NON-SPLIT REDUCTIVE GROUPS OVER Z Brian Conrad
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Generically Split Octonion Algebras and A1-Homotopy Theory
Generically split octonion algebras and A1-homotopy theory Aravind Asok∗ Marc Hoyoisy Matthias Wendtz Abstract 1 We study generically split octonion algebras over schemes using techniques of A -homotopy theory. By combining affine representability results with techniques of obstruction theory, we establish classifica- tion results over smooth affine schemes of small dimension. In particular, for smooth affine schemes over algebraically closed fields, we show that generically split octonion algebras may be classified by charac- teristic classes including the second Chern class and another “mod 3” invariant. We review Zorn’s “vector matrix” construction of octonion algebras, generalized to rings by various authors, and show that generi- cally split octonion algebras are always obtained from this construction over smooth affine schemes of low dimension. Finally, generalizing P. Gille’s analysis of octonion algebras with trivial norm form, we observe that generically split octonion algebras with trivial associated spinor bundle are automatically split in low dimensions. Contents 1 Introduction 1 2 Octonion algebras, algebra and geometry 6 2.1 Octonion algebras and Zorn’s vector matrices......................................7 2.2 Octonion algebras and G2-torsors............................................9 2.3 Homogeneous spaces related to G2 and octonions.................................... 11 1 3 A -homotopy sheaves of BNis G2 17 1 3.1 Some A -fiber sequences................................................. 17 1 3.2 Characteristic maps and strictly A -invariant sheaves.................................. 19 3.3 Realization and characteristic maps........................................... 21 1 3.4 Some low degree A -homotopy sheaves of BNis Spinn and BNis G2 .......................... 24 4 Classifying octonion algebras 29 4.1 Classifying generically split octonion algebras...................................... 30 4.2 When are octonion algebras isomorphic to Zorn algebras?............................... -
Galois-Equivariant Mckay Bijections for Primes Dividing $ Q-1$
GALOIS-EQUIVARIANT MCKAY BIJECTIONS FOR PRIMES DIVIDING q 1 − A. A. SCHAEFFER FRY Abstract. We prove that for most groups of Lie type, the bijections used by Malle and Sp¨ath in the proof of Isaacs–Malle–Navarro’s inductive McKay conditions for the prime 2 and odd primes dividing q − 1 are also equivariant with respect to certain Galois automorphisms. In particular, this shows that these bijections are candidates for proving Navarro–Sp¨ath–Vallejo’s recently-posited inductive Galois–McKay conditions. Mathematics Classification Number: 20C15, 20C33 Keywords: local-global conjectures, characters, McKay conjecture, Galois-McKay, finite simple groups, Lie type 1. Introduction One of the main sets of questions of interest in the representation theory of finite groups falls under the umbrella of the so-called local-global conjectures, which relate the character theory of a finite group G to that of certain local subgroups. Another rich topic in the area is the problem of determining how certain other groups (for example, the group of automorphisms Aut(G) of G, or the Galois group Gal(Q/Q)) act on the set of irreducible characters of G and the related question of determining the fields of values of characters of G. This paper concerns the McKay–Navarro conjecture, which incorporates both of these main problems. For a finite group G, the field Q(e2πi/|G|), obtained by adjoining the G th roots of unity in some | | algebraic closure Q of Q, is a splitting field for G. The group G := Gal(Q(e2πi/|G|)/Q) acts naturally on the set of irreducible ordinary characters, Irr(G), of G via χσ(g) := χ(g)σ for χ Irr(G), g G, σ G . -
Arxiv:Math/0511624V1
