Problems in the Classification Theory of Non-Associative Simple Algebras
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Center and Composition Conditions for Abel Equation
Center and Comp osition conditions for Ab el Equation Thesis for the degree of Do ctor of Philosophy by Michael Blinov Under the Sup ervision of professor Yosef Yomdin Department of Theoretical Mathematics The Weizmann Institute of Science Submitted to the Feinb erg Graduate Scho ol of the Weizmann Institute of Science Rehovot Israel June Acknowledgments I would like to thank my advisor Professor Yosef Yomdin for his great sup ervision constant supp ort and encouragement for guiding and stimulat ing my mathematical work I should also mention that my trips to scien tic meetings were supp orted from Y Yomdins Minerva Science Foundation grant I am grateful to JP Francoise F Pakovich E Shulman S Yakovenko and V Katsnelson for interesting discussions I thank Carla Scapinello Uruguay Jonatan Gutman Israel Michael Kiermaier Germany and Oleg Lavrovsky Canada for their results obtained during summer pro jects that I sup ervised I am also grateful to my mother for her patience supp ort and lo ve I wish to thank my mathematical teacher V Sap ozhnikov who gave the initial push to my mathematical research Finally I wish to thank the following for their friendship and in some cases love Lili Eugene Bom Elad Younghee Ira Rimma Yulia Felix Eduard Lena Dima Oksana Olga Alex and many more Abstract We consider the real vector eld f x y gx y on the real plane IR This vector eld describ es the dynamical system x f x y y g x y A classical CenterFo cus problem whichwas stated byHPoincare in s is nd conditions on f g under which all tra jectories -
Split Octonions
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 . -
Arxiv:1711.00219V2 [Math.OA] 9 Aug 2019 R,Fe Rbblt,Glbr Coefficients
CUMULANTS, SPREADABILITY AND THE CAMPBELL-BAKER-HAUSDORFF SERIES TAKAHIRO HASEBE AND FRANZ LEHNER Abstract. We define spreadability systems as a generalization of exchangeability systems in order to unify various notions of independence and cumulants known in noncommutative probability. In particular, our theory covers monotone independence and monotone cumulants which do not satisfy exchangeability. To this end we study generalized zeta and M¨obius functions in the context of the incidence algebra of the semilattice of ordered set partitions and prove an appropriate variant of Faa di Bruno’s theorem. With the aid of this machinery we show that our cumulants cover most of the previously known cumulants. Due to noncommutativity of independence the behaviour of these cumulants with respect to independent random variables is more complicated than in the exchangeable case and the appearance of Goldberg coefficients exhibits the role of the Campbell-Baker-Hausdorff series in this context. In a final section we exhibit an interpretation of the Campbell-Baker-Hausdorff series as a sum of cumulants in a particular spreadability system, thus providing a new derivation of the Goldberg coefficients. Contents 1. Introduction 2 2. Ordered set partitions 4 2.1. Set partitions 4 2.2. Ordered set partitions 5 2.3. Incidence algebras and multiplicative functions 8 2.4. Special functions in the poset of ordered set partitions 10 3. A generalized notion of independence related to spreadability systems 11 3.1. Notation and terminology 11 3.2. Spreadability systems 13 3.3. Examples from natural products of linear maps 14 3.4. E-independence 15 3.5. -
Euclidean Jordan Algebras for Optimization
Euclidean Jordan algebras for optimization Michael Orlitzky Euclidean Jordan algebras for optimization Michael Orlitzky Cover image by Vexels March 23, 2021 WARNING ¡ CUIDADO ! ACHTUNG This is a draft. It contains more or less correct proofs of statements that say more or less what they should. Beyond that, I make no promises. I’ve made many, many, many, many large-scale changes to notation without ever going back to proofread from page one. Caveat emptor. 1 Contents Contents2 Glossary5 1 Preface6 1.1 Notation notation........................... 10 I Fundamentals 11 2 Vector space structure 12 2.1 Algebraic building blocks...................... 12 2.2 Vector and Hilbert spaces...................... 17 2.3 Continuity and compactness..................... 26 2.4 Algebras................................ 29 2.5 Solutions to exercises......................... 33 3 Polynomials and power-associativity 35 3.1 Univariate polynomials........................ 35 3.2 Multivariate polynomials....................... 44 3.3 Polynomial ring embeddings..................... 56 3.4 Rational functions.......................... 58 3.5 Power-associative algebras...................... 60 3.6 Polynomial continuity........................ 63 3.7 Solutions to exercises......................... 68 4 Linear Algebra 72 4.1 Linear operators, matrix representation.............. 72 4.2 Eigenvalues and self-adjoint operators............... 76 4.3 Positive semi-definite operators................... 86 4.4 Characteristic and minimal polynomials.............. 87 4.5 Solutions -
