DOCSLIB.ORG

Split Octonions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Split Octonions Benjamin Prather Florida State University Department of Mathematics November 15, 2017 Group Algebras Split Octonions Let R be a ring (with unity). Prather Let G be a group. Loop Algebras Octonions Denition Moufang Loops An element of group algebra R[G] is the formal sum: Split-Octonions Analysis X Malcev Algebras rngn Summary gn2G Addition is component wise (as a free module). Multiplication follows from the products in R and G, distributivity and commutativity between R and G. Note: If G is innite only nitely many ri are non-zero. Group Algebras Split Octonions Prather Loop Algebras A group algebra is itself a ring. Octonions In particular, a group under multiplication. Moufang Loops Split-Octonions Analysis A set M with a binary operation is called a magma. Malcev Algebras Summary This process can be generalized to magmas. The resulting algebra often inherit the properties of M. Divisibility is a notable exception. Magmas Split Octonions Prather Loop Algebras Octonions Moufang Loops Split-Octonions Analysis Malcev Algebras Summary Loops Split Octonions Prather Loop Algebras In particular, loops are not associative. Octonions It is useful to dene some weaker properties. Moufang Loops power associative: the sub-algebra generated by Split-Octonions Analysis any one element is associative. Malcev Algebras diassociative: the sub-algebra generated by any Summary two elements is associative. associative: the sub-algebra generated by any three elements is associative. Loops Split Octonions Prather Loop Algebras Octonions Power associativity gives us (xx)x = x(xx). Moufang Loops This allows x n to be well dened. Split-Octonions Analysis −1 Malcev Algebras Diassociative loops have two sided inverses, x . −1 −1 Summary Diassociative gives us (xy)x = x(yx ). This allows conjugation to be well dened. Loop Algebras Split Octonions Prather Let L be a loop. Loop Algebras Denition Octonions An element of loop algebra is the formal sum: Moufang Loops R[L] Split-Octonions X Analysis rnln Malcev Algebras ln2L Summary Addition is component wise (as a free module). Multiplication follows from the products in R and L, distributivity and commutativity between R and L. Note: If L is innite only nitely many rn are non-zero. Octonion loop Split Octonions Let be the loop dened by this table. Prather O Loop Algebras e0 e1 e2 e3 e4 e5 e6 e7 Octonions e1 −e0 e3 −e2 e5 −e4 −e7 e6 Moufang Loops Split-Octonions e2 −e3 −e0 e1 e6 e7 −e4 −e5 Analysis e3 e2 −e1 −e0 e7 −e6 e5 −e4 Malcev Algebras e4 −e5 −e6 −e7 −e0 e1 e2 e3 Summary e5 e4 −e7 e6 −e1 −e0 −e3 e2 e6 e7 e4 −e5 −e2 e3 −e0 −e1 e7 −e6 e5 e4 −e3 −e2 e1 −e0 Note: There are really 16 elements, ± for each shown. 2 Formally, −en is really en, where = 1 and commutes. Other Representations Split Octonions Prather Loop Algebras Let e0 be the identity and in be indexed mod 7. Let 2 and ( ). Octonions in = −e0 inim = −imin m 6= n Moufang Loops Let inin+1 = in+3. This is isomorphic to O. Split-Octonions Analysis One such mapping is: 1 1, 2 2, 3 4. Malcev Algebras i = e i = e i = e This forces , , , . Summary i4 = e3 i5 = e6 i6 = −e7 i7 = e5 Also in+1in+3 = in and in+3in = in+1. This representation exhibits a seven fold symmetry in O. Other Representations Split Octonions Prather Loop Algebras 7 lines and 7 points. Octonions Central circle is a line. Moufang Loops Split-Octonions Three points on each line. Analysis Three lines at each point. Malcev Algebras Summary Points labeled with roots. Lines labeled to indicate positive products. Fano Plane Other Representations Split Octonions Prather Loop Algebras Octonions Moufang Loops Split-Octonions Analysis Malcev Algebras Summary John H. Conway Gino Fano Octonion Algebras Split Octonions Prather Loop Algebras (e1e2)e4 = e3e4 = e7 = −e1e6 = −e1(e2e4). Octonions Thus O is not associative. Moufang Loops Split-Octonions Analysis Also note that any two elements generate Malcev Algebras the associative quaternion group. Summary Thus O is diassociative. The anti-automorphism, q, toggles the sign of e1 to e7. Octonion Algebras Split Octonions Prather Let I = h1 + i if 1 6= −1 or h1i otherwise. Loop Algebras This formally associates with −1 in R, Octonions unless 2. Moufang Loops char(R) = is the loop algebra . is . Split-Octonions OR R[O]=I O OR Analysis Malcev Algebras N(q) = qq = qq Summary These are composition algebras, N(xy) = N(x)N(y). Over R only R, C, H and O are positive denite composition algebras. Discovery of the Octonions Split Octonions Prather Loop Algebras Octonions Moufang Loops Split-Octonions Analysis Malcev Algebras Summary William R. John T. Graves Aurther Cayley Hamilton Division Algebras Split Octonions Prather Theorem (Moufang's theorem) Loop Algebras Let an Loop L satisfy Octonions Moufang