Automorphism groups of polycyclic-by-finite groups and arithmetic groups Oliver Baues ∗ Fritz Grunewald † Mathematisches Institut II Mathematisches Institut Universit¨at Karlsruhe Heinrich-Heine-Universit¨at D-76128 Karlsruhe D-40225 D¨usseldorf May 18, 2005 Abstract We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ, 1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homo- topy theory. 2000 Mathematics Subject Classification: Primary 20F28, 20G30; Secondary 11F06, 14L27, 20F16, 20F34, 22E40, 55P10 arXiv:math/0511624v1 [math.GR] 25 Nov 2005 ∗e-mail: [email protected] †e-mail: [email protected] 1 Contents 1 Introduction 4 1.1 Themainresults ........................... 4 1.2 Outlineoftheproofsandmoreresults . 6 1.3 Cohomologicalrepresentations . 8 1.4 Applications to the groups of homotopy self-equivalences of spaces 9 2 Prerequisites on linear algebraic groups and arithmetic groups 12 2.1 Thegeneraltheory .......................... 12 2.2 Algebraicgroupsofautomorphisms. 15 3 The group of automorphisms of a solvable-by-finite linear algebraic group 17 3.1 The algebraic structure of Auta(H)................ -
Proofs, Sets, Functions, and More: Fundamentals of Mathematical Reasoning, with Engaging Examples from Algebra, Number Theory, and Analysis
Proofs, Sets, Functions, and More: Fundamentals of Mathematical Reasoning, with Engaging Examples from Algebra, Number Theory, and Analysis Mike Krebs James Pommersheim Anthony Shaheen FOR THE INSTRUCTOR TO READ i For the instructor to read We should put a section in the front of the book that organizes the organization of the book. It would be the instructor section that would have: { flow chart that shows which sections are prereqs for what sections. We can start making this now so we don't have to remember the flow later. { main organization and objects in each chapter { What a Cfu is and how to use it { Why we have the proofcomment formatting and what it is. { Applications sections and what they are { Other things that need to be pointed out. IDEA: Seperate each of the above into subsections that are labeled for ease of reading but not shown in the table of contents in the front of the book. ||||||||| main organization examples: ||||||| || In a course such as this, the student comes in contact with many abstract concepts, such as that of a set, a function, and an equivalence class of an equivalence relation. What is the best way to learn this material. We have come up with several rules that we want to follow in this book. Themes of the book: 1. The book has a few central mathematical objects that are used throughout the book. 2. Each central mathematical object from theme #1 must be a fundamental object in mathematics that appears in many areas of mathematics and its applications. -
14.2 Covers of Symmetric and Alternating Groups
Remark 206 In terms of Galois cohomology, there an exact sequence of alge- braic groups (over an algebrically closed field) 1 → GL1 → ΓV → OV → 1 We do not necessarily get an exact sequence when taking values in some subfield. If 1 → A → B → C → 1 is exact, 1 → A(K) → B(K) → C(K) is exact, but the map on the right need not be surjective. Instead what we get is 1 → H0(Gal(K¯ /K), A) → H0(Gal(K¯ /K), B) → H0(Gal(K¯ /K), C) → → H1(Gal(K¯ /K), A) → ··· 1 1 It turns out that H (Gal(K¯ /K), GL1) = 1. However, H (Gal(K¯ /K), ±1) = K×/(K×)2. So from 1 → GL1 → ΓV → OV → 1 we get × 1 1 → K → ΓV (K) → OV (K) → 1 = H (Gal(K¯ /K), GL1) However, taking 1 → µ2 → SpinV → SOV → 1 we get N × × 2 1 ¯ 1 → ±1 → SpinV (K) → SOV (K) −→ K /(K ) = H (K/K, µ2) so the non-surjectivity of N is some kind of higher Galois cohomology. Warning 207 SpinV → SOV is onto as a map of ALGEBRAIC GROUPS, but SpinV (K) → SOV (K) need NOT be onto. Example 208 Take O3(R) =∼ SO3(R)×{±1} as 3 is odd (in general O2n+1(R) =∼ SO2n+1(R) × {±1}). So we have a sequence 1 → ±1 → Spin3(R) → SO3(R) → 1. 0 × Notice that Spin3(R) ⊆ C3 (R) =∼ H, so Spin3(R) ⊆ H , and in fact we saw that it is S3. 14.2 Covers of symmetric and alternating groups The symmetric group on n letter can be embedded in the obvious way in On(R) as permutations of coordinates. -
The Evolution of Numbers
The Evolution of Numbers Counting Numbers Counting Numbers: {1, 2, 3, …} We use numbers to count: 1, 2, 3, 4, etc You can have "3 friends" A field can have "6 cows" Whole Numbers Whole numbers are the counting numbers plus zero. Whole Numbers: {0, 1, 2, 3, …} Negative Numbers We can count forward: 1, 2, 3, 4, ...... but what if we count backward: 3, 2, 1, 0, ... what happens next? The answer is: negative numbers: {…, -3, -2, -1} A negative number is any number less than zero. Integers If we include the negative numbers with the whole numbers, we have a new set of numbers that are called integers: {…, -3, -2, -1, 0, 1, 2, 3, …} The Integers include zero, the counting numbers, and the negative counting numbers, to make a list of numbers that stretch in either direction indefinitely. Rational Numbers A rational number is a number that can be written as a simple fraction (i.e. as a ratio). 