Spinor Algebras
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Unitary Group - Wikipedia
Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group Unitary group In mathematics, the unitary group of degree n, denoted U( n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL( n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group. In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group. The unitary group U( n) is a real Lie group of dimension n2. The Lie algebra of U( n) consists of n × n skew-Hermitian matrices, with the Lie bracket given by the commutator. The general unitary group (also called the group of unitary similitudes ) consists of all matrices A such that A∗A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix. Contents Properties Topology Related groups 2-out-of-3 property Special unitary and projective unitary groups G-structure: almost Hermitian Generalizations Indefinite forms Finite fields Degree-2 separable algebras Algebraic groups Unitary group of a quadratic module Polynomial invariants Classifying space See also Notes References Properties Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group 1 of 7 2/23/2018, 10:13 AM Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group homomorphism The kernel of this homomorphism is the set of unitary matrices with determinant 1. -
Chapter 1 BASICS of GROUP REPRESENTATIONS
Chapter 1 BASICS OF GROUP REPRESENTATIONS 1.1 Group representations A linear representation of a group G is identi¯ed by a module, that is by a couple (D; V ) ; (1.1.1) where V is a vector space, over a ¯eld that for us will always be R or C, called the carrier space, and D is an homomorphism from G to D(G) GL (V ), where GL(V ) is the space of invertible linear operators on V , see Fig. ??. Thus, we ha½ve a is a map g G D(g) GL(V ) ; (1.1.2) 8 2 7! 2 with the linear operators D(g) : v V D(g)v V (1.1.3) 2 7! 2 satisfying the properties D(g1g2) = D(g1)D(g2) ; D(g¡1 = [D(g)]¡1 ; D(e) = 1 : (1.1.4) The product (and the inverse) on the righ hand sides above are those appropriate for operators: D(g1)D(g2) corresponds to acting by D(g2) after having appplied D(g1). In the last line, 1 is the identity operator. We can consider both ¯nite and in¯nite-dimensional carrier spaces. In in¯nite-dimensional spaces, typically spaces of functions, linear operators will typically be linear di®erential operators. Example Consider the in¯nite-dimensional space of in¯nitely-derivable functions (\functions of class 1") Ã(x) de¯ned on R. Consider the group of translations by real numbers: x x + a, C 7! with a R. This group is evidently isomorphic to R; +. As we discussed in sec ??, these transformations2 induce an action D(a) on the functions Ã(x), de¯ned by the requirement that the transformed function in the translated point x + a equals the old original function in the point x, namely, that D(a)Ã(x) = Ã(x a) : (1.1.5) ¡ 1 Group representations 2 It follows that the translation group admits an in¯nite-dimensional representation over the space V of 1 functions given by the di®erential operators C d d a2 d2 D(a) = exp a = 1 a + + : : : : (1.1.6) ¡ dx ¡ dx 2 dx2 µ ¶ In fact, we have d a2 d2 dà a2 d2à D(a)Ã(x) = 1 a + + : : : Ã(x) = Ã(x) a (x) + (x) + : : : = Ã(x a) : ¡ dx 2 dx2 ¡ dx 2 dx2 ¡ · ¸ (1.1.7) In most of this chapter we will however be concerned with ¯nite-dimensional representations. -
Be the Integral Symplectic Group and S(G) Be the Set of All Positive Integers Which Can Occur As the Order of an Element in G
