This book aims to provide an overview of several topics in advanced Differential Geometry and Lie Group Theory, all of them stemming from mathematical problems in supersymmetric physical theories. It presents a mathematical illustration of the main development in geometry and symmetry theory that occurred under the fertilizing influence of supersymmetry/supergravity. The contents are mainly of mathematical nature, but each topic is introduced by historical information and enriched with motivations from high energy physics, which help the reader in getting a (...) deeper comprehension of the subject. (shrink)

In this paper we study a certain class of Lie algebras over commutative von Neumann algebras satisfying a certain finiteness condition. By using Boolean valued methods developed by Takeuti [8]-[11], we will establish the basic structure and representation theorems.

The elementary theory of a nontrivial free Lie algebra over a commutative integral domain is unstable and has the strict order property.

The rigidity of the real semisimple Lie algebra $${\mathfrak {so}}$$ so was first proved in a brief paper published by Fantappiè in 1954. The purpose of this note is to provide some historical context for this work and discuss why no further developments of this result were pursued by Italian mathematicians at the time.

In this paper, we develop the beginning of Lie-differential algebra, in the sense of Kolchin by using tools introduced by Hubert in [E. Hubert, Differential algebra for derivations with nontrivial commutation rules, J. Pure Appl. Algebra 200 163–190]. In particular it allows us to adapt the results of Tressl 3933–3951]) by showing the existence of a theory of Lie-differential fields of characteristic zero. This theory will serve as a model companion for every theory of large and Lie-differential (...) fields extending a model complete theory of pure fields. As an application, we introduce the Lie counterpart of classical theories of differential fields in several commuting derivations. (shrink)

This book presents the foundations of a general theory of algebras. Often called “universal algebra”, this theory provides a common framework for all algebraic systems, including groups, rings, modules, fields, and lattices. Each chapter is replete with useful illustrations and exercises that solidify the reader's understanding. The book begins by developing the main concepts and working tools of algebras and lattices, and continues with examples of classical algebraic systems like groups, semigroups, monoids, and categories. The essence of the book (...) lies in Chapter 4, which provides not only basic concepts and results of general algebra, but also the perspectives and intuitions shared by practitioners of the field. The book finishes with a study of possible uniqueness of factorizations of an algebra into a direct product of directly indecomposable algebras. There is enough material in this text for a two semester course sequence, but a one semester course could also focus primarily on Chapter 4, with additional topics selected from throughout the text. (shrink)

A simple model for the description of atomic and ionic species with spectra exhibiting a quantum defect is solved using the Lie algebra su(1, 1). The quantum defect of bound states is related to the phase shift of scattering states. The resonances are discussed in terms of the nonunitary representations of this algebra.

This book is intended for mathematicians. Its origins lie in a course of lectures given by an algebraist to a class which had just completed a sub stantial course on abstract algebra. Consequently, our treatment ofthe sub ject is algebraic. Although we assurne a reasonable level of sophistication in algebra, the text requires little more than the basic notions of group, ring, module, etc. A more detailed knowledge of algebra is required for some of . the exercises. (...) We also assurne a familiarity with the main ideas of set theory, including cardinal numbers and Zorn's Lemma. In this book, we carry out a mathematical study of the logic used in mathematics. We do this by constructing a mathematical model oflogic and applying mathematics to analyse the properties of the model. We therefore regard all our existing knowledge of mathematics as being applicable to the analysis of the model, and in particular we accept set theory as part of the meta-Ianguage. We are not attempting to construct a foundation on which all mathematics is to be based-rather, any conclusions to be drawn about the foundations of mathematics co me only by analogy with the model, and are to be regarded in much the same way as the conclusions drawn from any scientific theory. (shrink)

In this paper, we study the field of algebraic numbers with a set of elements of small height treated as a predicate. We prove that such structures are not simple and have the independence property. A real algebraic integer is called a Salem number if α and are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. We relate the simplicity of a (...) certain pair with Lehmer's conjecture and obtain a model‐theoretic characterization of Lehmer's conjecture for Salem numbers. (shrink)

All causal Lie products of solutions of the Klein-Gordon equation and the wave equation in Minkowski space are determined. The results shed light on the origin of the algebraic structures underlying quantum field theory.

