2,160 research outputs found
Lie pairs
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
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
In the present paper, dedicated to Yuri Manin, we investigate the general
notion of rings of -polynomials and relate this concept
to the known notion of number systems. The Riemann-Roch theorem for the ring
of the integers that we obtained recently uses the understanding of
as a ring of polynomials in one variable over the
absolute base , where . The absolute base
(the categorical version of the sphere spectrum) thus turns out to be a strong
candidate for the incarnation of the mysterious .Comment: Dedicated to Yuri Manin, 14 Figure
Counter-examples to a conjecture of Karpenko for spin groups
Consider the canonical morphism from the Chow ring of a smooth variety to
the associated graded ring of the topological filtration on the Grothendieck
ring of . In general, this morphism is not injective. However, Nikita
Karpenko conjectured that these two rings are isomorphic for a generically
twisted flag variety of a semisimple group . The conjecture was first
disproved by Nobuaki Yagita for with .
Later, another counter-example to the conjecture was given by Karpenko and the
first author for . In this note, we provide an infinite family of
counter-examples to Karpenko's conjecture for any -power integer greater
than . This generalizes Yagita's counter-example and its modification due to
Karpenko for .Comment: 28 page
Valuative lattices and spectra
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
The integral -motivic cohomology and the class module of a (rigid
analytically trivial) Anderson -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 -motive, namely, the function field
counterpart of Tate twists . We find out that
these modules are in relation with fundamental objects of function field
arithmetic: integral -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 for positive
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, 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
Based on an analogue for systems of partial isomorphisms between lower
sections in a complemented modular lattice we prove that principal right ideals
in a (von Neumann) regular ring are perspective if
is of finite height in . This is applied to derive, for
existence-varieties of regular rings, equivalence of
unit-regularity and direct finiteness, both conceived as a property shared by
all members of
Local symmetry groups for arbitrary wavevectors
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 ânon-universal high-symmetryâ 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
- âŠ