FINITE ORDER ELEMENTS IN THE INTEGRAL SYMPLECTIC GROUP KUMAR BALASUBRAMANIAN, M. RAM MURTY, AND KARAM DEO SHANKHADHAR Abstract For g 2 N, let G = Sp(2g; Z) be the integral symplectic group and S(g) be the set of all positive integers which can occur as the order of an element in G. In this paper, we show that S(g) is a bounded subset of R for all positive integers g. We also study the growth of the functions f(g) = jS(g)j, and h(g) = maxfm 2 N j m 2 S(g)g and show that they have at least exponential growth. 1. Introduction Given a group G and a positive integer m 2 N, it is natural to ask if there exists k 2 G such that o(k) = m, where o(k) denotes the order of the element k. In this paper, we make some observations about the collection of positive integers which can occur as orders of elements in G = Sp(2g; Z). Before we proceed further we set up some notations and briefly mention the questions studied in this paper. Let G = Sp(2g; Z) be the group of all 2g × 2g matrices with integral entries satisfying > A JA = J 0 I where A> is the transpose of the matrix A and J = g g . −Ig 0g α1 αk Throughout we write m = p1 : : : pk , where pi is a prime and αi > 0 for all i 2 f1; 2; : : : ; kg. We also assume that the primes pi are such that pi < pi+1 for 1 ≤ i < k. -
The Classical Groups and Domains 1. the Disk, Upper Half-Plane, SL 2(R
(June 8, 2018) The Classical Groups and Domains Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ The complex unit disk D = fz 2 C : jzj < 1g has four families of generalizations to bounded open subsets in Cn with groups acting transitively upon them. Such domains, defined more precisely below, are bounded symmetric domains. First, we recall some standard facts about the unit disk, the upper half-plane, the ambient complex projective line, and corresponding groups acting by linear fractional (M¨obius)transformations. Happily, many of the higher- dimensional bounded symmetric domains behave in a manner that is a simple extension of this simplest case. 1. The disk, upper half-plane, SL2(R), and U(1; 1) 2. Classical groups over C and over R 3. The four families of self-adjoint cones 4. The four families of classical domains 5. Harish-Chandra's and Borel's realization of domains 1. The disk, upper half-plane, SL2(R), and U(1; 1) The group a b GL ( ) = f : a; b; c; d 2 ; ad − bc 6= 0g 2 C c d C acts on the extended complex plane C [ 1 by linear fractional transformations a b az + b (z) = c d cz + d with the traditional natural convention about arithmetic with 1. But we can be more precise, in a form helpful for higher-dimensional cases: introduce homogeneous coordinates for the complex projective line P1, by defining P1 to be a set of cosets u 1 = f : not both u; v are 0g= × = 2 − f0g = × P v C C C where C× acts by scalar multiplication. -
Math 123—Algebra II
Math 123|Algebra II Lectures by Joe Harris Notes by Max Wang Harvard University, Spring 2012 Lecture 1: 1/23/12 . 1 Lecture 19: 3/7/12 . 26 Lecture 2: 1/25/12 . 2 Lecture 21: 3/19/12 . 27 Lecture 3: 1/27/12 . 3 Lecture 22: 3/21/12 . 31 Lecture 4: 1/30/12 . 4 Lecture 23: 3/23/12 . 33 Lecture 5: 2/1/12 . 6 Lecture 24: 3/26/12 . 34 Lecture 6: 2/3/12 . 8 Lecture 25: 3/28/12 . 36 Lecture 7: 2/6/12 . 9 Lecture 26: 3/30/12 . 37 Lecture 8: 2/8/12 . 10 Lecture 27: 4/2/12 . 38 Lecture 9: 2/10/12 . 12 Lecture 28: 4/4/12 . 40 Lecture 10: 2/13/12 . 13 Lecture 29: 4/6/12 . 41 Lecture 11: 2/15/12 . 15 Lecture 30: 4/9/12 . 42 Lecture 12: 2/17/16 . 17 Lecture 31: 4/11/12 . 44 Lecture 13: 2/22/12 . 19 Lecture 14: 2/24/12 . 21 Lecture 32: 4/13/12 . 45 Lecture 15: 2/27/12 . 22 Lecture 33: 4/16/12 . 46 Lecture 17: 3/2/12 . 23 Lecture 34: 4/18/12 . 47 Lecture 18: 3/5/12 . 25 Lecture 35: 4/20/12 . 48 Introduction Math 123 is the second in a two-course undergraduate series on abstract algebra offered at Harvard University. This instance of the course dealt with fields and Galois theory, representation theory of finite groups, and rings and modules. These notes were live-TEXed, then edited for correctness and clarity. -
The Symplectic Group
