DOCSLIB.ORG

Math 651 Homework 1 - Algebras and Groups Due 2/22/2013

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 651 Homework 1 - Algebras and Groups Due 2/22/2013 Math 651 Homework 1 - Algebras and Groups Due 2/22/2013 1) Consider the Lie Group SU(2), the group of 2 × 2 complex matrices A T with A A = I and det(A) = 1. The underlying set is z −w jzj2 + jwj2 = 1 (1) w z with the standard S3 topology. The usual basis for su(2) is 0 i 0 −1 i 0 X = Y = Z = (2) i 0 1 0 0 −i (which are each i times the Pauli matrices: X = iσx, etc.). a) Show this is the algebra of purely imaginary quaternions, under the commutator bracket. b) Extend X to a left-invariant field and Y to a right-invariant field, and show by computation that the Lie bracket between them is zero. c) Extending X, Y , Z to global left-invariant vector fields, give SU(2) the metric g(X; X) = g(Y; Y ) = g(Z; Z) = 1 and all other inner products zero. Show this is a bi-invariant metric. d) Pick > 0 and set g(X; X) = 2, leaving g(Y; Y ) = g(Z; Z) = 1. Show this is left-invariant but not bi-invariant. p 2) The realification of an n × n complex matrix A + −1B is its assignment it to the 2n × 2n matrix A −B (3) BA Any n × n quaternionic matrix can be written A + Bk where A and B are complex matrices. Its complexification is the 2n × 2n complex matrix A −B (4) B A a) Show that the realification of complex matrices and complexifica- tion of quaternionic matrices are algebra homomorphisms. In the 1 quaternionic case, show that quaternionic conjugation corresponds to complex conjugation and the transposition of the off-diagonal blocks (but not the transpositions of the blocks themselves), and that quaternionic transposition corresponds to transposition of the blocks themselves (so that quaternionic conjugate-transpose is the same as complex conjugate-transpose of the complexified matrix). b) The matrix groups Sp(n) are the n × n quaternion-valued matrices T Q that satisfy Q Q = I. Show that Sp(1) ≈ SU(2). c) We defined the the algebras sp(2n; R) and sp(2n; C) to be the 2 × 2 real (resp. complex) matrices X that satisfy XT J + JX = 0 (5) where 0 I J = : (6) −I 0 The Lie algebra sp(n) of the group Sp(n) defined above is necessarily a real Lie algebra. Via the complexification process, show that sp(n)⊗C ≈ sp(2n; C) (the real algebras sp(2n; R) and sp(n) are real forms of the complex algebra sp(2n; C)). 3) Let g be a finite-dimensional Lie algebra over C with radical r, and assume r is abelian. In this problem we will use the vanishing of H2 for semi- simple algebras to prove that g = h⊕r. As a side note, this is the essential step in Levi's theorem (that g = h n r whether or not r is abelian). a) Let g = g=r and show that r is a g-module. In the future, denote the canonical projection g ! g simply with a bar: x 7! x¯ 2 g. b) Let σ : g ! g be any vector-space splitting. This means that when x 2 g we have σ(¯x) =x ¯. Show that g(x; y) , σ([x; y]) − [σ(x); σ(y)] (7) is an element of V2 g∗⊗g, which can actually be considered an element of V2 g∗ ⊗ r. c) Show that the map g is closed, so g represents a class in H2(g; r). d) By the vanishing of H2(g; r), we know g is exact. Show that this results in a map η : g ! η so that g(x; y) = η([x; y]), and that σ −η : g ! g is a Lie algebra monomorphism, so provides the desired Lie algebra compliment to r. 2 4) In class we stated that H2(g;M) could be identified with the isomorphism classes of abelien extensions of g by M. We proved half of this assertion: that given a representative of a class in H2, we could construct such a Lie algebra extension, and if a different representative in the same class were chosen, the two extensions, though not the same, were isomorphic. Now assume M is a g-module and 0 M i h π g 0 (8) is an exact sequence of Lie algebras and so that the bracket in h is given by [x; i(m)] = i (π(x):m) (9) for m 2 M, and that [i(m); i(n)] = 0. Prove there is class in H2(g;M) that provides this extension. 5) Let G be a connected Lie algebra, meaning a manifold with a given dif- ferentiable structure, along with a group structure. Prove that G has a canonical analytic structure. You may proceed as you wish, but here is one practicable approach: Let Ω ⊂ g be a connected domain containing 0 which is small enough that exp :Ω ! U is a diffeomorphism, and let fpigi2Λ (where Λ ⊆ Z) be a set of points in G. Let Ui = Lpi U, and let πi : Ui ! Ω, given by −1 πi exp ◦ L −1 (10) , pi be the charts. Then show the following: a) The pi can be chosen so that fΩigi2Λ is a locally finite covering of G b) Set Uij = Ui \ Uj whenever i 6= j and the intersection is non-empty. Then both πi; πj : Uij ! Ω. Using the Cambpell-Baker-Hausdorff −1 formula, show that the transition maps πi ◦ πj are analytic. c) Indicate that the Lie algebra structure (itself determined by the dif- ferentiable structure of G) uniquely determines the analytic structure. 3.
