Three Fermion Generations with Two Unbroken Gauge Symmetries from the Complex Sedenions
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Notices of the American Mathematical Society
ISSN 0002-9920 of the American Mathematical Society February 2006 Volume 53, Number 2 Math Circles and Olympiads MSRI Asks: Is the U.S. Coming of Age? page 200 A System of Axioms of Set Theory for the Rationalists page 206 Durham Meeting page 299 San Francisco Meeting page 302 ICM Madrid 2006 (see page 213) > To mak• an antmat•d tub• plot Animated Tube Plot 1 Type an expression in one or :;)~~~G~~~t;~~i~~~~~~~~~~~~~:rtwo ' 2 Wrth the insertion point in the 3 Open the Plot Properties dialog the same variables Tl'le next animation shows • knot Plot 30 Animated + Tube Scientific Word ... version 5.5 Scientific Word"' offers the same features as Scientific WorkPlace, without the computer algebra system. Editors INTERNATIONAL Morris Weisfeld Editor-in-Chief Enrico Arbarello MATHEMATICS Joseph Bernstein Enrico Bombieri Richard E. Borcherds Alexei Borodin RESEARCH PAPERS Jean Bourgain Marc Burger James W. Cogdell http://www.hindawi.com/journals/imrp/ Tobias Colding Corrado De Concini IMRP provides very fast publication of lengthy research articles of high current interest in Percy Deift all areas of mathematics. All articles are fully refereed and are judged by their contribution Robbert Dijkgraaf to the advancement of the state of the science of mathematics. Issues are published as S. K. Donaldson frequently as necessary. Each issue will contain only one article. IMRP is expected to publish 400± pages in 2006. Yakov Eliashberg Edward Frenkel Articles of at least 50 pages are welcome and all articles are refereed and judged for Emmanuel Hebey correctness, interest, originality, depth, and applicability. Submissions are made by e-mail to Dennis Hejhal [email protected]. -
Gauging the Octonion Algebra
UM-P-92/60_» Gauging the octonion algebra A.K. Waldron and G.C. Joshi Research Centre for High Energy Physics, University of Melbourne, Parkville, Victoria 8052, Australia By considering representation theory for non-associative algebras we construct the fundamental and adjoint representations of the octonion algebra. We then show how these representations by associative matrices allow a consistent octonionic gauge theory to be realized. We find that non-associativity implies the existence of new terms in the transformation laws of fields and the kinetic term of an octonionic Lagrangian. PACS numbers: 11.30.Ly, 12.10.Dm, 12.40.-y. Typeset Using REVTEX 1 L INTRODUCTION The aim of this work is to genuinely gauge the octonion algebra as opposed to relating properties of this algebra back to the well known theory of Lie Groups and fibre bundles. Typically most attempts to utilise the octonion symmetry in physics have revolved around considerations of the automorphism group G2 of the octonions and Jordan matrix representations of the octonions [1]. Our approach is more simple since we provide a spinorial approach to the octonion symmetry. Previous to this work there were already several indications that this should be possible. To begin with the statement of the gauge principle itself uno theory shall depend on the labelling of the internal symmetry space coordinates" seems to be independent of the exact nature of the gauge algebra and so should apply equally to non-associative algebras. The octonion algebra is an alternative algebra (the associator {x-1,y,i} = 0 always) X -1 so that the transformation law for a gauge field TM —• T^, = UY^U~ — ^(c^C/)(/ is well defined for octonionic transformations U. -
Division Rings and V-Domains [4]
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 99, Number 3, March 1987 DIVISION RINGS AND V-DOMAINS RICHARD RESCO ABSTRACT. Let D be a division ring with center k and let fc(i) denote the field of rational functions over k. A square matrix r 6 Mn(D) is said to be totally transcendental over fc if the evaluation map e : k[x] —>Mn(D), e(f) = }(r), can be extended to fc(i). In this note it is shown that the tensor product D®k k(x) is a V-domain which has, up to isomorphism, a unique simple module iff any two totally transcendental matrices of the same order over D are similar. The result applies to the class of existentially closed division algebras and gives a partial solution to a problem posed by Cozzens and Faith. An associative ring R is called a right V-ring if every simple right fi-module is injective. While it is easy to see that any prime Goldie V-ring is necessarily simple [5, Lemma 5.15], the first examples of noetherian V-domains which are not artinian were constructed by Cozzens [4]. Now, if D is a division ring with center k and if k(x) is the field of rational functions over k, then an observation of Jacobson [7, p. 241] implies that the simple noetherian domain D (g)/.k(x) is a division ring iff the matrix ring Mn(D) is algebraic over k for every positive integer n. In their monograph on simple noetherian rings, Cozzens and Faith raise the question of whether this tensor product need be a V-domain [5, Problem (13), p. -
