Notes on Symmetric Bilinear Forms
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3 Bilinear Forms & Euclidean/Hermitian Spaces
3 Bilinear Forms & Euclidean/Hermitian Spaces Bilinear forms are a natural generalisation of linear forms and appear in many areas of mathematics. Just as linear algebra can be considered as the study of `degree one' mathematics, bilinear forms arise when we are considering `degree two' (or quadratic) mathematics. For example, an inner product is an example of a bilinear form and it is through inner products that we define the notion of length in n p 2 2 analytic geometry - recall that the length of a vector x 2 R is defined to be x1 + ... + xn and that this formula holds as a consequence of Pythagoras' Theorem. In addition, the `Hessian' matrix that is introduced in multivariable calculus can be considered as defining a bilinear form on tangent spaces and allows us to give well-defined notions of length and angle in tangent spaces to geometric objects. Through considering the properties of this bilinear form we are able to deduce geometric information - for example, the local nature of critical points of a geometric surface. In this final chapter we will give an introduction to arbitrary bilinear forms on K-vector spaces and then specialise to the case K 2 fR, Cg. By restricting our attention to thse number fields we can deduce some particularly nice classification theorems. We will also give an introduction to Euclidean spaces: these are R-vector spaces that are equipped with an inner product and for which we can `do Euclidean geometry', that is, all of the geometric Theorems of Euclid will hold true in any arbitrary Euclidean space. -
Embeddings of Integral Quadratic Forms Rick Miranda Colorado State
Embeddings of Integral Quadratic Forms Rick Miranda Colorado State University David R. Morrison University of California, Santa Barbara Copyright c 2009, Rick Miranda and David R. Morrison Preface The authors ran a seminar on Integral Quadratic Forms at the Institute for Advanced Study in the Spring of 1982, and worked on a book-length manuscript reporting on the topic throughout the 1980’s and early 1990’s. Some new results which are proved in the manuscript were announced in two brief papers in the Proceedings of the Japan Academy of Sciences in 1985 and 1986. We are making this preliminary version of the manuscript available at this time in the hope that it will be useful. Still to do before the manuscript is in final form: final editing of some portions, completion of the bibliography, and the addition of a chapter on the application to K3 surfaces. Rick Miranda David R. Morrison Fort Collins and Santa Barbara November, 2009 iii Contents Preface iii Chapter I. Quadratic Forms and Orthogonal Groups 1 1. Symmetric Bilinear Forms 1 2. Quadratic Forms 2 3. Quadratic Modules 4 4. Torsion Forms over Integral Domains 7 5. Orthogonality and Splitting 9 6. Homomorphisms 11 7. Examples 13 8. Change of Rings 22 9. Isometries 25 10. The Spinor Norm 29 11. Sign Structures and Orientations 31 Chapter II. Quadratic Forms over Integral Domains 35 1. Torsion Modules over a Principal Ideal Domain 35 2. The Functors ρk 37 3. The Discriminant of a Torsion Bilinear Form 40 4. The Discriminant of a Torsion Quadratic Form 45 5. -
Preliminary Material for M382d: Differential Topology
PRELIMINARY MATERIAL FOR M382D: DIFFERENTIAL TOPOLOGY DAVID BEN-ZVI Differential topology is the application of differential and integral calculus to study smooth curved spaces called manifolds. The course begins with the assumption that you have a solid working knowledge of calculus in flat space. This document sketches some important ideas and given you some problems to jog your memory and help you review the material. Don't panic if some of this material is unfamiliar, as I expect it will be for most of you. Simply try to work through as much as you can. [In the name of intellectual honesty, I mention that this handout, like much of the course, is cribbed from excellent course materials developed in previous semesters by Dan Freed.] Here, first, is a brief list of topics I expect you to know: Linear algebra: abstract vector spaces and linear maps, dual space, direct sum, subspaces and quotient spaces, bases, change of basis, matrix compu- tations, trace and determinant, bilinear forms, diagonalization of symmetric bilinear forms, inner product spaces Calculus: directional derivative, differential, (higher) partial derivatives, chain rule, second derivative as a symmetric bilinear form, Taylor's theo- rem, inverse and implicit function theorem, integration, change of variable formula, Fubini's theorem, vector calculus in 3 dimensions (div, grad, curl, Green and Stokes' theorems), fundamental theorem on ordinary differential equations You may or may not have studied all of these topics in your undergraduate years. Most of the topics listed here will come up at some point in the course. As for references, I suggest first that you review your undergraduate texts. -