Arxiv:1907.02135V1 [Math.RA] 3 Jul 2019
THE UNIVERSAL ENVELOPING ALGEBRA OF sl2 AND THE RACAH ALGEBRA SARAH BOCKTING-CONRAD AND HAU-WEN HUANG Abstract. Let F denote a field with char F 6= 2. The Racah algebra ℜ is the unital asso- ciative F-algebra defined by generators and relations in the following way. The generators are A, B, C, D. The relations assert that [A, B] = [B, C] = [C, A]=2D and each of the elements α = [A, D]+ AC − BA, β = [B,D]+ BA − CB, γ = [C,D]+ CB − AC is central in ℜ. Additionally the element δ = A + B + C is central in ℜ. In this paper we explore the relationship between the Racah algebra ℜ and the universal enveloping algebra U(sl2). Let a,b,c denote mutually commuting indeterminates. We show that there exists a unique F-algebra homomorphism ♮ : ℜ → F[a,b,c] ⊗F U(sl2) that sends A 7→ a(a + 1) ⊗ 1 + (b − c − a) ⊗ x + (a + b − c + 1) ⊗ y − 1 ⊗ xy, B 7→ b(b + 1) ⊗ 1 + (c − a − b) ⊗ y + (b + c − a + 1) ⊗ z − 1 ⊗ yz, C 7→ c(c + 1) ⊗ 1 + (a − b − c) ⊗ z + (c + a − b + 1) ⊗ x − 1 ⊗ zx, D 7→ 1 ⊗ (zyx + zx) + (c + b(c + a − b)) ⊗ x + (a + c(a + b − c)) ⊗ y +(b + a(b + c − a)) ⊗ z + (b − c) ⊗ xy + (c − a) ⊗ yz + (a − b) ⊗ zx, where x,y,z are the equitable generators for U(sl2). We additionally give the images of α, β, γ, δ, and certain Casimir elements of ℜ under ♮. We also show that the map ♮ is an injection and thus provides an embedding of ℜ into F[a,b,c] ⊗ U(sl2). -
Arxiv:1412.2365V1 [Math.AG]
POLARIZATION ALGEBRAS AND THEIR RELATIONS Ualbai Umirbaev1 Abstract. Using an approach to the Jacobian Conjecture by L.M. Dru˙zkowski and K. Rusek [12], G. Gorni and G. Zampieri [19], and A.V. Yagzhev [27], we describe a correspondence between finite dimensional symmetric algebras and homogeneous tuples of elements of polynomial algebras. We show that this correspondence closely relates Albert’s problem [10, Problem 1.1] in classical ring theory and the homogeneous de- pendence problem [13, page 145, Problem 7.1.5] in affine algebraic geometry related to the Jacobian Conjecture. We demonstrate these relations in concrete examples and formulate some open questions. Mathematics Subject Classification (2010): Primary 14R15, 17A40, 17A50; Sec- ondary 14R10, 17A36. Key words: the Jacobian Conjecture, polynomial mappings, homogeneous depen- dence, Engel algebras, nilpotent and solvable algebras. 1. Introduction The main objective of this paper is to connect two groups of specialists who are working on a closely connected problems and sometimes have intersections. Joining the efforts of these groups may be fruitful in studying the Jacobian Conjecture [13]. Namely, Albert’s problem [10, Problem 1.1] in classical ring theory and the homogeneous dependence prob- lem [13, page 145, Problem 7.1.5] in affine algebraic geometry are closely connected. Ring theory specialists are focused in studying only binary algebras and have some positive results in small dimensions. Affine algebraic geometry specialists have also some positive results in small dimensions and have some negative results for m-ary algebras if m ≥ 3. To make the subject more intriguing let’s start with two examples. -
Arxiv:1808.03808V1 [Math.RA]
THE UNIVERSALITY OF ONE HALF IN COMMUTATIVE NONASSOCIATIVE ALGEBRAS WITH IDENTITIES VLADIMIR G. TKACHEV Abstract. In this paper we will explain an interesting phenomenon which occurs in general nonasso- ciative algebras. More precisely, we establish that any finite-dimensional commutative nonassociative 1 algebra over a field satisfying an identity always contains 2 in its Peirce spectrum. We also show 1 that the corresponding 2 -Peirce module satisfies the Jordan type fusion laws. The present approach is based on an explicit representation of the Peirce polynomial for an arbitrary algebra identity. To work with fusion rules, we develop the concept of the Peirce symbol and show that it can be explicitly determined for a wide class of algebras. We also illustrate our approach by further applications to genetic algebras and algebra of minimal cones (the so-called Hsiang algebras). 