Pure Spinors to Associative Triples to Zero-Divisors
Pure Spinors to Associative Triples to Zero-Divisors Frank Dodd (Tony) Smith, Jr. - 2012 Abstract: Both Clifford Algebras and Cayley-Dickson Algebras can be used to construct Physics Models. Clifford and Cayley-Dickson Algebras have in common Real Numbers, Complex Numbers, and Quaternions, but in higher dimensions Clifford and Cayley-Dickson diverge. This paper is an attempt to explore the relationship between higher-dimensional Clifford and Cayley-Dickson Algebras by comparing Projective Pure Spinors of Clifford Algebras with Zero-Divisor structures of Cayley-Dickson Algebras. 1 Pure Spinors to Associative Triples to Zero-Divisors Frank Dodd (Tony) Smith, Jr. - 2012 Robert de Marrais and Guillermo Moreno, pioneers in studying Zero-Divisors, unfortunately have passed (2011 and 2006). Pure Spinors to Associative Triples ..... page 2 Associative Triples to Loops ............... page 3 Loops to Zero-Divisors ....................... page 6 New Phenomena at C32 = T32 ........... page 9 Zero-Divisor Annihilator Geometry .... page 17 Pure Spinors to Associative Triples Pure spinors are those spinors that can be represented by simple exterior wedge products of vectors. For Cl(2n) they can be described in terms of bivectors Spin(2n) and Spin(n) based on the twistor space Spin(2n)/U(n) = Spin(n) x Spin(n). Since (1/2)((1/2)(2n)(2n-1)-n^2) = (1/2)(2n^2-n-n^2) = = (1/2)(n(n-1)) Penrose and Rindler (Spinors and Spacetime v.2) describe Cl(2n) projective pure half-spinors as Spin(n) so that the Cl(2n) full space of pure half-spinors has dimension dim(Spin(n)) -
Theory of Trigintaduonion Emanation and Origins of Α and Π
Theory of Trigintaduonion Emanation and Origins of and Stephen Winters-Hilt Meta Logos Systems Aurora, Colorado Abstract In this paper a specific form of maximal information propagation is explored, from which the origins of , , and fractal reality are revealed. Introduction The new unification approach described here gives a precise derivation for the mysterious physics constant (the fine-structure constant) from the mathematical physics formalism providing maximal information propagation, with being the maximal perturbation amount. Furthermore, the new unification provides that the structure of the space of initial ‘propagation’ (with initial propagation being referred to as ‘emanation’) has a precise derivation, with a unit-norm perturbative limit that leads to an iterative-map-like computed (a limit that is precisely related to the Feigenbaum bifurcation constant and thus fractal). The computed can also, by a maximal information propagation argument, provide a derivation for the mathematical constant . The ideal constructs of planar geometry, and related such via complex analysis, give methods for calculation of to incredibly high precision (trillions of digits), thereby providing an indirect derivation of to similar precision. Propagation in 10 dimensions (chiral, fermionic) and 26 dimensions (bosonic) is indicated [1-3], in agreement with string theory. Furthermore a preliminary result showing a relation between the Feigenbaum bifurcation constant and , consistent with the hypercomplex algebras indicated in the Emanator Theory, suggest an individual object trajectory with 36=10+26 degrees of freedom (indicative of heterotic strings). The overall (any/all chirality) propagation degrees of freedom, 78, are also in agreement with the number of generators in string gauge symmetries [4]. -
An Introduction to Central Simple Algebras and Their Applications to Wireless Communication
Mathematical Surveys and Monographs Volume 191 An Introduction to Central Simple Algebras and Their Applications to Wireless Communication Grégory Berhuy Frédérique Oggier American Mathematical Society http://dx.doi.org/10.1090/surv/191 An Introduction to Central Simple Algebras and Their Applications to Wireless Communication Mathematical Surveys and Monographs Volume 191 An Introduction to Central Simple Algebras and Their Applications to Wireless Communication Grégory Berhuy Frédérique Oggier American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov Robert Guralnick MichaelI.Weinstein MichaelA.Singer 2010 Mathematics Subject Classification. Primary 12E15; Secondary 11T71, 16W10. For additional information and updates on this book, visit www.ams.org/bookpages/surv-191 Library of Congress Cataloging-in-Publication Data Berhuy, Gr´egory. An introduction to central simple algebras and their applications to wireless communications /Gr´egory Berhuy, Fr´ed´erique Oggier. pages cm. – (Mathematical surveys and monographs ; volume 191) Includes bibliographical references and index. ISBN 978-0-8218-4937-8 (alk. paper) 1. Division algebras. 2. Skew fields. I. Oggier, Fr´ed´erique. II. Title. QA247.45.B47 2013 512.3–dc23 2013009629 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
Subfields of Central Simple Algebras