Loops xy · zx = x · yz · x Split-Octonions x · y · xz = xyx · z Analysis Malcev Algebras xy · z · y = x · yzy Summary Let x; y and z be elements of L. Ruth If x(yz) = (xy)z, then x; y and z Moufang generate an associative sub-loop (i.e a group). Loops with these properties are Moufang loops. Moufang Loops Split Octonions Prather Loop Algebras All Moufang loops are diassociative. Octonions Thus the RHS are well dened. Moufang Loops Split-Octonions In particular, the units of are Moufang loops. Analysis OR Malcev Algebras Summary If K is a eld we can dene q−1 = q=N(q). This makes a division algebra. OK Projective spaces Split Octonions Prather Theorem (Desargues' theorem) Loop Algebras Octonions Two triangles are in perspective axially i they are in Moufang Loops perspective centrally. Split-Octonions Analysis This was proved for Euclidean spaces. Malcev Algebras It applies to all projective spaces, if dim 6= 2. Summary Hilbert found counter examples when dim = 2. Moufang showed that if an algebra is non-associative it's projective space is non-Desargues. In particular, she was interested in Cayley Planes. Projective spaces Split Octonions Prather Thus there is a non-Desarguesian octonionic projective Loop Algebras plane, but no higher octonionic projective spaces. Octonions Moufang Loops Split-Octonions Analysis Malcev Algebras Summary Girard Desargues David Hilbert Parker Loop Split Octonions Prather The Golay code is a 12 dimensional linear code 24 Loop Algebras over with a natural representation in . F2 F2 Octonions They can be viewed as sets of bits. Moufang Loops The Parker Loop is a Moufang loop created as Split-Octonions a double of the Golay Code, by adding a sign bit. Analysis Malcev Algebras Summary Let HW (x) be the cardinatity of code point x. x 2 = (−1)HW (x)=4 xy = (−1)HW (x\y)=2yx x(yz) = (−1)HW (x\y\z)(xy)z Note: The various sign choices here are isomorphic. Parker Loop Split Octonions Prather Parker showed that this is indeed a Moufang Loop. Loop Algebras Octonions Conway used this loop to simplify Griess' construction Moufang Loops of the Fischer-Griess Monster group. Split-Octonions (Griess's constructed it as the automorphism group of a Analysis Malcev Algebras commutative, non-associative, algebra). Summary The monster group plays a signicant role in the classication of nite simple groups. So Moufang loops play a role in one of the biggest, if not most important, theorems of modern mathematics. Parker Loop Split Octonions Prather Loop Algebras Octonions Moufang Loops Split-Octonions Analysis Malcev Algebras Summary Marcel J. E. Richard A. Parker Robert Griess Golay Paige Loops Split Octonions Prather Moufang loop are diassociative. Loop Algebras Thus conjugation is well dened. Octonions Moufang Loops This allows us to dene a normal sub-loop, Split-Octonions and simple Moufang loops in the usual way. Analysis Malcev Algebras Summary Paige showed that the units of split-octonion algebras over a nite eld are simple. Liebeck showed that their are no other cases. Finite simple Moufang loops are called Paige loops. Paige Loops Split Octonions Prather Loop Algebras Octonions Moufang Loops Split-Octonions Analysis Malcev Algebras Summary Lowell Paige Martin Liebeck Split-Octonions Split Octonions Prather Consider . OC Loop Algebras Clearly the real parts of this algebra form O. Octonions However, it contains another algebra over . Moufang Loops p R Specically, multiply each , 4 by 1. Split-Octonions en n ≥ − Analysis Malcev Algebras The resulting algebra has a split signature. Summary Thus they are called the split-octonions, O−. Rename as for 0 4, since 2 1. p en in < n < in = − Also, 1 as for 4, since 2 1. − en jn n ≥ jn = Split-Octonions Split Octonions Prather 2 2 2 2 2 2 2 2 Loop Algebras N(q) = qq = r0 + r1 + r2 + r3 − r4 − r5 − r6 − r7 − Octonions O not positive denite, as N(j4) = −1. Moufang Loops Split-Octonions − Infact, O has zero divisors, N(1 + j4) = 0. Analysis The zero divisors all have 0. Malcev Algebras N(q) = Summary O− has the composition property. The elements with N(q) 6= 0 are invertible. The invertible elements form a loop. Split Sub-Algebras Split Octonions Prather Loop Algebras q is a split root if N(q) = −1 (proper if N(q) = 1). Octonions Moufang Loops Any split root denes a split-complex algebra C−. Split-Octonions Any two orthogonal split roots dene a Analysis split-quaternion algebra, −. Malcev Algebras H Summary Any sub-algebra is either one of these, C or H. In particular, the algebra generated by a proper root and an orthogonal split root is also H−. Lorentz Transforms Split Octonions Prather − ∼ C = R × R. This algebra describes the Loop Algebras Lorentz transformations in one dimension. Octonions Moufang Loops − ∼ Split-Octonions H = M2(R).
Recommended publications
  • Moufang Loops and Alternative Algebras