Sp(n), THE SYMPLECTIC GROUP CONNIE FAN 1. Introduction t ¯ 1.1. Definition. Sp(n)= (n, H)= A Mn(H) A A = I is the symplectic group. O { 2 | } 1.2. Example. Sp(1) = z M1(H) z =1 { 2 || | } = z = a + ib + jc + kd a2 + b2 + c2 + d2 =1 { | } 3 ⇠= S 2. The Lie Algebra sp(n) t ¯ 2.1. Definition. AmatrixA Mn(H)isskew-symplecticif A + A =0. 2 t ¯ 2.2. Definition. sp(n)= A Mn(H) A + A =0 is the Lie algebra of Sp(n) with commutator bracket [A,B]{ 2 = AB -| BA. } Proof. This was proven in class. ⇤ 2.3. Fact. sp(n) is a real vector space. Proof. Let A, B sp(n)anda, b R 2 2 (aA + bB)+taA + bB = a(A + tA¯)+b(B + tB¯)=0 ⇤ 1 2.4. Fact. The dimension of sp(n)is(2n +1)n. Proof. Let A sp(n). 2 a11 + ib11 + jc11 + kd11 a12 + ib12 + jc12 + kd12 ... a1n + ib1n + jc1n + kd1n . A = . .. 0 . 1 an1 + ibn1 + jcn1 + kdn1 an2 + ibn2 + jcn2 + kdn2 ... ann + ibnn + jcnn + kdnn @ A Then A + tA¯ = 2a11 (a12 + a21)+i(b12 b21)+j(c12 c21)+k(d12 d21) ... (a1n + an1)+i(b1n bn1)+j(c1n cn1)+k(d1n dn1) − − − − − − (a21 + a12)+i(b21 b12)+j(c21 c12)+k(d21 d12)2a22 ... (a2n + an2)+i(b2n bn2)+j(c2n cn2)+k(d2n dn2) − − − − − − 0 . 1 . .. B . C B C B(a1n + an1)+i(b1n bn1)+j(c1n cn1)+k(d1n dn1) ... 2ann C @ − − − A t 2 A + A¯ =0,so: axx =0, x 0degreesoffreedom 8 ! n(n 1) − axy = ayx,x= y 2 degrees of freedom − 6 !n(n 1) − bxy = byx,x= y 2 degrees of freedom 6 ! n(n 1) − cxy = cyx,x= y 2 degrees of freedom 6 ! n(n 1) − dxy = dyx,x= y 2 degrees of freedom b ,c ,d unrestricted,6 ! x 3n degrees of freedom xx xx xx 8 ! In total, dim(sp(n)) = 2n2 + n = n(2n +1) ⇤ 2.5. -
21 Schur Indicator
21 Schur indicator Complex representation theory gives a good description of the homomorphisms from a group to a special unitary group. The Schur indicator describes when these are orthogonal or quaternionic, so it is also easy to describe all homomor- phisms to the compact orthogonal and symplectic groups (and a little bit of fiddling around with central extensions gives the homomorphisms to compact spin groups). So the homomorphisms to any compact classical group are well understood. However there seems to be no easy way to describe the homomor- phisms of a group to the exceptional compact Lie groups. We have several closely related problems: • Which irreducible representations have symmetric or alternating forms? • Classify the homomorphisms to orthogonal and symplectic groups • Classify the real and quaternionic representations given the complex ones. Suppose we have an irreducible representation V of a compact group G. We study the space of invariant bilinear forms on V . Obviously the bilinear forms correspond to maps from V to its dual, so the space of such forms is at most 1-dimensional, and is non-zero if and only if V is isomorphic to its dual, in other words if and only if the character of V is real. We also want to know when the bilinear form is symmetric or alternating. These two possibilities correspond to the symmetric or alternating square containing the 1-dimensional irreducible representation. So (1, χS2V − χΛ2V ) is 1, 0, or −1 depending on whether V has a symmetric bilinear form, no bilinear form, or an alternating bilinear form. By 2 the formulas for the symmetric and alternating squares this is given by G χg , and is called the Schur indicator. -
Master Thesis Written at the Max Planck Institute of Quantum Optics