Load more

Recommended publications
  • CLIFFORD ALGEBRAS and THEIR REPRESENTATIONS Introduction
    CLIFFORD ALGEBRAS AND THEIR REPRESENTATIONS Andrzej Trautman, Uniwersytet Warszawski, Warszawa, Poland Article accepted for publication in the Encyclopedia of Mathematical Physics c 2005 Elsevier Science Ltd ! Introduction Introductory and historical remarks Clifford (1878) introduced his ‘geometric algebras’ as a generalization of Grassmann alge- bras, complex numbers and quaternions. Lipschitz (1886) was the first to define groups constructed from ‘Clifford numbers’ and use them to represent rotations in a Euclidean ´ space. E. Cartan discovered representations of the Lie algebras son(C) and son(R), n > 2, that do not lift to representations of the orthogonal groups. In physics, Clifford algebras and spinors appear for the first time in Pauli’s nonrelativistic theory of the ‘magnetic elec- tron’. Dirac (1928), in his work on the relativistic wave equation of the electron, introduced matrices that provide a representation of the Clifford algebra of Minkowski space. Brauer and Weyl (1935) connected the Clifford and Dirac ideas with Cartan’s spinorial represen- tations of Lie algebras; they found, in any number of dimensions, the spinorial, projective representations of the orthogonal groups. Clifford algebras and spinors are implicit in Euclid’s solution of the Pythagorean equation x2 y2 + z2 = 0 which is equivalent to − y x z p = 2 p q (1) −z y + x q ! " ! " # $ so that x = q2 p2, y = p2 + q2, z = 2pq. If the numbers appearing in (1) are real, then − this equation can be interpreted as providing a representation of a vector (x, y, z) R3, null ∈ with respect to a quadratic form of signature (1, 2), as the ‘square’ of a spinor (p, q) R2.
    [Show full text]
  • On the Complexification of the Classical Geometries And
    On the complexication of the classical geometries and exceptional numb ers April Intro duction The classical groups On R Spn R and GLn C app ear as the isometry groups of sp ecial geome tries on a real vector space In fact the orthogonal group On R represents the linear isomorphisms of arealvector space V of dimension nleaving invariant a p ositive denite and symmetric bilinear form g on V The symplectic group Spn R represents the isometry group of a real vector space of dimension n leaving invariant a nondegenerate skewsymmetric bilinear form on V Finally GLn C represents the linear isomorphisms of a real vector space V of dimension nleaving invariant a complex structure on V ie an endomorphism J V V satisfying J These three geometries On R Spn R and GLn C in GLn Rintersect even pairwise in the unitary group Un C Considering now the relativeversions of these geometries on a real manifold of dimension n leads to the notions of a Riemannian manifold an almostsymplectic manifold and an almostcomplex manifold A symplectic manifold however is an almostsymplectic manifold X ie X is a manifold and is a nondegenerate form on X so that the form is closed Similarly an almostcomplex manifold X J is called complex if the torsion tensor N J vanishes These three geometries intersect in the notion of a Kahler manifold In view of the imp ortance of the complexication of the real Lie groupsfor instance in the structure theory and representation theory of semisimple real Lie groupswe consider here the question on the underlying geometrical
    [Show full text]
  • Patterns of Maximally Entangled States Within the Algebra of Biquaternions
    J. Phys. Commun. 4 (2020) 055018 https://doi.org/10.1088/2399-6528/ab9506 PAPER Patterns of maximally entangled states within the algebra of OPEN ACCESS biquaternions RECEIVED 21 March 2020 Lidia Obojska REVISED 16 May 2020 Siedlce University of Natural Sciences and Humanities, ul. 3 Maja 54, 08-110 Siedlce, Poland ACCEPTED FOR PUBLICATION E-mail: [email protected] 20 May 2020 Keywords: bipartite entanglement, biquaternions, density matrix, mereology PUBLISHED 28 May 2020 Original content from this Abstract work may be used under This paper proposes a description of maximally entangled bipartite states within the algebra of the terms of the Creative Commons Attribution 4.0 biquaternions–Ä . We assume that a bipartite entanglement is created in a process of splitting licence. one particle into a pair of two indiscernible, yet not identical particles. As a result, we describe an Any further distribution of this work must maintain entangled correlation by the use of a division relation and obtain twelve forms of biquaternions, attribution to the author(s) and the title of representing pure maximally entangled states. Additionally, we obtain other patterns, describing the work, journal citation mixed entangled states. Finally, we show that there are no other maximally entangled states in Ä and DOI. than those presented in this work. 1. Introduction In 1935, Einstein, Podolski and Rosen described a strange phenomenon, in which in its pure state, one particle behaves like a pair of two indistinguishable, yet not identical, particles [1]. This phenomenon, called by Einstein ’a spooky action at a distance’, has been investigated by many authors, who tried to explain this strange correlation [2–9].
