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A left-Alia algebra is a vector space together with a bilinear map satisfying the symmetric Jacobi identity. Motivated by invariant theory, we first construct a class of left-Alia algebras induced by twisted derivations. Then, we introduce the notions of Manin triples and bialgebras of left-Alia algebras. Via specific matched pairs of left-Alia algebras, we figure out the equivalence between Manin triples and bialgebras of left-Alia algebras. [ABSTRACT FROM AUTHOR]

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 in |${\mathcal {B}}^n$|⁠. 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 |$G$|⁠. [ABSTRACT FROM AUTHOR]

The goal of this paper is to construct a Frobenius splitting on |$G/U$| via the Poisson geometry of |$(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$|⁠ , where |$G$| is a simply connected semi-simple algebraic group defined over an algebraically closed field of characteristic |$p> 3$|⁠ , |$U$| is the uniradical of a Borel subgroup of |$G$|⁠ , and |$\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}}$| is the standard Poisson structure on |$G/U$|⁠. We first study the Poisson geometry of |$(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$|⁠. 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 subvarieties. Lastly, we apply our general theory to construct a Frobenius splitting on |$G/U$|⁠. [ABSTRACT FROM AUTHOR]

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 A 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 π G / Q on G/Q is presented, in each of the coordinate charts of A BS (G / Q) , as a symmetric Poisson CGL extension (or a certain localization thereof) in the sense of Goodearl–Yakimov, making (G / Q , π G / Q , A BS (G / Q)) into a Poisson–Ore variety. In addition, all coordinate functions in the Bott–Samelson atlas are shown to have complete Hamiltonian flows with respect to the Poisson structure π G / Q . Examples of G/Q include G itself, G/T, G/B, and G/N, where T ⊂ G is a maximal torus, B ⊂ G a Borel subgroup, and N the uniradical of B. [ABSTRACT FROM AUTHOR]

Phylogenomic research is accelerating the publication of landmark studies that aim to resolve deep divergences of major organismal groups. Meanwhile, systems for identifying and integrating the products of phylogenomic inference–such as newly supported clade concepts–have not kept pace. However, the ability to verbalize node concept congruence and conflict across multiple, in effect simultaneously endorsed phylogenomic hypotheses, is a prerequisite for building synthetic data environments for biological systematics and other domains impacted by these conflicting inferences. Here we develop a novel solution to the conflict verbalization challenge, based on a logic representation and reasoning approach that utilizes the language of Region Connection Calculus (RCC–5) to produce consistent alignments of node concepts endorsed by incongruent phylogenomic studies. The approach employs clade concept labels to individuate concepts used by each source, even if these carry identical names. Indirect RCC–5 modeling of intensional (property-based) node concept definitions, facilitated by the local relaxation of coverage constraints, allows parent concepts to attain congruence in spite of their differentially sampled children. To demonstrate the feasibility of this approach, we align two recent phylogenomic reconstructions of higher-level avian groups that entail strong conflict in the "neoavian explosion" region. According to our representations, this conflict is constituted by 26 instances of input "whole concept" overlap. These instances are further resolvable in the output labeling schemes and visualizations as "split concepts", which provide the labels and relations needed to build truly synthetic phylogenomic data environments. Because the RCC–5 alignments fundamentally reflect the trained, logic-enabled judgments of systematic experts, future designs for such environments need to promote a culture where experts routinely assess the intensionalities of node concepts published by our peers–even and especially when we are not in agreement with each other. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]