The Brauer Group Subfields of Algebras Subfields of Central Simple Algebras Patrick K. McFaddin University of Georgia Feb. 24, 2016 Patrick K. McFaddin University of Georgia Subfields of Central Simple Algebras The Brauer Group Subfields of Algebras Introduction Central simple algebras and the Brauer group have been well studied over the past century and have seen applications to class field theory, algebraic geometry, and physics. Since higher K-theory defined in '72, the theory of algebraic cycles have been utilized to study geometric objects associated to central simple algebras (with involution). This new machinery has provided a functorial viewpoint in which to study questions of arithmetic. Patrick K. McFaddin University of Georgia Subfields of Central Simple Algebras The Brauer Group Subfields of Algebras Central Simple Algebras Let F be a field. An F -algebra A is a ring with identity 1 such that A is an F -vector space and α(ab) = (αa)b = a(αb) for all α 2 F and a; b 2 A. The center of an algebra A is Z(A) = fa 2 A j ab = ba for every b 2 Ag: Definition A central simple algebra over F is an F -algebra whose only two-sided ideals are (0) and (1) and whose center is precisely F . Examples An F -central division algebra, i.e., an algebra in which every element has a multiplicative inverse. A matrix algebra Mn(F ) Patrick K. McFaddin University of Georgia Subfields of Central Simple Algebras The Brauer Group Subfields of Algebras Why Central Simple Algebras? Central simple algebras are a natural generalization of matrix algebras. -
A Brief History of Ring Theory
A Brief History of Ring Theory by Kristen Pollock Abstract Algebra II, Math 442 Loyola College, Spring 2005 A Brief History of Ring Theory Kristen Pollock 2 1. Introduction In order to fully define and examine an abstract ring, this essay will follow a procedure that is unlike a typical algebra textbook. That is, rather than initially offering just definitions, relevant examples will first be supplied so that the origins of a ring and its components can be better understood. Of course, this is the path that history has taken so what better way to proceed? First, it is important to understand that the abstract ring concept emerged from not one, but two theories: commutative ring theory and noncommutative ring the- ory. These two theories originated in different problems, were developed by different people and flourished in different directions. Still, these theories have much in com- mon and together form the foundation of today's ring theory. Specifically, modern commutative ring theory has its roots in problems of algebraic number theory and algebraic geometry. On the other hand, noncommutative ring theory originated from an attempt to expand the complex numbers to a variety of hypercomplex number systems. 2. Noncommutative Rings We will begin with noncommutative ring theory and its main originating ex- ample: the quaternions. According to Israel Kleiner's article \The Genesis of the Abstract Ring Concept," [2]. these numbers, created by Hamilton in 1843, are of the form a + bi + cj + dk (a; b; c; d 2 R) where addition is through its components 2 2 2 and multiplication is subject to the relations i =pj = k = ijk = −1. -
Some Notes on the Real Numbers and Other Division Algebras
Some Notes on the Real Numbers and other Division Algebras August 21, 2018 1 The Real Numbers Here is the standard classification of the real numbers (which I will denote by R). Q Z N Irrationals R 1 I personally think of the natural numbers as N = f0; 1; 2; 3; · · · g: Of course, not everyone agrees with this particular definition. From a modern perspective, 0 is the \most natural" number since all other numbers can be built out of it using machinery from set theory. Also, I have never used (or even seen) a symbol for the whole numbers in any mathematical paper. No one disagrees on the definition of the integers Z = f0; ±1; ±2; ±3; · · · g: The fancy Z stands for the German word \Zahlen" which means \numbers." To avoid the controversy over what exactly constitutes the natural numbers, many mathematicians will use Z≥0 to stand for the non-negative integers and Z>0 to stand for the positive integers. The rational numbers are given the fancy symbol Q (for Quotient): n a o = a; b 2 ; b > 0; a and b share no common prime factors : Q b Z The set of irrationals is not given any special symbol that I know of and is rather difficult to define rigorously. The usual notion that a number is irrational if it has a non-terminating, non-repeating decimal expansion is the easiest characterization, but that actually entails a lot of baggage about representations of numbers. For example, if a number has a non-terminating, non-repeating decimal expansion, will its expansion in base 7 also be non- terminating and non-repeating? The answer is yes, but it takes some work to prove that. -
1. Introduction 2. Real Octonion Division Algebra