1. Introduction Algebras whose associativity is replaced by identities were a central topic in mathematics in the 20th century, including the classical theory of Lie and Jordan algebras. Recall that an algebra is called Jordan if any two elements y,z ∈ A satisfy the following two identities: (1) zy − yz =0, (2) z((zz)y) − (zz)(zy)=0. The Peirce decomposition relative to an algebra idempotent is an important tool in the structure study of any nonassociative algebra. For example, the multiplication operator by an idempotent in a Jordan algebra is diagonalizable and the corresponding Peirce decomposition (relative to an idempotent c) into invariant subspaces 1 (3) A = Ac(0) ⊕ Ac(1) ⊕ Ac( 2 ) is compatible with the multiplication in the sense that the multiplication of eigenvectors is described by certain multiplication rules, also known as fusion laws. -
On a Class of Malcev-Admissible Algebras
47 J. Korean Math. Soc. Vo1.19, No. 1, 1982 ON A CLASS OF MALCEV-ADMISSIBLE ALGEBRAS by Youngso Ko and Hyo Chul Myung 1. Introduction For an algebra A over a field F, denote by A-the algebra with multiplication [x, y]=xy-yx defined on the same vector space as A, where juxtaposition xy is the product in A. Then A is termed Malcev-admissible if A-is a Malcev algebra; that is, the Malcev identity [[x, y], [x, z]]= [[[x, y], z], x]+[[[y, z], x], xJ+[[[z, xJ, xJ, yJ is satisfied for all x, y, zE A. An algebra A is called Lie-admissible if A-is a Lie algebra, namely, A- satisfies the Jacobi identity [[x, y], zJ + [[z, xJ, yJ + [[y, zJ, x] = 0 for x, y, zE A. It is well known that every Lie-admissible and alternative algebra is Malcev-admissible [8]. Thus a Malcev-admissible algebra is a natural generalization of Lie-admissible, alternative, and Malcev algebras. We assume hereafter that A is an algebra over F of characteristic =1= 2. Denote also by A+ the algebra with multiplication xoy= ~ (xy+yx) defined on the vector space A. The product xy in A is then given by xy= ~ [x, yJ+xoy. (1) Thus every Malcev algebra can occur as the attached algebra A-for a Malcev admissible algebra A by merely varying the commutative product xoy. This indicates that MaIcev-admissibility alone is too broad a condition to yield a fruitful structure theory. It has been shown that the flexible law (xy)x=x(yx) (2) for x, yEA is a useful restriction on A that imposes constraints on the product xoy and its relations with the product [, J. -
A View of Symplectic Lie Algebras from Quadratic Poisson Algebras
Bolet´ınde Matem´aticas 26(1) 1{30 (2019) 1 A view of symplectic Lie algebras from quadratic Poisson algebras Una mirada a las ´algebrasde Lie simplecticas desde las ´algebrasde Poisson cuadr´aticas Andr´esRia~no1;a, Armando Reyes1;b Abstract. Using the concept of double extension, Benayadi [2] showed how to construct a new quadratic algebra (g(A);T ) given a quadratic algebra (A;B). With both algebras and an invertible skew-symmetric algebra D over A, he en- dowed (A;B) with a simplectic structure through a bilinear form !, obtaining a simplectic algebra (g(A);T; Ω). Our purpose in this short communication is to show the construction given by Benayadi and present the complete develop- ment of each one of his assertions. We remark that this communication does not have original results and it was made as a result of the undergraduated work titled “Construcci´onde ´algebrasde Lie simpl´ecticasdesde ´algebrasde Poisson cuadr´aticas"[5],which was awarded as the best mathematics under- graduated thesis in the XXVI Contest at Universidad Nacional de Colombia, Sede Bogot´a.The work was written by the first author under the direction of the second author. Keywords: Lie algebra, Poisson algebra, quadratic algebra, symplectic alge- bra. Resumen. A partir del concepto de doble extensi´on,Benayadi [2] mostr´o c´omoconstruir una nueva ´algebracuadr´atica(g(A);T ), a partir de un ´algebra cuadr´aticadada (A;B). Con estas dos ´algebrasy una derivaci´oninvertible anti-sim´etrica D sobre A, ´eldot´oa (A;B) de una estructura simpl´ectica a traves de una forma bilineal !, obteniendo as´ı una ´algebrasimpl´ectica (g(A);T; Ω). -
Abraham Adrian Albert 1905-1972 by Nathan Jacobson