    [Show full text]
  • Majorana Spinors
    MAJORANA SPINORS JOSE´ FIGUEROA-O'FARRILL Contents 1. Complex, real and quaternionic representations 2 2. Some basis-dependent formulae 5 3. Clifford algebras and their spinors 6 4. Complex Clifford algebras and the Majorana condition 10 5. Examples 13 One dimension 13 Two dimensions 13 Three dimensions 14 Four dimensions 14 Six dimensions 15 Ten dimensions 16 Eleven dimensions 16 Twelve dimensions 16 ...and back! 16 Summary 19 References 19 These notes arose as an attempt to conceptualise the `symplectic Majorana{Weyl condition' in 5+1 dimensions; but have turned into a general discussion of spinors. Spinors play a crucial role in supersymmetry. Part of their versatility is that they come in many guises: `Dirac', `Majorana', `Weyl', `Majorana{Weyl', `symplectic Majorana', `symplectic Majorana{Weyl', and their `pseudo' counterparts. The tra- ditional physics approach to this topic is a mixed bag of tricks using disparate aspects of representation theory of finite groups. In these notes we will attempt to provide a uniform treatment based on the classification of Clifford algebras, a work dating back to the early 60s and all but ignored by the theoretical physics com- munity. Recent developments in superstring theory have made us re-examine the conditions for the existence of different kinds of spinors in spacetimes of arbitrary signature, and we believe that a discussion of this more uniform approach is timely and could be useful to the student meeting this topic for the first time or to the practitioner who has difficulty remembering the answer to questions like \when do symplectic Majorana{Weyl spinors exist?" The notes are organised as follows.
    [Show full text]
  • Maxwell's Equations
    Maxwell’s equations Daniel Henry Gottlieb August 1, 2004 Abstract We express Maxwell’s equations as a single equation, first using the divergence of a special type of matrix field to obtain the four current, and then the divergence of a special matrix to obtain the Electromagnetic field. These two equations give rise to a remarkable dual set of equations in which the operators become the matrices and the vectors become the fields. The decoupling of the equations into the wave equation is very simple and natural. The divergence of the stress energy tensor gives the Lorentz Law in a very natural way. We compare this approach to the related descriptions of Maxwell’s equations by biquaternions and Clifford algebras. 1 Introduction Maxwell’s equations have been expressed in many forms in the century and a half since their dis- covery. The original equations were 16 in number. The vector forms, written below, consist of 4 equations. The differential form versions consists of two equations; see [Misner, Thorne and Wheeler(1973); see equations 4.10, 4.11]. See also [Steven Parrott(1987) page 98 -100 ] The ap- plication of quaternions, and their complexification, the biquaternions, results in a version of one equation. William E. Baylis (1999) equation 3.8, is an example. In this work, we obtain one Maxwell equation, (10), representing the electromagnetic field as a matrix and the divergence as a vector multiplying the field matrix. But we also obtain a remarkable dual formulation of Maxwell’s equation, (15), wherein the operator is now the matrix and the field is now the vector.
    [Show full text]
  • 5 Spinor Calculus
    5 Spinor Calculus 5.1 From triads and Euler angles to spinors. A heuristic introduction. As mentioned already in Section 3.4.3, it is an obvious idea to enrich the Pauli algebra formalism by introducing the complex vector space V(2; C) on which the matrices operate. The two-component complex vectors are traditionally called spinors28. We wish to show that they give rise to a wide range of applications. In fact we shall introduce the spinor concept as a natural answer to a problem that arises in the context of rotational motion. In Section 3 we have considered rotations as operations performed on a vector space. Whereas this approach enabled us to give a group-theoretical definition of the magnetic field, a vector is not an appropriate construct to account for the rotation of an orientable object. The simplest mathematical model suitable for this purpose is a Cartesian (orthogonal) three-frame, briefly, a triad. The problem is to consider two triads with coinciding origins, and the rotation of the object frame is described with respect to the space frame. The triads are represented in terms of their respective unit vectors: the space frame as Σs(x^1; x^2; x^3) and the object frame as Σc(e^1; e^2; e^3). Here c stands for “corpus,” since o for “object” seems ambiguous. We choose the frames to be right-handed. These orientable objects are not pointlike, and their parametrization offers novel problems. In this sense we may refer to triads as “higher objects,” by contrast to points which are “lower objects.” The thought that comes most easily to mind is to consider the nine direction cosines e^i · x^k but this is impractical, because of the six relations connecting these parameters.