Limit Roots of Lorentzian Coxeter Systems
UNIVERSITY OF CALIFORNIA Santa Barbara Limit Roots of Lorentzian Coxeter Systems A Thesis submitted in partial satisfaction of the requirements for the degree of Master of Arts in Mathematics by Nicolas R. Brody Committee in Charge: Professor Jon McCammond, Chair Professor Darren Long Professor Stephen Bigelow June 2016 The Thesis of Nicolas R. Brody is approved: Professor Darren Long Professor Stephen Bigelow Professor Jon McCammond, Committee Chairperson May 2016 Limit Roots of Lorentzian Coxeter Systems Copyright c 2016 by Nicolas R. Brody iii Acknowledgements My time at UC Santa Barbara has been inundated with wonderful friends, teachers, mentors, and mathematicians. I am especially indebted to Jon McCammond, Jordan Schettler, and Maribel Cachadia. A very special thanks to Jean-Phillippe Labb for sharing his code which was used to produce Figures 7.1, 7.3, and 7.4. This thesis is dedicated to my three roommates: Viki, Luna, and Layla. I hope we can all be roommates again soon. iv Abstract Limit Roots of Lorentzian Coxeter Systems Nicolas R. Brody A reflection in a real vector space equipped with a positive definite symmetric bilinear form is any automorphism that sends some nonzero vector v to its negative and pointwise fixes its orthogonal complement, and a finite reflection group is a group generated by such transformations that acts properly on the sphere of unit vectors. We note two important classes of groups which occur as finite reflection groups: for a 2-dimensional vector space, we recover precisely the finite dihedral groups as reflection groups, and permuting basis vectors in an n-dimensional vector space gives a way of viewing a symmetric group as a reflection group. -
Krammer Representations for Complex Braid Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 371 (2012) 175–206 Contents lists available at SciVerse ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Krammer representations for complex braid groups Ivan Marin Institut de Mathématiques de Jussieu, Université Paris 7, 175 rue du Chevaleret, F-75013 Paris, France article info abstract Article history: Let B be the generalized braid group associated to some finite Received 5 April 2012 complex reflection group W . We define a representation of B Availableonline30August2012 of dimension the number of reflections of the corresponding Communicated by Michel Broué reflection group, which generalizes the Krammer representation of the classical braid groups, and is thus a good candidate in view MSC: 20C99 of proving the linearity of B. We decompose this representation 20F36 in irreducible components and compute its Zariski closure, as well as its restriction to parabolic subgroups. We prove that it is Keywords: faithful when W is a Coxeter group of type ADE and odd dihedral Braid groups types, and conjecture its faithfulness when W has a single class Complex reflection groups of reflections. If true, this conjecture would imply various group- Linear representations theoretic properties for these groups, that we prove separately to Knizhnik–Zamolodchikov systems be true for the other groups. Krammer representation © 2012 Elsevier Inc. All rights reserved. 1. Introduction Let k be a field of characteristic 0 and E a finite-dimensional k-vector space. A reflection in E is an element s ∈ GL(E) such that s2 = 1 and Ker(s − 1) is a hyperplane of E. -
On the Classification of Integral Quadratic Forms
15 On the Classification of Integral Quadratic Forms J. H. Conway and N. J. A. Sloane This chapter gives an account of the classification of integral quadratic forms. It is particularly designed for readers who wish to be able to do explicit calculations. Novel features include an elementary system of rational invariants (defined without using the Hilbert norm residue symbol), an improved notation for the genus of a form, an efficient way to compute the number of spinor genera in a genus, and some conditions which imply that there is only one class in a genus. We give tables of the binary forms with - 100 ~ det ~ 50, the indecomposable ternary forms with Idetl ~ 50, the genera of forms with Idetl ~ 11, the genera of p elementary forms for all p, and the positive definite forms with determinant 2 up to dimension 18 and determinant 3 up to dimension 17. 1. Introduction The project of classifying integral quadratic forms has a long history, to which many mathematicians have contributed. The binary (or two dimensional) forms were comprehensively discussed by Gauss. Gauss and later workers also made substantial inroads into the problem of ternary and higher-dimensional forms. The greatest advances since then have been the beautiful development of the theory of rational quadratic forms (Minkowski, Hasse, Witt), and Eichler's complete classification of indefinite forms in dimension 3 or higher in terms of the notion of spinor genus. Definite forms correspond to lattices in Euclidean space. For small dimensions they can be classified using Minkowski's generalization of Gauss's notion of reduced form, but this method rapidly becomes impracticable when the dimension reaches 6 or 7. -