    Moufang Loops and Alternative Algebras

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 2, Pages 313{316 S 0002-9939(03)07260-5 Article electronically published on August 28, 2003 MOUFANG LOOPS AND ALTERNATIVE ALGEBRAS IVAN P. SHESTAKOV (Communicated by Lance W. Small) Abstract. Let O be the algebra O of classical real octonions or the (split) algebra of octonions over the finite field GF (p2);p>2. Then the quotient loop O∗=Z ∗ of the Moufang loop O∗ of all invertible elements of the algebra O modulo its center Z∗ is not embedded into a loop of invertible elements of any alternative algebra. It is well known that for an alternative algebra A with the unit 1 the set U(A) of all invertible elements of A forms a Moufang loop with respect to multiplication [3]. But, as far as the author knows, the following question remains open (see, for example, [1]). Question 1. Is it true that any Moufang loop can be imbedded into a loop of type U(A) for a suitable unital alternative algebra A? A positive answer to this question was announced in [5]. Here we show that, in fact, the answer to this question is negative: the Moufang loop U(O)=R∗ for the algebra O of classical octonions over the real field R andthesimilarloopforthe algebra of octonions over the finite field GF (p2);p>2; are not imbeddable into the loops of type U(A). Below O denotes an algebra of octonions (or Cayley{Dickson algebra) over a field F of characteristic =2,6 O∗ = U(O) is the Moufang loop of all the invertible elements of O, L = O∗=F ∗.
  • Arxiv:1902.01706V2 [Math.RA] 1 Apr 2019 O Emti Eut E,Freape 6 ,1,1,1] He N Dimensional 14]

    Arxiv:1902.01706V2 [Math.RA] 1 Apr 2019 O Emti Eut E,Freape 6 ,1,1,1] He N Dimensional 14]