    [Show full text]
  • Inner Product Spaces
    CHAPTER 6 Woman teaching geometry, from a fourteenth-century edition of Euclid’s geometry book. Inner Product Spaces In making the definition of a vector space, we generalized the linear structure (addition and scalar multiplication) of R2 and R3. We ignored other important features, such as the notions of length and angle. These ideas are embedded in the concept we now investigate, inner products. Our standing assumptions are as follows: 6.1 Notation F, V F denotes R or C. V denotes a vector space over F. LEARNING OBJECTIVES FOR THIS CHAPTER Cauchy–Schwarz Inequality Gram–Schmidt Procedure linear functionals on inner product spaces calculating minimum distance to a subspace Linear Algebra Done Right, third edition, by Sheldon Axler 164 CHAPTER 6 Inner Product Spaces 6.A Inner Products and Norms Inner Products To motivate the concept of inner prod- 2 3 x1 , x 2 uct, think of vectors in R and R as x arrows with initial point at the origin. x R2 R3 H L The length of a vector in or is called the norm of x, denoted x . 2 k k Thus for x .x1; x2/ R , we have The length of this vector x is p D2 2 2 x x1 x2 . p 2 2 x1 x2 . k k D C 3 C Similarly, if x .x1; x2; x3/ R , p 2D 2 2 2 then x x1 x2 x3 . k k D C C Even though we cannot draw pictures in higher dimensions, the gener- n n alization to R is obvious: we define the norm of x .x1; : : : ; xn/ R D 2 by p 2 2 x x1 xn : k k D C C The norm is not linear on Rn.
    [Show full text]
  • 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]).
    [Show full text]
  • Chapter 1: Complex Numbers Lecture Notes Math Section
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Almae Matris Studiorum Campus Chapter 1: Complex Numbers Lecture notes Math Section 1.1: Definition of Complex Numbers Definition of a complex number A complex number is a number that can be expressed in the form z = a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation i2 = −1. In this expression, a is the real part Re(z) and b is the imaginary part Im(z) of the complex number. The complex number a + bi can be identified with the point (a; b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. Ex.1 Understanding complex numbers Write the real part and the imaginary part of the following complex numbers and plot each number in the complex plane. (1) i (2) 4 + 2i (3) 1 − 3i (4) −2 Section 1.2: Operations with Complex Numbers Addition and subtraction of two complex numbers To add/subtract two complex numbers we add/subtract each part separately: (a + bi) + (c + di) = (a + c) + (b + d)i and (a + bi) − (c + di) = (a − c) + (b − d)i Ex.1 Addition and subtraction of complex numbers (1) (9 + i) + (2 − 3i) (2) (−2 + 4i) − (6 + 3i) (3) (i) − (−11 + 2i) (4) (1 + i) + (4 + 9i) Multiplication of two complex numbers To multiply two complex numbers we proceed as follows: (a + bi)(c + di) = ac + adi + bci + bdi2 = ac + adi + bci − bd = (ac − bd) + (ad + bc)i Ex.2 Multiplication of complex numbers (1) (3 + 2i)(1 + 7i) (2) (i + 1)2 (3) (−4 + 3i)(2 − 5i) 1 Chapter 1: Complex Numbers Lecture notes Math Conjugate of a complex number The complex conjugate of the complex number z = a + bi is defined to be z¯ = a − bi.
