arXiv subjects

Wolfgang Bertram

Publications and source records attributed to Wolfgang Bertram.

At least 19 recordsLinked to original sources

arXiv2025

On Conway's Numbers and Games, the Von Neumann Universe, and Pure Set Theory

We take up Dedekind's question ''Was sind und was sollen die Zahlen?'' (''What are numbers, and would should they be?''), with the aim to describe the place that Conway's (Surreal) Numbers and Games take, or deserve to take, in the whole of mathematics. Rather than just reviewing the work of Conway, and subsequent one by Gonshor, Alling, Ehrlich, and others, we propose a new setting which puts the theory of surreal numbers onto the firm ground of ''pure'' set theory. This approach is closely related to Gonshor's one by ''sign expansions'', but appears to be significantly simpler and clearer, and hopefully may contribute to realizing that ''surreal'' numbers are by no means surrealistic, goofy or wacky. They could, and probably should, play a central role in mathematics. We discuss the interplay between the various approaches to surreal numbers, and analyze the link with Conway's original approach via Combinatorial Game Theory (CGT). To clarify this, we propose to call pure set theory the algebraic theory of pure sets, or in other terms, of the algebraic structures of the von Neumann universe. This topic may be interesting in its own right: it puts CGT into a broad context which has a strong ''quantum flavor'', and where Conway's numbers (as well as their analogue, the nimbers) arise naturally.

math.LO↗

arXiv2024

On group and loop spheres

We investigate the problem of defining group or loop structures on spheres, where by ''sphere'' we mean the level set q(x) = c of a general K-valued quadratic form q, for an invertible scalar c. When K is a field and q non-degenerate, then this corresponds to the classical theory of composition algebras; in particular, for K = R and positive definite forms, we obtain the sequence of the four real division algebras R, C, H (quaternions), O (octonions). Our theory is more general, allowing that K is merely a ring, and the form q possibly degenerate. To achieve this goal, we give a more geometric formulation, replacing the theory of binary composition algebras by ternary algebraic structures, thus defining categories of group spherical and of Moufang spherical spaces. In particular, we develop a theory of ternary Moufang loops, and show how it is related to the Albert-Cayley-Dickson construction and to generalized ternary octonion algebras. At the bottom, a starting point of the whole theory is the (elementary) result that every 2-dimensional quadratic space carries a canonical structure of commutative group spherical space.

math.GR↗

arXiv2021

Graded sets, graded groups, and Clifford algebras

We define a general notion of centrally $Γ$-graded sets and groups and of their graded products, and prove some basic results about the corresponding categories: most importantly, they form braided monoidal categories. Here, $Γ$ is an arbitrary (generalized) ring. The case $Γ$ = Z/2Z is studied in detail: it is related to Clifford algebras and their discrete Clifford groups (also called Salingaros Vee groups).

math.CT↗

arXiv2021

A functorial approach to differential calculus

We show that differential calculus (in its usual form, or in the general form of topological differential calculus) can be fully imdedded into a functor category (functors from a small category of anchord tangent algebras to anchored sets). To prepare this approach, we define a new, symmetric, presentation of differential calculus, whose main feature is the central r{ô}le played by the anchor map, which we study in detail. Our aim for developing this theory is twofold: (1) define a setting for calculus over any commutative ring, including finite rings; (2) define a setting that can be generalized to categories of graded rings (super differential calculus).

math.AG↗

arXiv2020

Distributive lattices, associative geometries: the arithmetic case

We prove an identity for five arguments, valid in the lattice of natural numbers with gcd and lcm as lattice operations. More generally, this identity characterizes arbitrary distributive lattices. Fixing three of the five arguments, we always get associative products, and thus every distributive lattice carries many semigroup structures. In the arithmetic case, we explicitly compute multiplication tables of such semigroups and describe some of their properties. Many of them are periodic, and can be seen as "non-commutative analogs" of the rings Z/nZ.