    The algebraic and geometric classification of nilpotent binary Lie algebras 1,2 Hani Abdelwahaba, Antonio Jesus´ Calderon´ b & Ivan Kaygorodovc a Department of Mathematics, Faculty of Sciences, Mansoura University, Mansoura, Egypt b Department of Mathematics, Faculty of Sciences, University of Cadiz, Cadiz, Spain c CMCC, Universidade Federal do ABC. Santo Andr´e, Brasil E-mail addresses: Hani Abdelwahab ([email protected]) Antonio Jes´us Calder´on ([email protected]) Ivan Kaygorodov ([email protected]) Abstract: We give a complete algebraic classification of nilpotent binary Lie algebras of dimension at most 6 over an arbitrary base field of char- acteristic not 2 and a complete geometric classification of nilpotent binary Lie algebras of dimension 6 over C. As an application, we give an algebraic and geometric classification of nilpotent anticommutative CD-algebras of dimension at most 6. Keywords: Nilpotent algebras, binary Lie algebras, Malcev algebras, CD- algebras, Lie algebras, algebraic classification, geometric classification, degeneration. MSC2010: 17D10, 17D30. INTRODUCTION There are many results related to both the algebraic and geometric classi- fication of small dimensional algebras in the varieties of Jordan, Lie, Leib- niz, Zinbiel algebras; for algebraic results see, for example, [2,5,8,14,16]; arXiv:1902.01706v2 [math.RA] 1 Apr 2019 for geometric results see, for example, [6,9,11,12,14]. Here we give an algebraic and geometric classification of low dimensional nilpotent binary Lie algebras. Malcev defined binary Lie algebras as algebras such that every two- generated subalgebra is a Lie algebra [17]. Identities of the variety of bi- nary Lie algebras were described by Gainov [4].
  • Malcev Algebras

    Malcev Algebras

    MALCEVALGEBRAS BY ARTHUR A. SAGLEO) 1. Introduction. This paper is an investigation of a class of nonassociative algebras which generalizes the class of Lie algebras. These algebras satisfy certain identities that were suggested to Malcev [7] when he used the com- mutator of two elements as a new multiplicative operation for an alternative algebra. As a means of establishing some notation for the present paper, a brief sketch of this development will be given here. If ^4 is any algebra (associative or not) over a field P, the original product of two elements will be denoted by juxtaposition, xy, and the following nota- tion will be adopted for elements x, y, z of A : (1.1) Commutator, xoy = (x, y) = xy — yx; (1.2) Associator, (x, y, z) = (xy)z — x(yz) ; (1.3) Jacobian, J(x, y, z) = (xy)z + (yz)x + (zx)y. An alternative algebra A is a nonassociative algebra such that for any elements xi, x2, x3 oí A, the associator (¡Ci,x2, x3) "alternates" ; that is, (xi, x2, xs) = t(xiv xh, xh) for any permutation i%,i2, i%of 1, 2, 3 where « is 1 in case the permutation is even, — 1 in case the permutation is odd. If we introduce a new product into an alternative algebra A by means of a commutator x o y, we obtain for any x, y, z of A J(x, y, z)o = (x o y) o z + (y o z) o x + (z o y) o x = 6(x, y, z). The new algebra thus obtained will be denoted by A(~K Using the preceding identity with the known identities of an alternative algebra [2] and the fact that the Jacobian is a skew-symmetric function in /l(-), we see that /1(_) satis- fies the identities (1.4) ïoï = 0, (1.5) (x o y) o (x o z) = ((x o y) o z) o x + ((y o z) o x) o x + ((z o x) o x) o y Presented to the Society, August 31, 1960; received by the editors March 14, 1961.
  • Gauging the Octonion Algebra