Ludwig-Maximilians-Universitat¨ Munchen¨ and Technische Universitat¨ Munchen¨ A time-dependent Gaussian variational description of Lattice Gauge Theories Master thesis written at the Max Planck Institute of Quantum Optics September 2017 Pablo Sala de Torres-Solanot Ludwig-Maximilians-Universitat¨ Munchen¨ Technische Universitat¨ Munchen¨ Max-Planck-Institut fur¨ Quantenoptik A time-dependent Gaussian variational description of Lattice Gauge Theories First reviewer: Prof. Dr. Ignacio Cirac Second reviewer: Dr. Tao Shi Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science within the programme Theoretical and Mathematical Physics Munich, September 2017 Pablo Sala de Torres-Solanot Abstract In this thesis we pursue the novel idea of using Gaussian states for high energy physics and explore some of the first necessary steps towards such a direction, via a time- dependent variational method. While these states have often been applied to con- densed matter systems via Hartree-Fock, BCS or generalized Hartree-Fock theory, this is not the case in high energy physics. Their advantage is the fact that they are completely characterized, via Wick's theorem, by their two-point correlation func- tions (and one-point averages in bosonic systems as well), e.g., all possible pairings between fermionic operators. These are collected in the so-called covariance matrix which then becomes the most relevant object in their description. In order to reach this goal, we set the theory on a lattice by making use of Lat- tice Gauge Theories (LGT). This method, introduced by Kenneth Wilson, has been extensively used for the study of gauge theories in non-perturbative regimes since it allows new analytical methods as for example strong coupling expansions, as well as the use of Monte Carlo simulations and more recently, Tensor Networks techniques as well. -
Differential Topology: Exercise Sheet 3
Prof. Dr. M. Wolf WS 2018/19 M. Heinze Sheet 3 Differential Topology: Exercise Sheet 3 Exercises (for Nov. 21th and 22th) 3.1 Smooth maps Prove the following: (a) A map f : M ! N between smooth manifolds (M; A); (N; B) is smooth if and only if for all x 2 M there are pairs (U; φ) 2 A; (V; ) 2 B such that x 2 U, f(U) ⊂ V and ◦ f ◦ φ−1 : φ(U) ! (V ) is smooth. (b) Compositions of smooth maps between subsets of smooth manifolds are smooth. Solution: (a) As the charts of a smooth structure cover the manifold the above property is certainly necessary for smoothness of f : M ! N. To show that it is also sufficient, consider two arbitrary charts U;~ φ~ 2 A and V;~ ~ 2 B. We need to prove that the map ~ ◦ f ◦ φ~−1 : φ~(U~) ! ~(V~ ) is smooth. Therefore consider an arbitrary y 2 φ~(U~) such that there is some x 2 M such that x = φ~−1(y). By assumption there are charts (U; φ) 2 A; (V; ) 2 B such that x 2 U, −1 f(U) ⊂ V and ◦ f ◦ φ : φ(U) ! (V ) is smooth. Now we choose U0;V0 such ~ ~ that x 2 U0 ⊂ U \ U and f(x) 2 V0 ⊂ V \ V and therefore we can write ~ ~−1 ~ −1 −1 ~−1 ◦ f ◦ φ j = ◦ ◦ ◦ f ◦ φ ◦ φ ◦ φ : (1) φ~(U0) As a composition of smooth maps, we showed that ~ ◦ f ◦ φ~−1j is smooth. As φ~(U0) x 2 M was chosen arbitrarily we proved that ~ ◦ f ◦ φ~−1 is a smooth map. -
Hamiltonian and Symplectic Symmetries: an Introduction
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 3, July 2017, Pages 383–436 http://dx.doi.org/10.1090/bull/1572 Article electronically published on March 6, 2017 HAMILTONIAN AND SYMPLECTIC SYMMETRIES: AN INTRODUCTION ALVARO´ PELAYO In memory of Professor J.J. Duistermaat (1942–2010) Abstract. Classical mechanical systems are modeled by a symplectic mani- fold (M,ω), and their symmetries are encoded in the action of a Lie group G on M by diffeomorphisms which preserve ω. These actions, which are called sym- plectic, have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact Abelian Lie groups that are, in addition, of Hamiltonian type, i.e., they also satisfy Hamilton’s equations. Since then a number of connections with combinatorics, finite- dimensional integrable Hamiltonian systems, more general symplectic actions, and topology have flourished. In this paper we review classical and recent re- sults on Hamiltonian and non-Hamiltonian symplectic group actions roughly starting from the results of these authors. This paper also serves as a quick introduction to the basics of symplectic geometry. 1. Introduction Symplectic geometry is concerned with the study of a notion of signed area, rather than length, distance, or volume. It can be, as we will see, less intuitive than Euclidean or metric geometry and it is taking mathematicians many years to understand its intricacies (which is work in progress). The word “symplectic” goes back to the 1946 book [164] by Hermann Weyl (1885–1955) on classical groups. -