    [Show full text]
  • Unifying the Hyperbolic and Spherical 2-Body Problem with Biquaternions
    Unifying the Hyperbolic and Spherical 2-Body Problem with Biquaternions Philip Arathoon December 2020 Abstract The 2-body problem on the sphere and hyperbolic space are both real forms of holo- morphic Hamiltonian systems defined on the complex sphere. This admits a natural description in terms of biquaternions and allows us to address questions concerning the hyperbolic system by complexifying it and treating it as the complexification of a spherical system. In this way, results for the 2-body problem on the sphere are readily translated to the hyperbolic case. For instance, we implement this idea to completely classify the relative equilibria for the 2-body problem on hyperbolic 3-space for a strictly attractive potential. Background and outline The case of the 2-body problem on the 3-sphere has recently been considered by the author in [1]. This treatment takes advantage of the fact that S3 is a group and that the action of SO(4) on S3 is generated by the left and right multiplication of S3 on itself. This allows for a reduction in stages, first reducing by the left multiplication, and then reducing an intermediate space by the residual right-action. An advantage of this reduction-by-stages is that it allows for a fairly straightforward derivation of the relative equilibria solutions: the relative equilibria may first be classified in the intermediate reduced space and then reconstructed on the original phase space. For the 2-body problem on hyperbolic space the same idea does not apply. Hyperbolic 3-space H3 cannot be endowed with an isometric group structure and the symmetry group SO(1; 3) does not arise as a direct product of two groups.
    [Show full text]
  • Complex Inner Product Spaces
    MATH 355 Supplemental Notes Complex Inner Product Spaces Complex Inner Product Spaces The Cn spaces The prototypical (and most important) real vector spaces are the Euclidean spaces Rn. Any study of complex vector spaces will similar begin with Cn. As a set, Cn contains vectors of length n whose entries are complex numbers. Thus, 2 i ` 3 5i C3, » ´ fi P i — ffi – fl 5, 1 is an element found both in R2 and C2 (and, indeed, all of Rn is found in Cn), and 0, 0, 0, 0 p ´ q p q serves as the zero element in C4. Addition and scalar multiplication in Cn is done in the analogous way to how they are performed in Rn, except that now the scalars are allowed to be nonreal numbers. Thus, to rescale the vector 3 i, 2 3i by 1 3i, we have p ` ´ ´ q ´ 3 i 1 3i 3 i 6 8i 1 3i ` p ´ qp ` q ´ . p ´ q « 2 3iff “ « 1 3i 2 3i ff “ « 11 3iff ´ ´ p ´ qp´ ´ q ´ ` Given the notation 3 2i for the complex conjugate 3 2i of 3 2i, we adopt a similar notation ` ´ ` when we want to take the complex conjugate simultaneously of all entries in a vector. Thus, 3 4i 3 4i ´ ` » 2i fi » 2i fi if z , then z ´ . “ “ — 2 5iffi — 2 5iffi —´ ` ffi —´ ´ ffi — 1 ffi — 1 ffi — ´ ffi — ´ ffi – fl – fl Both z and z are vectors in C4. In general, if the entries of z are all real numbers, then z z. “ The inner product in Cn In Rn, the length of a vector x ?x x is a real, nonnegative number.
    [Show full text]
  • MATH 304 Linear Algebra Lecture 25: Complex Eigenvalues and Eigenvectors
    MATH 304 Linear Algebra Lecture 25: Complex eigenvalues and eigenvectors. Orthogonal matrices. Rotations in space. Complex numbers C: complex numbers. Complex number: z = x + iy, where x, y R and i 2 = 1. ∈ − i = √ 1: imaginary unit − Alternative notation: z = x + yi. x = real part of z, iy = imaginary part of z y = 0 = z = x (real number) ⇒ x = 0 = z = iy (purely imaginary number) ⇒ We add, subtract, and multiply complex numbers as polynomials in i (but keep in mind that i 2 = 1). − If z1 = x1 + iy1 and z2 = x2 + iy2, then z1 + z2 = (x1 + x2) + i(y1 + y2), z z = (x x ) + i(y y ), 1 − 2 1 − 2 1 − 2 z z = (x x y y ) + i(x y + x y ). 1 2 1 2 − 1 2 1 2 2 1 Given z = x + iy, the complex conjugate of z is z¯ = x iy. The modulus of z is z = x 2 + y 2. − | | zz¯ = (x + iy)(x iy) = x 2 (iy)2 = x 2 +py 2 = z 2. − − | | 1 z¯ 1 x iy z− = ,(x + iy) = − . z 2 − x 2 + y 2 | | Geometric representation Any complex number z = x + iy is represented by the vector/point (x, y) R2. ∈ y r φ 0 x 0 x = r cos φ, y = r sin φ = z = r(cos φ + i sin φ) = reiφ ⇒ iφ1 iφ2 If z1 = r1e and z2 = r2e , then i(φ1+φ2) i(φ1 φ2) z1z2 = r1r2e , z1/z2 = (r1/r2)e − . Fundamental Theorem of Algebra Any polynomial of degree n 1, with complex ≥ coefficients, has exactly n roots (counting with multiplicities).
    [Show full text]