    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.
  • Hypercomplex Numbers

    Hypercomplex Numbers

    Hypercomplex numbers Johanna R¨am¨o Queen Mary, University of London [email protected] We have gradually expanded the set of numbers we use: first from finger counting to the whole set of positive integers, then to positive rationals, ir- rational reals, negatives and finally to complex numbers. It has not always been easy to accept new numbers. Negative numbers were rejected for cen- turies, and complex numbers, the square roots of negative numbers, were considered impossible. Complex numbers behave like ordinary numbers. You can add, subtract, multiply and divide them, and on top of that, do some nice things which you cannot do with real numbers. Complex numbers are now accepted, and have many important applications in mathematics and physics. Scientists could not live without complex numbers. What if we take the next step? What comes after the complex numbers? Is there a bigger set of numbers that has the same nice properties as the real numbers and the complex numbers? The answer is yes. In fact, there are two (and only two) bigger number systems that resemble real and complex numbers, and their discovery has been almost as dramatic as the discovery of complex numbers was. 1 Complex numbers Complex numbers where discovered in the 15th century when Italian math- ematicians tried to find a general solution to the cubic equation x3 + ax2 + bx + c = 0: At that time, mathematicians did not publish their results but kept them secret. They made their living by challenging each other to public contests of 1 problem solving in which the winner got money and fame.
  • MOUFANG LOOPS THAT SHARE ASSOCIATOR and THREE QUARTERS of THEIR MULTIPLICATION TABLES 1. Introduction Moufang Loops, I.E., Loops

    MOUFANG LOOPS THAT SHARE ASSOCIATOR and THREE QUARTERS of THEIR MULTIPLICATION TABLES 1. Introduction Moufang Loops, I.E., Loops

    MOUFANG LOOPS THAT SHARE ASSOCIATOR AND THREE QUARTERS OF THEIR MULTIPLICATION TABLES ALES· DRAPAL¶ AND PETR VOJTECHOVSK· Y¶ Abstract. Two constructions due to Dr¶apalproduce a group by modifying exactly one quarter of the Cayley table of another group. We present these constructions in a compact way, and generalize them to Moufang loops, using loop extensions. Both constructions preserve associators, the associator subloop, and the nucleus. We con- jecture that two Moufang 2-loops of ¯nite order n with equivalent associator can be connected by a series of constructions similar to ours, and o®er empirical evidence that this is so for n = 16, 24, 32; the only interesting cases with n · 32. We further investigate the way the constructions a®ect code loops and loops of type M(G; 2). The paper closes with several conjectures and research questions concerning the distance of Moufang loops, classi¯cation of small Moufang loops, and generalizations of the two constructions. MSC2000: Primary: 20N05. Secondary: 20D60, 05B15. 1. Introduction Moufang loops, i.e., loops satisfying the Moufang identity ((xy)x)z = x(y(xz)), are surely the most extensively studied loops. Despite this fact, the classi¯cation of Moufang loops is ¯nished only for orders less than 64, and several ingenious constructions are needed to obtain all these loops. The purpose of this paper is to initiate a new approach to ¯nite Moufang 2-loops. Namely, we intend to decide whether all Moufang 2-loops of given order with equivalent associator can be obtained from just one of them, using only group-theoretical constructions.
  • An Octonion Model for Physics

    An Octonion Model for Physics

    An octonion model for physics Paul C. Kainen¤ Department of Mathematics Georgetown University Washington, DC 20057 USA [email protected] Abstract The no-zero-divisor division algebra of highest possible dimension over the reals is taken as a model for various physical and mathematical phenomena mostly related to the Four Color Conjecture. A geometric form of associativity is the common thread. Keywords: Geometric algebra, the Four Color Conjecture, rooted cu- bic plane trees, Catalan numbers, quaternions, octaves, quantum alge- bra, gravity, waves, associativity 1 Introduction We explore some consequences of octonion arithmetic for a hidden variables model of quantum theory and ideas regarding the propagation of gravity. This approach may also have utility for number theory and combinatorial topology. Octonion models are currently the focus of much work in the physics com- munity. See, e.g., Okubo [29], Gursey [14], Ward [36] and Dixon [11]. Yaglom [37, pp. 94, 107] referred to an \octonion boom" and it seems to be acceler- ating. Historically speaking, the inventors/discoverers of the quaternions, oc- tonions and related algebras (Hamilton, Cayley, Graves, Grassmann, Jordan, Cli®ord and others) were working from a physical point-of-view and wanted ¤4th Conference on Emergence, Coherence, Hierarchy, and Organization (ECHO 4), Odense, Denmark, 2000 1 their abstractions to be helpful in solving natural problems [37]. Thus, a con- nection between physics and octonions is a reasonable though not yet fully justi¯ed suspicion. It is easy to see the allure of octaves since there are many phenomena in the elementary particle realm which have 8-fold symmetries.
  • A Guide to Self-Distributive Quasigroups, Or Latin Quandles