Pr´e-Publica¸c~oesdo Departamento de Matem´atica Universidade de Coimbra Preprint Number 19{02 EIGENVALUES OF MATRICES RELATED TO THE OCTONIONS ROGERIO´ SERODIO,^ PATR´ICIA DAMAS BEITES AND JOSE´ VITORIA´ Abstract: A pseudo real matrix representation of an octonion, which is based on two real matrix representations of a quaternion, is considered. We study how some operations defined on the octonions change the set of eigenvalues of the matrix obtained if these operations are performed after or before the matrix representation. The established results could be of particular interest to researchers working on estimation algorithms involving such operations. Keywords: Octonions, quaternions, real matrix representations, eigenvalues. Math. Subject Classification (2010): 11R52, 15A18. 1. Introduction Due to nonassociativity, the real octonion division algebra is not alge- braically isomorphic to a real matrix algebra. Despite this fact, pseudo real matrix representations of an octonion may be introduced, as in [1], through real matrix representations of a quaternion. In this work, the left matrix representation of an octonion over R, as called by Tian in [1], is considered. For the sake of completeness, some definitions and results, in particular on this pseudo representation, are recalled in Section 2. Using the mentioned representation, results concerning eigenvalues of ma- trices related to the octonions are established in Section 3. Previous research on this subject, although not explicitly applying real matrix representations of a quaternion, can be seen in [2]. 2. Real octonion division algebra Consider the real octonion division algebra O, that is, the usual real vector 8 space R , with canonical basis fe0; : : : ; e7g, equipped with the multiplication given by the relations eiej = −δije0 + "ijkek; Received January 28, 2019. -
1 Introduction 2 Central Simple Algebras
Alexandre Puttick Etale´ Cohomology December 13, 2018 Brauer groups and Galois Cohomology 1 Introduction The main goal of the next talks is to prove the following theorem: Theorem 1.1. Let K be a field extension of transcendence degree 1 over an algebraically 2 closed field k. Then Het´ (Spec K; Gm) = 0. sep Let k be an arbitrary field, and fix a separable closure k of k, and let Gk := Gal(ksep=k). The first step is to show Theorem 1.2 (Corollary 4.10). There is a natural bijection 2 sep × ∼ H Gk; (k ) = Br(k); (1.1) where Br(k) is the Brauer group of k. This is aim of the current talk. The main references are [1, Chapter IV] and [2, Tag 073W]. 2 Central simple algebras 2.1 Basic definitions and properties Let k be a field. In what follows, we use the term k-algebra to refer to an associative unital k-algebra which is finite dimensional as a k-vector space. In particular, we do not assume that k-algebras are commutative. Definition 2.1. A k-algebra is called simple if it contains no proper two sided ideals other than (0). Definition 2.2. A k-algebra A is said to be central if its center Z(A) is equal to k. If A is also simple, we say that it is central simple. We say a k-algebra D is a division algebra if every non-zero element has a multi- plicative inverse, i.e., for every a 2 D r f0g, there exists a b 2 D such that ab = 1 = ba. -
The Cayley-Dickson Construction in ACL2
The Cayley-Dickson Construction in ACL2 John Cowles Ruben Gamboa University of Wyoming Laramie, WY fcowles,[email protected] The Cayley-Dickson Construction is a generalization of the familiar construction of the complex numbers from pairs of real numbers. The complex numbers can be viewed as two-dimensional vectors equipped with a multiplication. The construction can be used to construct, not only the two-dimensional Complex Numbers, but also the four-dimensional Quaternions and the eight-dimensional Octonions. Each of these vector spaces has a vector multiplication, v1 • v2, that satisfies: 1. Each nonzero vector, v, has a multiplicative inverse v−1. 2. For the Euclidean length of a vector jvj, jv1 • v2j = jv1j · jv2j Real numbers can also be viewed as (one-dimensional) vectors with the above two properties. ACL2(r) is used to explore this question: Given a vector space, equipped with a multiplication, satisfying the Euclidean length condition 2, given above. Make pairs of vectors into “new” vectors with a multiplication. When do the newly constructed vectors also satisfy condition 2? 1 Cayley-Dickson Construction Given a vector space, with vector addition, v1 + v2; vector minus −v; a zero vector~0; scalar multiplica- tion by real number a, a ◦ v; a unit vector~1; and vector multiplication v1 • v2; satisfying the Euclidean length condition jv1 • v2j = jv1j · jv2j (2). 2 2 Define the norm of vector v by kvk = jvj . Since jv1 • v2j = jv1j · jv2j is equivalent to jv1 • v2j = 2 2 jv1j · jv2j , the Euclidean length condition is equivalent to kv1 • v2k = kv1k · kv2k: Recall the dot (or inner) product, of n-dimensional vectors, is defined by (x1;:::;xn) (y1;:::;yn) = x1 · y1 + ··· + xn · yn n = ∑ xi · yi i=1 Then Euclidean length and norm of vector v are given by p jvj = v v kvk = v v: Except for vector multiplication, it is easy to treat ordered pairs of vectors, (v1;v2), as vectors: 1.