math.GR↗

arXiv2018

An Essay on the Completion of Quantum Theory. II: Unitary Time Evolution

In this second part of the `essay on the completion of quantum theory' we define the {\em unitary setting of completed quantum mechanics}, by adding as intrinsic data to those from Part I (arXiv:1711.08643) the choice of a north pole N and south pole S in the geometric space. Then we explain that, in the unitary setting, a complete observable corresponds to a right (or left) invariant vector field (Hamiltonian field) on the geometric space, and {\em unitary time evolution} is the flow of such a vector field. This interpretation is in fact nothing but the Lie group-Lie group algebra correspondence, for a geometric space that can be interpreted as the Cayley transform of the usual, Hermitian operator space. In order to clarify the geometric nature of this setting, we realize the Cayley transform as a member of a natural octahedral group that can be associated to any triple of pairwise transversal elements.

math-ph↗

arXiv2018

Cyclic orders defined by ordered jordan algebras

We define a general notion of partially ordered Jordan algebra (over a partially ordered ring), and we show that the Jordan geometry associated to such a Jordan algebra admits a natural invariant partial cyclic order, whose intervals are modelled on the symmetric cone of the Jordan algebra. We define and describe, by affine images of intervals, the interval topology on the Jordan geometry, and we outline a reserch program aiming at generalizing main features of the theory of classical symmetric cones and bounded symmetric domains.

math.RA↗

arXiv2017

An Essay on the Completion of Quantum Theory. I: General Setting

We propose a geometric setting of the axiomatic mathematical formalism of quantum theory. Guided by the idea that understanding the mathematical structures of these axioms is of similar importance as was historically the process of understanding the axioms of geometry, we complete the spaces of observables and of states in a similar way as in classical geometry linear or affine spaces are completed by projective spaces. In this sense, our theory can be considered as a "completion of usual linear quantum theory" , such that the usual theory appears as the special case where a reference frame is fixed once and for all. In the present first part, this general setting is explained. Dynamics (time evolution) will be discussed in subsequent work.

math-ph↗

arXiv2017

Lie Calculus

We explain that general differential calculus and Lie theory have a common foundation: Lie Calculus is differential calculus, seen from the point of view of Lie theory, by making use of the groupoid concept as link between them. Higher order theory naturally involves higher algebra (n-fold groupoids).(conceptual, topological) differential calculus, groupoids, higher algebra($n$-fold groupoids), Lie group, Lie groupoid, tangent groupoid, cubes of rings

math.GR↗

arXiv2016

A precise and general notion of manifold

We give a completely formalized definition of a notion of " general manifold ". It turns out that " gluing data " form an equivalence-partially ordered set (e-pos), which is a special instance of an ordered groupoid. We state and prove reconstruction theorems, allowing to reconstruct general manifolds and their mor-phisms from such gluing data. To describe morphisms between manifolds, the notion of natural relations between groupoids is introduced, which emphasizes the close analogy with natural transformations of general category theory.

math.CT↗

arXiv2015

Conceptual differential calculus part ii: Cubic higher order calculus

Following the programme set out in Part I of this work, we develop a conceptual higher order differential calculus. The '' local linear algebra '' defined in Part I is generalized by '' higher order local linear algebra ''. The underlying combinatorial object of such higher algebra is the natural n-dimensional hyper-cube, and so we qualify this calculus as '' cubic ''. More precisely, we define two versions of conceptual cubic calculus: '' full '' and '' symmetric cubic ''. The theory thus initiated sheds new light on several foundational issues.

math.DG↗

arXiv2015

Conceptual Differential Calculus. I: First Order Local Linear Algebra

We give a rigorous formulation of the intuitive idea that a differentiable map should be thesame thing as a locally, or infinitesimally, linear map: just as a linear map respects the operations of addition and multiplication by scalars ina vector space or module, a locally linear map is defined to be a map respecting two canonical operationsliving "over" its domain of definition.These two operations are composition laws of a canonical groupoid and of a scaled action category, respectively,fitting together into a canonical double category. Local linear algebra (of first order) is the study of such double categories and of their morphisms; it is a purely algebraic and conceptual (i.e., categorical and chart-independent) version of first order differential calculus. In subsequent work, the higher order theory (using higher multiple categories) will be investigated.