We define a Lie differential field as a field of characteristic 0 with an action, as derivations on , of some given Lie algebra . We assume that is a finite-dimensional vector space over some sub-field given in advance. As an example take the field of rational functions on a smooth algebraic variety, with .For every simple extension of Lie differential fields we find a finite system of differential equations that characterizes it. We then define, using first-order conditions, a (...) collection of allowed systems of differential equations s.t. the above characteristic systems are allowed. We prove that for every allowed system there exists a generic solution in some extension, and this solution is unique .We construct the model completion of the theory of Lie differential fields by adding axioms stating that every allowed system has almost generic solutions. The construction is a generalization of Blum's axioms for DCF0. We also show that this model completion is ω-stable. (shrink)

This book is the first to systematically develop, explain, and apply diagrammatic projection operators to construct all semi-simple Lie algebras, both classical and exceptional.

The CPT theorem states that any causal, Lorentz-invariant, thermodynamically well-behaved quantum field theory must also be invariant under a reflection symmetry that reverses the direction of time, flips spatial parity, and conjugates charge. Although its physical basis remains obscure, CPT symmetry appears to be necessary in order to unify quantum mechanics with relativity. This paper attempts to decipher the physical reasoning behind proofs of the CPT theorem in algebraic quantum field theory. Ultimately, CPT symmetry is linked to a systematic reversal (...) of the C*-algebraic Lie product that encodes the generating relationship between observables and symmetries. In any physically reasonable relativistic quantum field theory it is always possible to systematically reverse this generating relationship while preserving the dynamics, spectra, and localization properties of physical systems. Rather than the product of three separate reflections, CPT symmetry is revealed to be a single global reflection of the theory’s state space. (shrink)

Bundles of C*-algebras can be used to represent limits of physical theories whose algebraic structure depends on the value of a parameter. The primary example is the ℏ→0 limit of the C*-algebras of physical quantities in quantum theories, represented in the framework of strict deformation quantization. In this paper, we understand such limiting procedures in terms of the extension of a bundle of C*-algebras to some limiting value of a parameter. We prove existence and uniqueness results for such extensions. Moreover, (...) we show that such extensions are functorial for the C*-product, dynamical automorphisms, and the Lie bracket (in the ℏ→0 case) on the fiber C*-algebras. (shrink)

Just as Kaplansky [4] has introduced the notion of an AW*-module as a generalization of a complex Hilbert space, we introduce the notion of an AL*-algebra, which is a generalization of that of an L*-algebra invented by Schue [9, 10]. By using Boolean valued methods developed by Ozawa [6–8], Takeuti [11–13] and others, we establish its basic properties including a fundamental structure theorem. This paper should be regarded as a continuation or our previous paper [5], the familiarity with (...) which is presupposed. MSC: 03C90, 03E40, 17B65, 46L10. (shrink)

Recently the interest in Bohm realist interpretation of quantum mechanics has grown. The important advantage of this approach lies in the possibility to introduce non-locality ab initio, and not as an "unexpected host". In this book the authors give a detailed analysis of quantum potential, the non-locality term and its role in quantum cosmology and information. The different approaches to the quantum potential are analysed, starting from the original attempt to introduce a realism of particles trajectories (influenced by de Broglie's (...) pilot wave) to the recent dynamic interpretation provided by Goldstein, Durr, Tumulka and Zanghì, and the geometrodynamic picture, with suggestion about quantum gravity. Finally we focus on the algebraic reading of Hiley and Birkbeck school, that analyse the meaning of the non-local structure of the world, bringing important consequences for the space, time and information concepts. (shrink)

