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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 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.

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.

Coordinated multi-platform information operations are implemented in a variety of contexts on social media, including state-run disinformation campaigns, marketing strategies, and social activism. Characterized by the promotion of messages via multi-platform coordination, in which multiple user accounts, within a short time, post content advancing a shared informational agenda on multiple platforms, they contribute to an already confusing and manipulated information ecosystem. To make things worse, reliable datasets that contain ground truth information about such operations are virtually nonexistent. This paper presents a multi-modal approach that identifies the social media messages potentially engaged in a coordinated information campaign across multiple platforms. Our approach incorporates textual content, temporal information and the underlying network of user and messages posted to identify groups of messages with unusual coordination patterns across multiple social media platforms. We apply our approach to content posted on four platforms related to the Syrian Civil Defence organization known as the White Helmets: Twitter, Facebook, Reddit, and YouTube. Results show that our approach identifies social media posts that link to news YouTube channels with similar factuality score, which is often an indication of coordinated operations., Comment: To appear in the 5th Multidisciplinary International Symposium on Disinformation in Open Online Media (MISDOOM 2023)

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

Tuberculosis (TB) is still considered a leading cause of death and a substantial threat to global child health. Both TB infection and disease are curable using antibiotics. However, most children who die of TB are never diagnosed or treated. In clinical practice, experienced physicians assess TB by examining chest X-rays (CXR). Pediatric CXR has specific challenges compared to adult CXR, which makes TB diagnosis in children more difficult. Computer-aided diagnosis systems supported by Artificial Intelligence have shown performance comparable to experienced radiologist TB readings, which could ease mass TB screening and reduce clinical burden. We propose a multi-view deep learning-based solution which, by following a proposed template, aims to automatically regionalize and extract lung and mediastinal regions of interest from pediatric CXR images where key TB findings may be present. Experimental results have shown accurate region extraction, which can be used for further analysis to confirm TB finding presence and severity assessment. Code publicly available at https://github.com/dani-capellan/pTB_LungRegionExtractor., Comment: This work has been accepted at the SPIE Medical Imaging 2023, Image Processing conference

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.

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.

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.

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

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 $\mathfrak{g}$.

A quadratic Lie algebra is a Lie algebra endowed with a symmetric, invariant and non degenerate bilinear form; such a bilinear form is called an invariant metric. The aim of this work is to describe the general structure of those central extensions of quadratic Lie algebras which in turn have invariantmetrics. The structure is such that the central extensions can be described algebraically in terms of the original quadratic Lie algebra, and geometrically in terms of the direct sum decompositions that the invariant metrics involvedgive rise to.