math.CT↗

arXiv2014

Universal Associative Geometry

We generalize parts of the theory of associative geometries developed by Kinyon and the author in the framework of universal algebra: we prove that certain associoid structures, such as pregroupoids and principal equivalence relations, have a natural prolongation from a set to its the power set. We reinvestigate the case of homogeneous pregroupoids (corresponding to the projective geometry of a group) from the point of view of pairs of commuting principal equivalence relations. We use the ternary approach to groupoids developed by Anders Kock, and the torsors defined by our construction can be seen as a generalisation of the known groups of bisections of a groupoid.

math.CT↗

arXiv2014

Jordan Geometries - an Approach by Inversions

Jordan geometries are defined as spaces equipped with point reflections depending on triples of points, exchanging two of the points and fixing the third. In a similar way, symmetric spaces have been defined by Loos (Symmetric Spaces I, 1969) as spaces equipped with point reflections depending on a point and fixing this point; therefore the theories of Jordan geometries and of symmetric spaces are closely related to each other -- in order to describe this link, the notion of symmetry actions of torsors and of symmetric spaces is introduced. Jordan geometries give rise both to symmetry actions of certain abelian torsors and of certain symmetric spaces, which in a sense are dual to each other. By using an algebraic differential calculus generalizing the classical Weil functors (see arxiv:1402.2619), we attach a tangent object to such geometries, namely a Jordan pair, respectively a Jordan algebra. The present approach works equally well over base rings in which 2 is not invertible (and in particular over the integers), and hence can be seen as a globalization of quadratic Jordan pairs; it also has a very transparent relation with the theory of associative geometries developped by M. Kinyon and the author.

math.RA↗

arXiv2014

Weil Spaces and Weil-Lie Groups

We define Weil spaces, Weil manifolds, Weil varieties and Weil Lie groups over an arbitrary commutative base ring K (in particular, over discrete rings such as the integers), and we develop the basic theory of such spaces, leading up the definition of a Lie algebra attached to a Weil Lie group. By definition, the category of Weil spaces is the category of functors from K-Weil algebras to sets; thus our notion of Weil space is similar to, but weaker than the one of Weil topos defined by E. Dubuc (1979). In view of recent result on Weil functors for manifolds over general topological base fields or rings by A. Souvay, this generality is the suitable context to formulate and to prove general results of infinitesimal differential geometry, as started by the approach developed in Bertram, Mem. AMS 900.

math.GR↗

arXiv2013

Commutative and Non-commutative Parallelogram Geometry: an Experimental Approach

By "parallelogram geometry" we mean the elementary, "commutative", geometry corresponding to vector addition, and by "trapezoid geometry" a certain "non-commutative deformation" of the former. This text presents an elementary approach via exercises using dynamical software (such as geogebra), hopefully accessible to a wide mathematical audience, from undergraduate students and high school teachers to researchers, proceeding in three steps: (1) experimental geometry, (2) algebra (linear algebra and elementary group theory), and (3) axiomatic geometry.

math.HO↗

arXiv2012

Torsors and ternary Moufang loops arising in projective geometry

We give an interpretation of the construction of torsors from preceding work (Bertram, Kinyon: Associative Geometries. I, J. Lie Theory 20) in terms of classical projective geometry. For the Desarguesian case, this leads to a reformulation of certain results from lot.cit., whereas for the Moufang case the result is new. But even in the Desarguesian case it sheds new light on the relation between the lattice structure and the algebraic structures of a projective space.

math.GR↗

arXiv2012

Homotopes of Symmetric Spaces I. Construction by Algebras with Two Involutions

We investigate a special kind of contraction of symmetric spaces (respectively, of Lie triple systems), called homotopy. In this first part of a series of two papers we construct such contractions for classical symmetric spaces in an elementary way by using associative algebras with several involutions. This construction shows a remarkable duality between the underlying "space" and the "deformation parameter".

math.DG↗