Lie Groups and Algebras 1 Intro 2 the Adjoint Representation and The
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DIFFERENTIAL GEOMETRY FINAL PROJECT 1. Introduction for This
DIFFERENTIAL GEOMETRY FINAL PROJECT KOUNDINYA VAJJHA 1. Introduction For this project, we outline the basic structure theory of Lie groups relating them to the concept of Lie algebras. Roughly, a Lie algebra encodes the \infinitesimal” structure of a Lie group, but is simpler, being a vector space rather than a nonlinear manifold. At the local level at least, the Fundamental Theorems of Lie allow one to reconstruct the group from the algebra. 2. The Category of Local (Lie) Groups The correspondence between Lie groups and Lie algebras will be local in nature, the only portion of the Lie group that will be of importance is that portion of the group close to the group identity 1. To formalize this locality, we introduce local groups: Definition 2.1 (Local group). A local topological group is a topological space G, with an identity element 1 2 G, a partially defined but continuous multiplication operation · :Ω ! G for some domain Ω ⊂ G × G, a partially defined but continuous inversion operation ()−1 :Λ ! G with Λ ⊂ G, obeying the following axioms: • Ω is an open neighbourhood of G × f1g Sf1g × G and Λ is an open neighbourhood of 1. • (Local associativity) If it happens that for elements g; h; k 2 G g · (h · k) and (g · h) · k are both well-defined, then they are equal. • (Identity) For all g 2 G, g · 1 = 1 · g = g. • (Local inverse) If g 2 G and g−1 is well-defined in G, then g · g−1 = g−1 · g = 1. A local group is said to be symmetric if Λ = G, that is, every element g 2 G has an inverse in G.A local Lie group is a local group in which the underlying topological space is a smooth manifold and where the associated maps are smooth maps on their domain of definition. -
10 Group Theory and Standard Model
Physics 129b Lecture 18 Caltech, 03/05/20 10 Group Theory and Standard Model Group theory played a big role in the development of the Standard model, which explains the origin of all fundamental particles we see in nature. In order to understand how that works, we need to learn about a new Lie group: SU(3). 10.1 SU(3) and more about Lie groups SU(3) is the group of special (det U = 1) unitary (UU y = I) matrices of dimension three. What are the generators of SU(3)? If we want three dimensional matrices X such that U = eiθX is unitary (eigenvalues of absolute value 1), then X need to be Hermitian (real eigenvalue). Moreover, if U has determinant 1, X has to be traceless. Therefore, the generators of SU(3) are the set of traceless Hermitian matrices of dimension 3. Let's count how many independent parameters we need to characterize this set of matrices (what is the dimension of the Lie algebra). 3 × 3 complex matrices contains 18 real parameters. If it is to be Hermitian, then the number of parameters reduces by a half to 9. If we further impose traceless-ness, then the number of parameter reduces to 8. Therefore, the generator of SU(3) forms an 8 dimensional vector space. We can choose a basis for this eight dimensional vector space as 00 1 01 00 −i 01 01 0 01 00 0 11 λ1 = @1 0 0A ; λ2 = @i 0 0A ; λ3 = @0 −1 0A ; λ4 = @0 0 0A (1) 0 0 0 0 0 0 0 0 0 1 0 0 00 0 −i1 00 0 01 00 0 0 1 01 0 0 1 1 λ5 = @0 0 0 A ; λ6 = @0 0 1A ; λ7 = @0 0 −iA ; λ8 = p @0 1 0 A (2) i 0 0 0 1 0 0 i 0 3 0 0 −2 They are called the Gell-Mann matrices. -
Universal Enveloping Algebras and Some Applications in Physics
Universal enveloping algebras and some applications in physics Xavier BEKAERT Institut des Hautes Etudes´ Scientifiques 35, route de Chartres 91440 – Bures-sur-Yvette (France) Octobre 2005 IHES/P/05/26 IHES/P/05/26 Universal enveloping algebras and some applications in physics Xavier Bekaert Institut des Hautes Etudes´ Scientifiques Le Bois-Marie, 35 route de Chartres 91440 Bures-sur-Yvette, France [email protected] Abstract These notes are intended to provide a self-contained and peda- gogical introduction to the universal enveloping algebras and some of their uses in mathematical physics. After reviewing their abstract definitions and properties, the focus is put on their relevance in Weyl calculus, in representation theory and their appearance as higher sym- metries of physical systems. Lecture given at the first Modave Summer School in Mathematical Physics (Belgium, June 2005). These lecture notes are written by a layman in abstract algebra and are aimed for other aliens to this vast and dry planet, therefore many basic definitions are reviewed. Indeed, physicists may be unfamiliar with the daily- life terminology of mathematicians and translation rules might prove to be useful in order to have access to the mathematical literature. Each definition is particularized to the finite-dimensional case to gain some intuition and make contact between the abstract definitions and familiar objects. The lecture notes are divided into four sections. In the first section, several examples of associative algebras that will be used throughout the text are provided. Associative and Lie algebras are also compared in order to motivate the introduction of enveloping algebras. The Baker-Campbell- Haussdorff formula is presented since it is used in the second section where the definitions and main elementary results on universal enveloping algebras (such as the Poincar´e-Birkhoff-Witt) are reviewed in details. -