    A Guide to Self-Distributive Quasigroups, Or Latin Quandles

    A GUIDE TO SELF-DISTRIBUTIVE QUASIGROUPS, OR LATIN QUANDLES DAVID STANOVSKY´ Abstract. We present an overview of the theory of self-distributive quasigroups, both in the two- sided and one-sided cases, and relate the older results to the modern theory of quandles, to which self-distributive quasigroups are a special case. Most attention is paid to the representation results (loop isotopy, linear representation, homogeneous representation), as the main tool to investigate self-distributive quasigroups. 1. Introduction 1.1. The origins of self-distributivity. Self-distributivity is such a natural concept: given a binary operation on a set A, fix one parameter, say the left one, and consider the mappings ∗ La(x) = a x, called left translations. If all such mappings are endomorphisms of the algebraic structure (A,∗ ), the operation is called left self-distributive (the prefix self- is usually omitted). Equationally,∗ the property says a (x y) = (a x) (a y) ∗ ∗ ∗ ∗ ∗ for every a, x, y A, and we see that distributes over itself. Self-distributivity∈ was pinpointed already∗ in the late 19th century works of logicians Peirce and Schr¨oder [69, 76], and ever since, it keeps appearing in a natural way throughout mathematics, perhaps most notably in low dimensional topology (knot and braid invariants) [12, 15, 63], in the theory of symmetric spaces [57] and in set theory (Laver’s groupoids of elementary embeddings) [15]. Recently, Moskovich expressed an interesting statement on his blog [60] that while associativity caters to the classical world of space and time, distributivity is, perhaps, the setting for the emerging world of information.
  • Universal Enveloping Algebras for Simple Malcev Algebras

    Universal Enveloping Algebras for Simple Malcev Algebras

    Pr´e-Publica¸c˜oes do Departamento de Matem´atica Universidade de Coimbra Preprint Number 10–32 UNIVERSAL ENVELOPING ALGEBRAS FOR SIMPLE MALCEV ALGEBRAS ELISABETE BARREIRO AND SARA MADARIAGA Abstract: It is our goal to present an universal enveloping algebra for the simple (non-Lie) 7-dimensional Malcev algebra M(α, β, γ) in order to obtain an universal enveloping algebra for simple Malcev algebras and to study its properties. Keywords: Simple Malcev algebra; universal enveloping algebra. Introduction The universal enveloping algebra of an algebra is a basic tool in the study of its representation theory. Given a algebra A, let us denote by A− the algebra obtained from A by replacing the product xy by the commutator [x, y]= xy yx, for elements x, y A. It is known that if A is an associative − ∈ algebra one obtains a Lie algebra A− and, conversely, the Poincar´e-Birkhoff- Witt Theorem establishes that any Lie algebra is isomorphic to a subalgebra of A− for some associative algebra A ([1], [2]). If we start with an alternative algebra A (alternative means that x(xy)= 2 2 x y and (yx)x = yx , for x, y A) then A− is a Malcev algebra. Since any associative algebra is in particular∈ an alternative algebra, then the Mal- cev algebra A− is a natural generalization of the Lie algebra A−. In [4], P´erez-Izquierdo and Shestakov presented an enveloping algebra of Malcev algebras (constructed in a more general way), generalizing the classical no- tion of universal enveloping algebra for Lie algebras. They proved that for every Malcev algebra M there exist an algebra U(M) and an injective Malcev algebra homomorphism ι : M U(M)− such that the image of M lies into the alternative nucleus of U(M−→), being U(M) a universal object with respect to such homomorphism.
  • Truly Hypercomplex Numbers