A. THE TEXT The main importance of these two treatises lies in the insight they provide into Spinoza's conception of the relation between mathematics and certain disciplines not touched upon elsewhere in his major writings. The mathematics they involve are not the as those of the Ethics however, and the precise connection same between the geometrical order of this work and these excursions into optics and probability is by no means obvious. Add to this difficulty the knotty problems presented by (...) their editorial his tory, dating and scientific background, and it is not perhaps surprising that in spite of the fact that they provide such an excellent illustration of Spinoza's reaction to certain important developments in the history of physics and mathematics, they should not, so far, have attracted much attention. They were first published in 1687 by Levyn van Dyck (d. 1695), official printer to the town council in The Hague. Printing anything by Spinoza was not without its risks, and it is probably significant that during the same year van Dyck should also have published a lengthy and elaborate refutation of Spinozism by 1 the pious and eccentric physician ]. F. Helvetius. Spinoza's name was omitted from the title-page, possibly because the editor or publisher thought that his reputation as an atheist might prejudice the sale of the booklet, and it was not until 1860 that the Amsterdam bookseller Frederik Muller (1817-1881) identified him as its author. (shrink)

We construct, by model-theoretic methods, several exponentiations on the universal enveloping algebra U of the Lie algebra.

A novel conceptual framework is introduced for the Complexity Levels Theory in a Categorical Ontology of Space and Time. This conceptual and formal construction is intended for ontological studies of Emergent Biosystems, Super-complex Dynamics, Evolution and Human Consciousness. A claim is defended concerning the universal representation of an item’s essence in categorical terms. As an essential example, relational structures of living organisms are well represented by applying the important categorical concept of natural transformations to biomolecular reactions and relational structures that (...) emerge from the latter in living systems. Thus, several relational theories of living systems can be represented by natural transformations of organismic, relational structures. The ascent of man and other living organisms through adaptation, is viewed in novel categorical terms, such as variable biogroupoid representations of evolving species. Such precise but flexible evolutionary concepts will allow the further development of the unifying theme of local-to-global approaches to highly complex systems in order to represent novel patterns of relations that emerge in super- and ultra-complex systems in terms of compositions of local procedures. Solutions to such local-to-global problems in highly complex systems with ‘broken symmetry’ might be possible to be reached with the help of higher homotopy theorems in algebraic topology such as the generalized van Kampen theorems (HHvKT). Categories of many-valued, Łukasiewicz-Moisil (LM) logic algebras provide useful concepts for representing the intrinsic dynamic ‘asymmetry’ of genetic networks in organismic development and evolution, as well as to derive novel results for (non-commutative) Quantum Logics. Furthermore, as recently pointed out by Baianu and Poli (Theory and applications of ontology, vol 1. Springer, Berlin, in press), LM-logic algebras may also provide the appropriate framework for future developments of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. As shown in the preceding two papers in this issue, a paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics—which is presently unfolding in physics, mathematics, life and cognitive sciences—may be implemented through realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics. (shrink)

We look at a hierarchical arrangement of ideals in an MV -algebra. The principal classes of ideals studied are the maximals, the primes, the local and perfect ideals and the semi-locals. Beyond these special classes of ideals are the general ideals. Herein we study some relationships among these classes and, more specifically, the products of ideals of these classes. Among the results obtained are the square of a prime ideal is a local ideal, the finite product of prime ideals (...) is a semilocal ideal. We also show that the set of local ideals shares many of the properties of the class of prime ideals, as the Lying Over Theorem and the Going Up Theorem. (shrink)

The formulas for the Lie covariant differentiation of spinors are deduced from an algebraic viewpoint. The Lie covariant derivative of the spinor connection is calculated, and is given a geometric meaning. A theorem about the Lie covariant derivative of an operator in spin space that was stated in Part I of this work is discussed.