2.5 is rational, because it can be written as the ratio 5/2 7 is rational, because it can be written as the ratio 7/1 0.333... (3 repeating) is also rational, because it can be written as the ratio 1/3 More formally we say: A rational number is a number that can be written in the form p/q where p and q are integers and q is not equal to zero. Example: If p is 3 and q is 2, then: p/q = 3/2 = 1.5 is a rational number Rational Numbers include: all integers all fractions Irrational Numbers An irrational number is a number that cannot be written as a simple fraction. -
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. -
Control Number: FD-00133
Math Released Set 2015 Algebra 1 PBA Item #13 Two Real Numbers Defined M44105 Prompt Rubric Task is worth a total of 3 points. M44105 Rubric Score Description 3 Student response includes the following 3 elements. • Reasoning component = 3 points o Correct identification of a as rational and b as irrational o Correct identification that the product is irrational o Correct reasoning used to determine rational and irrational numbers Sample Student Response: A rational number can be written as a ratio. In other words, a number that can be written as a simple fraction. a = 0.444444444444... can be written as 4 . Thus, a is a 9 rational number. All numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction. b = 0.354355435554... cannot be written as a fraction, so it is irrational. The product of an irrational number and a nonzero rational number is always irrational, so the product of a and b is irrational. You can also see it is irrational with my calculations: 4 (.354355435554...)= .15749... 9 .15749... is irrational. 2 Student response includes 2 of the 3 elements. 1 Student response includes 1 of the 3 elements. 0 Student response is incorrect or irrelevant. Anchor Set A1 – A8 A1 Score Point 3 Annotations Anchor Paper 1 Score Point 3 This response receives full credit. The student includes each of the three required elements: • Correct identification of a as rational and b as irrational (The number represented by a is rational . The number represented by b would be irrational). -
O/Zp OCTONION ALGEBRA and ITS MATRIX REPRESENTATIONS
Palestine Journal of Mathematics Vol. 6(1)(2017) , 307313 © Palestine Polytechnic University-PPU 2017 O Z OCTONION ALGEBRA AND ITS MATRIX = p REPRESENTATIONS Serpil HALICI and Adnan KARATA¸S Communicated by Ayman Badawi MSC 2010 Classications: Primary 17A20; Secondary 16G99. Keywords and phrases: Quaternion algebra, Octonion algebra, Cayley-Dickson process, Matrix representation. Abstract. Since O is a non-associative algebra over R, this real division algebra can not be algebraically isomorphic to any matrix algebras over the real number eld R. In this study using H with Cayley-Dickson process we obtain octonion algebra. Firstly, We investigate octonion algebra over Zp. Then, we use the left and right matrix representations of H to construct repre- sentation for octonion algebra. Furthermore, we get the matrix representations of O Z with the = p help of Cayley-Dickson process. 1 Introduction Let us summarize the notations needed to understand octonionic algebra. There are only four normed division algebras R, C, H and O [2], [6]. The octonion algebra is a non-commutative, non-associative but alternative algebra which discovered in 1843 by John T. Graves. Octonions have many applications in quantum logic, special relativity, supersymmetry, etc. Due to the non-associativity, representing octonions by matrices seems impossible. Nevertheless, one can overcome these problems by introducing left (or right) octonionic operators and xing the direc- tion of action. In this study we investigate matrix representations of octonion division algebra O Z . Now, lets = p start with the quaternion algebra H to construct the octonion algebra O. It is well known that any octonion a can be written by the Cayley-Dickson process as follows [7]. -
The Finite Simple Groups Have Complemented Subgroup Lattices