    Truly Hypercomplex Numbers

    TRULY HYPERCOMPLEX NUMBERS: UNIFICATION OF NUMBERS AND VECTORS Redouane BOUHENNACHE (pronounce Redwan Boohennash) Independent Exploration Geophysical Engineer / Geophysicist 14, rue du 1er Novembre, Beni-Guecha Centre, 43019 Wilaya de Mila, Algeria E-mail: [email protected] First written: 21 July 2014 Revised: 17 May 2015 Abstract Since the beginning of the quest of hypercomplex numbers in the late eighteenth century, many hypercomplex number systems have been proposed but none of them succeeded in extending the concept of complex numbers to higher dimensions. This paper provides a definitive solution to this problem by defining the truly hypercomplex numbers of dimension N ≥ 3. The secret lies in the definition of the multiplicative law and its properties. This law is based on spherical and hyperspherical coordinates. These numbers which I call spherical and hyperspherical hypercomplex numbers define Abelian groups over addition and multiplication. Nevertheless, the multiplicative law generally does not distribute over addition, thus the set of these numbers equipped with addition and multiplication does not form a mathematical field. However, such numbers are expected to have a tremendous utility in mathematics and in science in general. Keywords Hypercomplex numbers; Spherical; Hyperspherical; Unification of numbers and vectors Note This paper (or say preprint or e-print) has been submitted, under the title “Spherical and Hyperspherical Hypercomplex Numbers: Merging Numbers and Vectors into Just One Mathematical Entity”, to the following journals: Bulletin of Mathematical Sciences on 08 August 2014, Hypercomplex Numbers in Geometry and Physics (HNGP) on 13 August 2014 and has been accepted for publication on 29 April 2015 in issue No.
  • Quandles an Introduction to the Algebra of Knots

    Quandles an Introduction to the Algebra of Knots

    STUDENT MATHEMATICAL LIBRARY Volume 74 Quandles An Introduction to the Algebra of Knots Mohamed Elhamdadi Sam Nelson Quandles An Introduction to the Algebra of Knots http://dx.doi.org/10.1090/stml/074 STUDENT MATHEMATICAL LIBRARY Volume 74 Quandles An Introduction to the Algebra of Knots Mohamed Elhamdadi Sam Nelson American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 57M25, 55M25, 20N05, 20B05, 55N35, 57M05, 57M27, 20N02, 57Q45. For additional information and updates on this book, visit www.ams.org/bookpages/stml-74 Library of Congress Cataloging-in-Publication Data Elhamdadi, Mohamed, 1968– Quandles: an introduction to the algebra of knots / Mohamed Elhamdadi, Sam Nelson. pages cm. – (Student mathematical library ; volume 74) Includes bibliographical references and index. ISBN 978-1-4704-2213-4 (alk. paper) 1. Knot theory. 2. Low-dimensional topology. I. Nelson, Sam, 1974– II. Title. III. Title: Algebra of Knots. QA612.2.E44 2015 514.2242–dc23 2015012551 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 select pages 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. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service.
  • Classification of Left Octonion Modules

    Classification of Left Octonion Modules

    Classification of left octonion modules Qinghai Huo, Yong Li, Guangbin Ren Abstract It is natural to study octonion Hilbert spaces as the recently swift development of the theory of quaternion Hilbert spaces. In order to do this, it is important to study first its algebraic structure, namely, octonion modules. In this article, we provide complete classification of left octonion modules. In contrast to the quaternionic setting, we encounter some new phenomena. That is, a submodule generated by one element m may be the whole module and may be not in the form Om. This motivates us to introduce some new notions such as associative elements, conjugate associative elements, cyclic elements. We can characterize octonion modules in terms of these notions. It turns out that octonions admit two distinct structures of octonion modules, and moreover, the direct sum of their several copies exhaust all octonion modules with finite dimensions. Keywords: Octonion module; associative element; cyclic element; Cℓ7-module. AMS Subject Classifications: 17A05 Contents 1 introduction 1 2 Preliminaries 3 2.1 The algebra of the octonions O .............................. 3 2.2 Universal Clifford algebra . ... 4 3 O-modules 6 4 The structure of left O-moudles 10 4.1 Finite dimensional O-modules............................... 10 4.2 Structure of general left O-modules............................ 13 4.3 Cyclic elements in left O-module ............................. 15 arXiv:1911.08282v2 [math.RA] 21 Nov 2019 1 introduction The theory of quaternion Hilbert spaces brings the classical theory of functional analysis into the non-commutative realm (see [10, 16, 17, 20, 21]). It arises some new notions such as spherical spectrum, which has potential applications in quantum mechanics (see [4, 6]).