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Left-Alia algebras are a class of algebras with symmetric Jacobi identities. They contain several typical types of algebras as subclasses, and are closely related to the invariant theory. In this paper, we study the construction theory of left-Alia bialgebras. We introduce the notion of the left-Alia Yang-Baxter equation. We show that an antisymmetric solution of the left-Alia Yang-Baxter equation gives rise to a left-Alia bialgebra that we call triangular. The notions of relative Rota-Baxter operators of left-Alia algebras and pre-left-Alia algebras are introduced to provide antisymmetric solutions of the left-Alia Yang-Baxter equation., Comment: arXiv admin note: text overlap with arXiv:2403.05339

We first classify all Nijenhuis operators on 2-dimensional complex pre-Lie algebras and 3-dimensional complex associative algebras. Moreover, applying the obtained classification, we provide solutions of classical Yang-Baxter equation of the semi-direct product sub-adjacent Lie algebras.

In this paper, we study the Manin triples associated to $n$-Lie bialgebras. We introduce the concept of operad matrices for $n$-Lie bialgebras. In particular, by studying a special case of operad matrices, it leads to the notion of local cocycle $n$-Lie bialgebras. Furthermore, we establish a one-to-one correspondence between the double of $n$-Lie bialgebras and Manin triples of $n$-Lie algebras.

A polyuble of a Manin triple can be regarded as the ``$n$-th power'' of it, which plays an important rule in the study of Poisson geometry, mathematical physics and Lie theory. In this paper, we first construct an isomorphism between the $mn$-ble and the $n$-ubles of $m$-uble by colored graph and point out it is unique. Then, we construct a class of Poisson homogeneous spaces and obtain a class of Poisson homeomorphisms between them based on the first main result. Last, we apply first two main results to multi-flag varieties as well as multi-double flag varieties and construct a class of global Poisson isomorphisms between them as well as their $T$-leaves.

Let $G$ be a connected complex semi-simple Lie group and ${\mathcal{B}}$ its flag variety. For every positive integer $n$, we introduce a Poisson groupoid over ${\mathcal{B}}^n$, called the $n$th total configuration Poisson groupoid of flags of $G$, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells. Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to $G$.

Let $G$ be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous $G$-spaces $G/Q$, we construct a finite atlas ${\mathcal{A}}_{\rm BS}(G/Q)$ on $G/Q$, called the Bott-Samelson atlas, and we prove that all of its coordinate functions are positive with respect to the Lusztig positive structure on $G/Q$. We also show that the standard Poisson structure $\pi_{G/Q}$ on $G/Q$ is presented, in each of the coordinate charts of ${\mathcal{A}}_{\rm BS}(G/Q)$, as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl-Yakimov, making $(G/Q, \pi_{G/Q}, {\mathcal{A}}_{\rm BS}(G/Q))$ into a Poisson-Ore variety. Examples of $G/Q$ include $G$ itself, $G/T$, $G/B$, and $G/N$, where $T \subset G$ is a maximal torus, $B \subset G$ a Borel subgroup, and $N$ the uniradical of $B$.

The goal of this paper is to construct a Frobenius splitting on $G/U$ via the Poisson geometry of $(G/U,\pi_{G/U})$, where $G$ is a semi-simple algebraic group of classical type defined over an algebraically closed field of characteristic $p > 3$, $U$ is the uniradical of a Borel subgroup of $G$ and $\pi_{G/U}$ is the standard Poisson structure on $G/U$. We first study the Poisson geometry of $(G/U,\pi_{G/U})$. Then, we develop a general theory for Frobenius splittings on $\mathbb{T}$-Poisson varieties, where $\mathbb{T}$ is an algebraic torus. In particular, we prove that compatibly split subvarieties of Frobenius splittings constructed in this way must be $\mathbb{T}$-Poisson sub-varieties. Lastly, we apply our general theory to construct a Frobenius splitting on $G/U$.

Classification standards such as the Mammal Species of the World (MSW) aim to unify name usages at the global scale, but may nevertheless experience significant levels of taxonomic change from one edition to the next. This circumstance challenges the biodiversity and phylogenetic data communities to develop more granular identifiers to track taxonomic congruence and incongruence in ways that both humans and machines can process, i.e., to logically represent taxonomic provenance across multiple classification hierarchies. Here we show that reasoning over taxonomic provenance is feasible for two classifications of primates corresponding to the second and third MSW editions. Our approach entails three main components: (1) individuation of name usages as taxonomic concepts, (2) articulation of concepts via human-asserted Region Connection Calculus (RCC-5) relationships, and (3) the use of an Answer Set Programming toolkit to infer and visualize logically consistent alignments of these taxonomic input constraints. Our use case entails the Primates sec. Groves (1993; MSW2 - 317 taxonomic concepts; 233 at the species level) and Primates sec. Groves (2005; MSW3 - 483 taxonomic concepts; 376 at the species level). Using 402 concept-to-concept input articulations, the reasoning process yields a single, consistent alignment, and infers 153,111 Maximally Informative Relations that constitute a comprehensive provenance resolution map for every concept pair in the Primates sec. MSW2/MSW3. The entire alignment and various partitions facilitate quantitative analyses of name/meaning dissociation, revealing that approximately one in three paired name usages across treatments is not reliable - in the sense of the same name identifying congruent taxonomic meanings. We conclude with an optimistic outlook for logic-based provenance tools in next-generation biodiversity and phylogeny data platforms., Comment: Partial manuscript; figures down-sampled

Classifications and phylogenetic inferences of organismal groups change in light of new insights. Over time these changes can result in an imperfect tracking of taxonomic perspectives through the re-/use of Code-compliant or informal names. To mitigate these limitations, we introduce a novel approach for aligning taxonomies through the interaction of human experts and logic reasoners. We explore the performance of this approach with the Perelleschus use case of Franz & Cardona-Duque (2013). The use case includes six taxonomies published from 1936 to 2013, 54 taxonomic concepts (i.e., circumscriptions of names individuated according to their respective source publications), and 75 expert-asserted Region Connection Calculus articulations (e.g., congruence, proper inclusion, overlap, or exclusion). An Open Source reasoning toolkit is used to analyze 13 paired Perelleschus taxonomy alignments under heterogeneous constraints and interpretations. The reasoning workflow optimizes the logical consistency and expressiveness of the input and infers the set of maximally informative relations among the entailed taxonomic concepts. The latter are then used to produce merge visualizations that represent all congruent and non-congruent taxonomic elements among the aligned input trees. In this small use case with 6-53 input concepts per alignment, the information gained through the reasoning process is on average one order of magnitude greater than in the input. The approach offers scalable solutions for tracking provenance among succeeding taxonomic perspectives that may have differential biases in naming conventions, phylogenetic resolution, ingroup and outgroup sampling, or ostensive (member-referencing) versus intensional (property-referencing) concepts and articulations., Comment: 30 pages, 16 figures

We introduce Euler/X, a toolkit for logic-based taxonomy integration. Given two taxonomies and a set of alignment constraints between them, Euler/X provides tools for detecting, explaining, and reconciling inconsistencies; finding all possible merges between (consistent) taxonomies; and visualizing merge results. Euler/X employs a number of different underlying reasoning systems, including first-order reasoners (Prover9 and Mace4), answer set programming (DLV and Potassco), and RCC reasoners (PyRCC8). We demonstrate the features of Euler/X and provide experimental results showing its feasibility on various synthetic and real-world examples., Comment: 8 pages, 14 figures, WFLP 2013