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 6, November 1974 ABRAHAM ADRIAN ALBERT 1905-1972 BY NATHAN JACOBSON Adrian Albert, one of the foremost algebraists of the world and Presi dent of the American Mathematical Society from 1965 to 1967, died on June 6, 1972. For almost a year before his death it had become apparent to his friends that his manner had altered from its customary vigor to one which was rather subdued. At first they attributed this to a letdown which might have resulted from Albert's having recently relinquished a very demanding administrative position (Dean of the Division of Physical Sciences at the University of Chicago) that he had held for a number of years. Eventually it became known that he was gravely ill of physical causes that had their origin in diabetes with which he had been afflicted for many years. Albert was a first generation American and a second generation Ameri can mathematician following that of E. H. Moore, Oswald Veblen, L. E. Dickson and G. D. Birkhoff. His mother came to the United States from Kiev and his father came from England.1 The father had run away from his home in Yilna at the age of fourteen, and on arriving in England, he discarded his family name (which remains unknown) and took in its place the name Albert after the prince consort of Queen Victoria. Albert's father was something of a scholar, with a deep interest in English literature. He taught school for a while in England but after coming to the United States he became a salesman, a shopkeeper, and a manufacturer. -
Dynatomic Polynomials, Necklace Operators, and Universal Relations for Dynamical Units
DYNATOMIC POLYNOMIALS, NECKLACE OPERATORS, AND UNIVERSAL RELATIONS FOR DYNAMICAL UNITS JOHN R. DOYLE, PAUL FILI, AND TREVOR HYDE 1. INTRODUCTION Let f(x) 2 K[x] be a polynomial with coefficients in a field K. For an integer k ≥ 0, we denote by f k(x) the k-fold iterated composition of f with itself. The dth dynatomic polynomial Φf;d(x) 2 K[x] of f is defined by the product Y d=e µ(e) Φf;d(x) := (f (x) − x) ; ejd where µ is the standard number-theoretic Möbius function on N. We refer the reader to [12, §4.1] for background on dynatomic polynomials. For generic f(x), the dth dynatomic polyno- mial Φf;d(x) vanishes at precisely the periodic points of f with primitive period d. In this paper we consider the polynomial equation Φf;d(x) = 1 and show that it often has f-preperiodic solu- tions determined by arithmetic properties of d, independent of f. Moreover, these f-preperiodic solutions are detected by cyclotomic factors of the dth necklace polynomial: 1 X M (x) = µ(e)xd=e 2 [x]: d d Q ejd Our results extend earlier work of Morton and Silverman [8], but our techniques are quite dif- ferent and apply more broadly. We begin by recalling some notation and terminology. The cocore of a positive integer d is d=d0 where d0 is the largest squarefree factor of d. If m ≥ 0 and n ≥ 1, then the (m; n)th generalized dynatomic polynomial Φf;m;n(x) of f(x) is defined by Φf;0;n(x) := Φf;n(x) and m Φf;n(f (x)) Φf;m;n(x) := m−1 Φf;n(f (x)) for m ≥ 1. -
Finding Octonion Algebras in Associative Algebras John R
proceedings of the american mathematical society Volume 104, Number 4, December 1988 FINDING OCTONION ALGEBRAS IN ASSOCIATIVE ALGEBRAS JOHN R. FAULKNER (Communicated by Donald S. Passman) ABSTRACT. An embedding of octonion algebras in central simple degree three associative algebras is given. Using standard operations on trace 0 elements of a simple 9-dimensional associa- tive algebra (such as 3 x 3 matrices) over a field of characteristic not 3 containing a primitive cube root of 1, we obtain in the Theorem the somewhat surprising embedding of an octonion algebra in an associative algebra. One clue to this em- bedding is that the Tits construction in [4] applied to F © cf (cf split octonion) or to F3 (3 x 3 matrices) gives the same 27-dimensional exceptional Jordan algebra, suggesting that F © (f is related to F3. A similar embedding of the 27-dimensional exceptional Jordan algebra JF as trace 0 elements in the degree 4 special Jordan algebra J4 of 4 x 4 hermitian quaternionic matrices was previously known (see [1 and 2]). In this case, a relation between F ® fí and ^4 was suggested by the Cayley-Dickson process in [1] which applied to F © f or to %\ gives the same 56-dimensional structurable algebra. Our approach is to show that the set An of trace 0 elements of a separable degree 3 alternative algebra A carries a nondegenerate quadratic form S permitting composition S(x * y) — S(x)S(y), from which it follows that an isotope of (An,* ) is a unital composition algebra. If A has dimension 9, then Aq has dimension 8, and the composition algebra must be octonion.