb{F}D$, is completely reducible, in analogy to the well known Theorem of Weyl from the classical Lie theory. We apply this result to characterize the non-solvable Lie algebras having an invertible generalized derivation of the type of $D$. Finally, using root space decomposition techniques we provide an intrinsic proof of the fact that $\mathfrak{sl}_2(\mathbb{F})$ is the only simple Lie algebra admitting non-trivial Hom-Lie structures.

We study and classify the 3-dimensional Hom-Lie algebras over $\mathbb{C}$. We provide first a complete set of representatives for the isomorphism classes of skew-symmetric bilinear products defined on a 3-dimensional complex vector space $\mathfrak{g}$. The well known Lie brackets for the 3-dimensional Lie algebras are included into appropriate isomorphism classes of such products representatives. For each product representative, we provide a complete set of canonical forms for the linear maps $\mathfrak{g} \to \mathfrak{g}$ that turn $g$ into a Hom-Lie algebra, thus characterizing the corresponding isomorphism classes. As by-products, Hom-Lie algebras for which the linear maps $\mathfrak{g} \to \mathfrak{g}$ are not homomorphisms for their products, are exhibited. Examples also arise of non-isomorphic families of HomLie algebras which share, however, a fixed Lie-algebra product on $\mathfrak{g}$. In particular, this is the case for the complex simple Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. Similarly, there are isomorphism classes for which their skew-symmetric bilinear products can never be Lie algebra brackets on $\mathfrak{g}$.