Pacific Journal of Mathematics THE FINITE SIMPLE GROUPS HAVE COMPLEMENTED SUBGROUP LATTICES Mauro Costantini and Giovanni Zacher Volume 213 No. 2 February 2004 PACIFIC JOURNAL OF MATHEMATICS Vol. 213, No. 2, 2004 THE FINITE SIMPLE GROUPS HAVE COMPLEMENTED SUBGROUP LATTICES Mauro Costantini and Giovanni Zacher We prove that the lattice of subgroups of every finite simple group is a complemented lattice. 1. Introduction. A group G is called a K-group (a complemented group) if its subgroup lattice is a complemented lattice, i.e., for a given H ≤ G there exists a X ≤ G such that hH, Xi = G and H ∧X = 1. The main purpose of this Note is to answer a long-standing open question in finite group theory, by proving that: Every finite simple group is a K-group. In this context, it was known that the alternating groups, the projective special linear groups and the Suzuki groups are K-groups ([P]). Our proof relies on the FSGC-theorem and on structural properties of the maximal subgroups in finite simple groups. The rest of this paper is divided into four sections. In Section 2 we collect some criteria for a subgroup of a group G to have a complement and recall some useful known results. In Section 3 we deal with the classical groups, in 4 with the exceptional groups of Lie type and in Section 5 with the sporadic groups. With reference to notation and terminology, we shall follow closely those in use in [P] and [S]. All groups are meant to be finite. 2. Preliminaries. -
Arxiv:2012.09644V1 [Math.RA] 15 Dec 2020 Oitv -Iesoa Rs.Nnsoitv -Iesoa)Comp Form 8-Dimensional) Norm Nonassociative Associated Whose (Resp
SPLITTING OF QUATERNIONS AND OCTONIONS OVER PURELY INSEPARABLE EXTENSIONS IN CHARACTERISTIC 2 DETLEV W. HOFFMANN Abstract. We give examples of quaternion and octonion division algebras over a field F of characteristic 2 that split over a purely inseparable extension E of F of degree 4 but that do not split over any subextension of F inside E of lower exponent,≥ or, in the case of octonions, over any simple subextension of F inside E. Thus, we get a negative answer to a question posed by Bernhard M¨uhlherr and Richard Weiss. We study this question in terms of the isotropy behaviour of the associated norm forms. 1. Introduction Recall that a quaternion (resp. octonion) algebra A over a field F is a an as- sociative 4-dimensional (resp. nonassociative 8-dimensional) composition algebra whose associated norm form nA is a nondegenerate quadratic form defined on the F -vector space A. This norm form has the property that A is a division algebra iff nA is anisotropic. Furthermore, nA determines A in the sense that if A′ is an- other quaternion (resp. octonion) algebra, then A ∼= A′ as F -algebras iff nA ∼= nA′ (i.e., the norm forms are isometric). There is a vast amount of literature on such algebras. In this note, we consider only base fields of characteristic 2, a case that, compared to characteristic not 2, gives rise to additional subtleties that make the theory in some sense richer but also more intricate and complicated. For example, when conisdering subfields of such algebras one has to distingush between separable and inseparable quadratic extensions. -
Automorphism Groups of Locally Compact Reductive Groups
PROCEEDINGS OF THE AMERICAN MATHEMATICALSOCIETY Volume 106, Number 2, June 1989 AUTOMORPHISM GROUPS OF LOCALLY COMPACT REDUCTIVE GROUPS T. S. WU (Communicated by David G Ebin) Dedicated to Mr. Chu. Ming-Lun on his seventieth birthday Abstract. A topological group G is reductive if every continuous finite di- mensional G-module is semi-simple. We study the structure of those locally compact reductive groups which are the extension of their identity components by compact groups. We then study the automorphism groups of such groups in connection with the groups of inner automorphisms. Proposition. Let G be a locally compact reductive group such that G/Gq is compact. Then /(Go) is dense in Aq(G) . Let G be a locally compact topological group. Let A(G) be the group of all bi-continuous automorphisms of G. Then A(G) has the natural topology (the so-called Birkhoff topology or ^-topology [1, 3, 4]), so that it becomes a topo- logical group. We shall always adopt such topology in the following discussion. When G is compact, it is well known that the identity component A0(G) of A(G) is the group of all inner automorphisms induced by elements from the identity component G0 of G, i.e., A0(G) = I(G0). This fact is very useful in the study of the structure of locally compact groups. On the other hand, it is also well known that when G is a semi-simple Lie group with finitely many connected components A0(G) - I(G0). The latter fact had been generalized to more general groups ([3]).