2,160 research outputs found

    Lie pairs

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    Extending the theory of systems, we introduce a theory of Lie semialgebra ``pairs'' which parallels the classical theory of Lie algebras, but with a ``null set'' replacing 0. A selection of examples is given. These Lie pairs comprise two categories in addition to the universal algebraic definition, one with ``weak Lie morphisms'' preserving null sums, and the other with ``âȘŻ-morphisms'' preserving a surpassing relation âȘŻ that replaces equality. We provide versions of the PBW (Poincare-Birkhoff-Witt) Theorem in these three categories

    Every topos has an optimal noetherian form

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    The search, of almost a century long, for a unified axiomatic framework for establishing homomorphism theorems of classical algebra (such as Noether isomorphism theorems and homological diagram lemmas) has led to the notion of a `noetherian form', which is a generalization of an abelian category suitable to encompass categories of non-abelian algebraic structures (such as non-abelian groups, or rings with identity, or cocommutative Hopf algebras over any field, and many others). In this paper, we show that, surprisingly, even the category of sets, and more generally, any topos, fits under the framework of a noetherian form. Moreover, we give an intrinsic characterization of such noetherian form and show that it is very closely related to the known noetherian form of a semi-abelian category. In fact, we show that for a pointed category having finite products and sums, the existence of the type of noetherian form that any topos possesses is equivalent to the category being semi-abelian (this result is unexpected since only trivial toposes can be semi-abelian). We also show that these noetherian forms are optimal, in a suitable sense.Comment: 66 pages, submitted for publicatio

    On the metaphysics of F1\mathbb F_1

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    In the present paper, dedicated to Yuri Manin, we investigate the general notion of rings of S[ÎŒn,+]\mathbb S[\mu_{n,+}]-polynomials and relate this concept to the known notion of number systems. The Riemann-Roch theorem for the ring Z\mathbb Z of the integers that we obtained recently uses the understanding of Z\mathbb Z as a ring of polynomials S[X]\mathbb S[X] in one variable over the absolute base S\mathbb S, where 1+1=X+X21+1=X+X^2. The absolute base S\mathbb S (the categorical version of the sphere spectrum) thus turns out to be a strong candidate for the incarnation of the mysterious F1\mathbb F_1.Comment: Dedicated to Yuri Manin, 14 Figure

    Counter-examples to a conjecture of Karpenko for spin groups

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    Consider the canonical morphism from the Chow ring of a smooth variety XX to the associated graded ring of the topological filtration on the Grothendieck ring of XX. In general, this morphism is not injective. However, Nikita Karpenko conjectured that these two rings are isomorphic for a generically twisted flag variety XX of a semisimple group GG. The conjecture was first disproved by Nobuaki Yagita for G=Spin(2n+1)G=\mathop{\mathrm{Spin}}(2n+1) with n=8,9n=8, 9. Later, another counter-example to the conjecture was given by Karpenko and the first author for n=10n=10. In this note, we provide an infinite family of counter-examples to Karpenko's conjecture for any 22-power integer nn greater than 44. This generalizes Yagita's counter-example and its modification due to Karpenko for n=8n=8.Comment: 28 page

    Valuative lattices and spectra

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    The first part of the present article consists in a survey about the dynamical constructive method designed using dynamical theories and dynamical algebraic structures. Dynamical methods uncovers a hidden computational content for numerous abstract objects of classical mathematics, which seem a priori inaccessible constructively, e.g., the algebraic closure of a (discrete) field. When a proof in classical mathematics uses these abstract objects and results in a concrete outcome, dynamical methods generally make possible to discover an algorithm for this concrete outcome. The second part of the article applies this dynamical method to the theory of divisibility. We compare two notions of valuative spectra present in the literature and we introduce a third notion, which is implicit in an article devoted to the dynamical theory of algebraically closed discrete valued fields. The two first notions are respectively due to Huber \& Knebusch and to Coquand. We prove that the corresponding valuative lattices are essentially the same. We establish formal Valuativestellens\"atze corresponding to these theories, and we compare the various resulting notions of valuative dimensions.Comment: This file contains also a French version of the paper. English version appears in the Proceedings of Graz Conference on Rings and Factorizations 2021. Title: Algebraic, Number Theoretic, and Topological Aspects of Ring Theory. Editors: Jean-Luc Chabert, Marco Fontana, Sophie Frisch, Sarah Glaz, Keith Johnson. Springer 2023 ISBN 978-3-031-28846-3 DOI 10.1007/978-3-031-28847-

    Carlitz twists: their motivic cohomology, regulators, zeta values and polylogarithms

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    The integral tt-motivic cohomology and the class module of a (rigid analytically trivial) Anderson tt-motive were introduced by the first author in [Gaz22b]. This paper is devoted to their determination in the particular case of tensor powers of the Carlitz tt-motive, namely, the function field counterpart A‟(n)\underline{A}(n) of Tate twists Z(n)\mathbb{Z}(n). We find out that these modules are in relation with fundamental objects of function field arithmetic: integral tt-motivic cohomology governs linear relations among Carlitz polylogarithms, its torsion is expressed in terms of the denominator of Bernoulli-Carlitz numbers and the Fitting ideal of class modules is a special zeta value. We also express the regulator of A‟(n)\underline{A}(n) for positive nn in terms of generalized Carlitz polylogarithms; after establishing their algebraic relations using difference Galois theory together with the Anderson-Brownawell-Papanikolas criterion, we prove that the regulator is an isomorphism if, and only if, nn is prime to the characteristic.Comment: This represents a significant update compared to the previous version. Notably, our late conjecture regarding the Fitting ideal of the torsion in the class module, expressed as Carlitz zeta values, has been established (Thm. C). Additionally, we clarified the connection between integral t-motivic cohomology and Carlitz polylogs (Thm. A) and determined the regulator rank

    Perspectivity in complemented modular lattices and regular rings

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    Based on an analogue for systems of partial isomorphisms between lower sections in a complemented modular lattice we prove that principal right ideals aR≅bRaR \cong bR in a (von Neumann) regular ring RR are perspective if aR∩bRaR \cap bR is of finite height in L(R)L(R). This is applied to derive, for existence-varieties V\mathcal{V} of regular rings, equivalence of unit-regularity and direct finiteness, both conceived as a property shared by all members of V\mathcal{V}

    Local symmetry groups for arbitrary wavevectors

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    We present an algorithm for the determination of the local symmetry group for arbitrary k-points in 3D Brillouin zones. First, we test our implementation against tabulated results available for standard high-symmetry points (given by universal fractional coordinates). Then, to showcase the general applicability of our methodology, we produce the irreducible representations for the ‘non-universal high-symmetry’ points, first reported by Setyawan and Curtarolo (2010 Comput. Mater. Sci. 49 299). The present method can be regarded as a first step for the determination of elementary band decompositions and symmetry-enforced constraints in crystalline topological materials.</p
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