Contents 5 Bilinear and Quadratic Forms
Linear Algebra (part 5): Bilinear and Quadratic Forms (by Evan Dummit, 2020, v. 1.00) Contents 5 Bilinear and Quadratic Forms 1 5.1 Bilinear Forms . 1 5.1.1 Denition, Associated Matrices, Basic Properties . 1 5.1.2 Symmetric Bilinear Forms and Diagonalization . 3 5.2 Quadratic Forms . 5 5.2.1 Denition and Basic Properties . 6 5.2.2 Quadratic Forms Over Rn: Diagonalization of Quadratic Varieties . 7 5.2.3 Quadratic Forms Over Rn: The Second Derivatives Test . 9 5.2.4 Quadratic Forms Over Rn: Sylvester's Law of Inertia . 11 5 Bilinear and Quadratic Forms In this chapter, we will discuss bilinear and quadratic forms. Bilinear forms are simply linear transformations that are linear in more than one variable, and they will allow us to extend our study of linear phenomena. They are closely related to quadratic forms, which are (classically speaking) homogeneous quadratic polynomials in multiple variables. Despite the fact that quadratic forms are not linear, we can (perhaps surprisingly) still use many of the tools of linear algebra to study them. 5.1 Bilinear Forms • We begin by discussing basic properties of bilinear forms on an arbitrary vector space. 5.1.1 Denition, Associated Matrices, Basic Properties • Let V be a vector space over the eld F . • Denition: A function Φ: V × V ! F is a bilinear form on V if it is linear in each variable when the other variable is xed. Explicitly, this means Φ(v1 + αv2; y) = Φ(v1; w) + αΦ(v2; w) and Φ(v; w1 + αw2) = Φ(v; w1) + αΦ(v; w2) for arbitrary vi; wi 2 V and α 2 F . -
Symmetric Powers of Symmetric Bilinear Forms 11
SYMMETRIC POWERS OF SYMMETRIC BILINEAR FORMS SEAN¶ MCGARRAGHY Abstract. We study symmetric powers of classes of symmetric bilinear forms in the Witt-Grothendieck ring of a ¯eld of characteristic not equal to 2, and derive their basic properties and compute their classical invariants. We relate these to earlier results on exterior powers of such forms. 1991 AMS Subject Classi¯cation: 11E04, 11E81 Keywords: exterior power, symmetric power, pre-¸-ring, ¸-ring, ¸-ring-augmentation, symmetric bilinear form, Witt ring, Witt-Grothendieck ring. 1. Introduction Throughout this paper, K will be a ¯eld of characteristic di®erent from 2. Given a ¯nite-dimensional K-vector space V and a non-negative integer k, denote by kV the k-fold exterior power of V , and by SkV the k-fold symmetric power of V . It is well-known that the ring of isomorphism classes of ¯nite-dimensional K-vector spaces underV direct sum and tensor product is a ¸-ring, with the exterior powers acting as the ¸- operations (associated to elementary symmetric polynomials) and the symmetric powers being the \dual" s-operations (associated to complete homogeneous polynomials). The exterior and symmetric powers are in fact functors on the category of ¯nite-dimensional K-vector spaces and K-linear maps, special cases of the Schur functors which arise in representation theory (see, for example, [2] or [3]). A natural question to ask is whether we may de¯ne such \Schur powers" of symmetric bilinear and quadratic forms as well. Let ' be a symmetric bilinear form on an n-dimensional vector space V over K. Bourbaki (see [1, Ch. -
Geometric Algebra for Mathematicians
Geometric Algebra for Mathematicians Kalle Rutanen 09.04.2013 (this is an incomplete draft) 1 Contents 1 Preface 4 1.1 Relation to other texts . 4 1.2 Acknowledgements . 6 2 Preliminaries 11 2.1 Basic definitions . 11 2.2 Vector spaces . 12 2.3 Bilinear spaces . 21 2.4 Finite-dimensional bilinear spaces . 27 2.5 Real bilinear spaces . 29 2.6 Tensor product . 32 2.7 Algebras . 33 2.8 Tensor algebra . 35 2.9 Magnitude . 35 2.10 Topological spaces . 37 2.11 Topological vector spaces . 39 2.12 Norm spaces . 40 2.13 Total derivative . 44 3 Symmetry groups 50 3.1 General linear group . 50 3.2 Scaling linear group . 51 3.3 Orthogonal linear group . 52 3.4 Conformal linear group . 55 3.5 Affine groups . 55 4 Algebraic structure 56 4.1 Clifford algebra . 56 4.2 Morphisms . 58 4.3 Inverse . 59 4.4 Grading . 60 4.5 Exterior product . 64 4.6 Blades . 70 4.7 Scalar product . 72 4.8 Contraction . 74 4.9 Interplay . 80 4.10 Span . 81 4.11 Dual . 87 4.12 Commutator product . 89 4.13 Orthogonality . 90 4.14 Magnitude . 93 4.15 Exponential and trigonometric functions . 94 2 5 Linear transformations 97 5.1 Determinant . 97 5.2 Pinors and unit-versors . 98 5.3 Spinors and rotors . 104 5.4 Projections . 105 5.5 Reflections . 105 5.6 Adjoint . 107 6 Geometric applications 107 6.1 Cramer’s rule . 108 6.2 Gram-Schmidt orthogonalization . 108 6.3 Alternating forms and determinant . -
Math 272Y: Rational Lattices and Their Theta Functions 9 September 2019: Lattice Basics