Symplectic Group and Heisenberg Group in P-Adic Quantum Mechanics
SYMPLECTIC GROUP AND HEISENBERG GROUP IN p-ADIC QUANTUM MECHANICS SEN HU & ZHI HU ABSTRACT. This paper treats mathematically some problems in p-adic quantum mechanics. We first deal with p-adic symplectic group corresponding to the symmetry on the classical phase space. By the filtrations of isotropic subspaces and almost self-dual lattices in the p-adic symplectic vector space, we explicitly give the expressions of parabolic subgroups, maximal compact subgroups and corresponding Iwasawa decompositions of some symplectic groups. For a triple of Lagrangian subspaces, we associated it with a quadratic form whose Hasse invariant is calculated. Next we study the various equivalent realizations of unique irreducible and admissible representation of p-adic Heisenberg group. For the Schrödinger representation, we can define Weyl operator and its kernel function, while for the induced representations from the characters of maximal abelian subgroups of Heisenberg group generated by the isotropic subspaces or self-dual lattice in the p-adic symplectic vector space, we calculate the Maslov index defined via the intertwining operators corresponding to the representation transformation operators in quantum mechanics. CONTENTS 1. Introduction 1 2. Brief Review of p-adic Analysis 2 3. p-adic Symplectic Group 3 3.1. Isotropic subspaces and parabolic subgroups 3 3.2. Lagrangian subspaces and quadratic forms 4 3.3. Latices and maximal compact subgroups 6 4. p-adic Heisenberg Group 8 4.1. Schrödinger representations and Weyl operators 8 4.2. Induced representations and Maslov indices 11 References 14 1. INTRODUCTION It seems that there is no prior reason for opposing the fascinating idea that at the very small (Planck) scale the geometry of the spacetime should be non-Archimedean. -
20 Representation Theory
20 Representation theory A representation of a group is an action of a group on something, in other words a homomorphism from the group to the group of automorphisms of something. The most obvious representations are permutation representations: actions of a group on a (discrete) set. For example the group of rotations of a cube has order 24, and has several obvious permutation representations: it can act on faces, edges, vertices, diagonals, pairs of opposite faces, and so on. For another example, the group SL2(R) acts on the real projective line, and on the upper half plane. These sets carry topologies, which we should take into account when defining representations. Let us try to classify all permutation representations of some group G . First of all, if the group G acts on some set X, then we can write X as the disjoint union of the orbits of G on X, and G is transitive on each of these orbits. So this reduces the classification of all permutation representations to that of transitive permutation representations. Next, given a transitive permutation representation of G on a set X, fix a point x ∈ X and write Gx for the subgroup fixing x. Then as a permutation representation, X is isomorphic to the action of G on the set of cosets G/Gx. If we choose a different point y, then Gx is conjugate to Gy (using some element of G taking x to y), so we see that transitive permutation representations up to isomorphism correspond to conjugacy classes of subgroups of G.