§1. Introduction. Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the ‘large-scale structure’ of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow rather easily from saturation properties of ultrapowers, and in this survey, we want to present two applications of the van den Dries-Wilkie approach. Using ultrapowers (...) we obtain an explicit description of the asymptotic cone of a semisimple Lie group. From this description, using semi-algebraic groups and non-standard methods, we can give a short proof of the Margulis Conjecture. In a second application, we use set theory to answer a question of Gromov.§2. Definitions. The intuitive idea behind Gromov's concept of an asymptotic cone was to look at a given metric space from an ‘infinite distance’, so that large-scale patterns should become visible. In his original definition this was done by gradually scaling down the metric by factors 1/nfornϵ ℕ. In the approach by van den Dries and Wilkie, this idea was captured by ultrapowers. Their construction is more general in the sense that the asymptotic cone exists for any metric space, whereas in Gromov's original definition, the asymptotic cone existed only for a rather restricted class of spaces. (shrink)

Du silex à l'internet l'Homme dans toute sa complexité.

A “mental” multiplication scheme is given for the super hypercomplex numbers, which extend the 16-element Dirac algebra to 32 elements by appending the complex octonions. This extends the 5-vectors of relativity to 9-vectors. The problems with nonassociativity, for the group structures and wave equation covariance, are explored.

The World Medical Association’s revised Declaration of Helsinki endorses the view that all trial participants in every country are entitled to the worldwide best standard of care. In this paper the authors show that this requirement has been rejected by every national and international committee that has examined this issue. They argue that the consensus view now holds that it is ethically permissible, in some circumstances, to provide research participants less than the worldwide best care. Finally, the authors show that (...) there is also consensus regarding the broad conditions under which this is acceptable. (shrink)

This paper analyses Steven Vogel's claim that his account of a post-natural environmental philosophy solves the dualism problem within the field. Through what I will call a novel critique of social constructionism, this paper examines whether Vogel's attempt succeeds or whether it reinforces the problem he wants to solve. Could the ontological foundations of social constructivism themselves be in conflict with Vogel's stated aim of overcoming the human/nature dualism? The last part of the paper focuses on the significance and role (...) of a post-natural environmental philosophy for the field of environmental philosophy in general. (shrink)

With a specific example from Norway and inspiration from Sara Ahmed’s The Promise of Happiness, this article demonstrates how today’s educational rhetoric lacks the language and will to recognise a key pedagogical dimension in education: what happens when the normative ambitions of education and students meet. At best, teaching students life skills to mitigate their mental health issues is naive. Inspired by Ahmed, such an initiative might actually work against its purpose. At a time when educational outcomes are emphasised in (...) local and international political contexts, I argue that the task of philosophy of education should be 1) to reclaim the significance of the pedagogical dimension in education and 2) to philosophise on what negative emotions such as unhappiness require of education. (shrink)

This article explores possibilities for establishing dialogues between feminism and constructivism in the field of technology studies. Based on an overview of Norwegian feminist debates about technology, it indicates several points where feminism and constructivism meet and can mutually benefit from each other. The article critically examines feminist studies questioning the problems of technological determinism, social deternacnism, and essentialism. It criticizes constructivism for a lack of concern for gender and politics but holds that it is still possible to use theoretical (...) tools from constructivism in feminist analyses. Fruitful dialogues require the application of the principle of symmetry to the dcalogues and sharing some common ground and mutual recognation of each other's strengths and weaknesses. (shrink)

ArgumentDuring the 1950s it became apparent that antibiotics could not conquer all microbes, and a series of tests were developed to assess the susceptibility of microbes to antibiotics. This article explores the development and standardization of one such testing procedure which became dominant in the Nordic region, and how the project eventually failed in the late 1970s. The standardization procedures amounted to a comprehensive scheme, standardizing not only the materials used, but also the methods and the interpretation of the results. (...) Focusing on Sweden and Norway in particular, the article shows how this comprehensive standardization procedure accounted for several co-dependent factors and demanded collaboration within and across laboratories. Whereas literature on standardization has focused mostly on how facts and artefacts move within and across laboratories, I argue for the importance of also attending to regions and territories. More particularly, while arguing that the practices, ideals, and politics related to what have been called the “Nordic welfare state” were contributing to the design of the standardized procedure in the laboratory, I also argue that Scandinavia was drawn together as a unified region with and by these very same practices. (shrink)