Math 272y: Rational Lattices and their Theta Functions 9 September 2019: Lattice basics This first chapter is mainly for fixing definitions and notations. The mathematical content is mostly what one might expect from adapting to Zn the familiar structure of symmetric pairings on Rn. Still there are a few novel features, such as the notion of an even lattice and the connection with the Fermat–Pell equation. By a lattice we mean a finitely-generated free abelian group L together with a symmetric bilinear pairing L × L ! R. The rank of the lattice is the rank of L, which is the integer n ≥ 0 such that L =∼ Zn. The bilinear form is often denoted h·; ·i. The lattice is said to be rational if h·; ·i takes values in Q, and integral if h·; ·i takes values in Z. The associated quadratic form Q : L ! R is defined by Q(x) = hx; xi; the well-known “polar- ization” identity 2hx; yi = hx + y; x + yi − hx; xi − hy; yi = Q(x + y) − Q(x) − Q(y) (1) lets us recover h·; ·i from Q. It follows from (1) that the lattice is rational if and only if Q takes rational values. Note that it is not true that the lattice is integral if and only if Q takes integral values: certainly if hx; yi 2 Z for all x; y 2 L then hx; xi 2 Z for all x 2 L, but in the reverse direction we can only conclude that h·; ·i takes half-integral values because of the factor 2 1 T 2 −1 of 2 in (1). -
Arxiv:2010.09374V1 [Math.AG] 19 Oct 2020 Lsdfils N Eoescasclcut Over Counts Classical Recovers One fields
Applications to A1-enumerative geometry of the A1-degree Sabrina Pauli Kirsten Wickelgren Abstract These are lecture notes from the conference Arithmetic Topology at the Pacific Institute of Mathemat- ical Sciences on applications of Morel’s A1-degree to questions in enumerative geometry. Additionally, we give a new dynamic interpretation of the A1-Milnor number inspired by the first named author’s enrichment of dynamic intersection numbers. 1 Introduction A1-homotopy theory provides a powerful framework to apply tools from algebraic topology to schemes. In these notes, we discuss Morel’s A1-degree, giving the analog of the Brouwer degree in classical topology, and applications to enumerative geometry. Instead of the integers, the A1-degree takes values in bilinear forms, or more precisely, in the Grothendieck-Witt ring GW(k) of a field k, defined to be the group completion of isomorphism classes of symmetric, non-degenerate bilinear k-forms. This can result in an enumeration of algebro-geometric objects valued in GW(k), giving an A1-enumerative geometry over non-algebraically closed fields. One recovers classical counts over C using the rank homomorphism GW(k) Z, but GW(k) can contain more information. This information can record arithmetic-geometric properties→ of the objects being enumerated over field extensions of k. In more detail, we start with the classical Brouwer degree. We introduce enough A1-homotopy theory to describe Morel’s degree and use the Eisenbud-Khimshiashvili-Levine signature formula to give context for the degree and a formula for the local A1-degree. The latter is from joint work of Jesse Kass and the second-named author. -
Hyperquadrics
CHAPTER VII HYPERQUADRICS Conic sections have played an important role in projective geometry almost since the beginning of the subject. In this chapter we shall begin by defining suitable projective versions of conics in the plane, quadrics in 3-space, and more generally hyperquadrics in n-space. We shall also discuss tangents to such figures from several different viewpoints, prove a geometric classification for conics similar to familiar classifications for ordinary conics and quadrics in R2 and R3, and we shall derive an enhanced duality principle for projective spaces and hyperquadrics. Finally, we shall use a mixture of synthetic and analytic methods to prove a famous classical theorem due to B. Pascal (1623{1662)1 on hexagons inscribed in plane conics, a dual theorem due to C. Brianchon (1783{1864),2 and several other closely related results. 1. Definitions The three familiar curves which we call the \conic sections" have a long history ... It seems that they will always hold a place in the curriculum. The beginner in analytic geometry will take up these curves after he has studied the circle. Whoever looks at a circle will continue to see an ellipse, unless his eye is on the axis of the curve. The earth will continue to follow a nearly elliptical orbit around the sun, projectiles will approximate parabolic orbits, [and] a shaded light will illuminate a hyperbolic arch. | J. L. Coolidge (1873{1954) In classical Greek geometry, conic sections were first described synthetically as intersections of a plane and a cone. On the other hand, today such curves are usually viewed as sets of points (x; y) in the Cartesian plane which satisfy a nontrivial quadratic equation of the form Ax2 + 2Bxy + Cy2 + 2D + 2E + F = 0 where at least one of A; B; C is nonzero.