Representation Theory
M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23. -
Special Unitary Group - Wikipedia
Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Special unitary group In mathematics, the special unitary group of degree n, denoted SU( n), is the Lie group of n×n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U( n), consisting of all n×n unitary matrices. As a compact classical group, U( n) is the group that preserves the standard inner product on Cn.[nb 1] It is itself a subgroup of the general linear group, SU( n) ⊂ U( n) ⊂ GL( n, C). The SU( n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.[1] The simplest case, SU(1) , is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+ I, − I}. [nb 2] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. Contents Properties Lie algebra Fundamental representation Adjoint representation The group SU(2) Diffeomorphism with S 3 Isomorphism with unit quaternions Lie Algebra The group SU(3) Topology Representation theory Lie algebra Lie algebra structure Generalized special unitary group Example Important subgroups See also 1 of 10 2/22/2018, 8:54 PM Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Remarks Notes References Properties The special unitary group SU( n) is a real Lie group (though not a complex Lie group). -
General Formulas of the Structure Constants in the $\Mathfrak {Su}(N
General Formulas of the Structure Constants in the su(N) Lie Algebra Duncan Bossion1, ∗ and Pengfei Huo1,2, † 1Department of Chemistry, University of Rochester, Rochester, New York, 14627 2Institute of Optics, University of Rochester, Rochester, New York, 14627 We provide the analytic expressions of the totally symmetric and anti-symmetric structure con- stants in the su(N) Lie algebra. The derivation is based on a relation linking the index of a generator to the indexes of its non-null elements. The closed formulas obtained to compute the values of the structure constants are simple expressions involving those indexes and can be analytically evaluated without any need of the expression of the generators. We hope that these expressions can be widely used for analytical and computational interest in Physics. The su(N) Lie algebra and their corresponding Lie The indexes Snm, Anm and Dn indicate generators corre- groups are widely used in fundamental physics, partic- sponding to the symmetric, anti-symmetric, and diago- ularly in the Standard Model of particle physics [1, 2]. nal matrices, respectively. The explicit relation between su 1 The (2) Lie algebra is used describe the spin- 2 system. a projection operator m n and the generators can be ˆ ~ found in Ref. [5]. Note| thatih these| generators are traceless Its generators, the spin operators, are j = 2 σj with the S ˆ ˆ ˆ ~2 Pauli matrices Tr[ i] = 0, as well as orthonormal Tr[ i j ]= 2 δij . TheseS higher dimensional su(N) LieS algebraS are com- 0 1 0 i 1 0 σ = , σ = , σ = . -
Group Theory - QMII 2017
Group Theory - QMII 2017 Reminder Last time we said that a group element of a matrix lie group can be written as an exponent: a U = eiαaX ; a = 1; :::; N: We called Xa the generators, we have N of them, they span a basis for the Lie algebra, and they can be found by taking the derivative with respect to αa at α = 0. The generators are closed under the Lie product (∼ [·; ·]), and related by the structure constants [Xa;Xb] = ifabcXc: (1) We end with the Jacobi identity [Xa; [Xb;Xc]] + [Xb; [Xc;Xa]] + [Xc; [Xa;Xb]] = 0: (2) 1 Representations of Lie Algebra 1.1 The adjoint rep. part- I One of the most important representations is the adjoint. There are two equivalent ways to defined it. Here we follow the definition by Georgi. By plugging Eq. (1) into the Jacobi identity Eq. (2) we get the following: fbcdfade + fabdfcde + fcadfbde = 0 (3) Proof: 0 = [Xa; [Xb;Xc]] + [Xb; [Xc;Xa]] + [Xc; [Xa;Xb]] = ifbcd [Xa;Xd] + ifcad [Xb;Xd] + ifabd [Xc;Xd] = − (fbcdfade + fcadfbde + fabdfcde) Xe (4) 1 Now we can define a set of matrices Ta s.t. [Ta]bc = −ifabc: (5) Then by using the relation of the structure constants Eq. (3) we get [Ta;Tb] = ifabcTc: (6) Proof: [Ta;Tb]ce = [Ta]cd[Tb]de − [Tb]cd[Ta]de = −facdfbde + fbcdfade = fcadfbde + fbcdfade = −fabdfcde = fabdfdce = ifabd[Td]ce (7) which means that: [Ta;Tb] = ifabdTd: (8) Therefore the structure constants themselves generate a representation. This representation is called the adjoint representation. The adjoint of su(2) We already found that the structure constants of su(2) are given by the Levi-Civita tensor. -
Groups and Representations the Material Here Is Partly in Appendix a and B of the Book
Groups and representations The material here is partly in Appendix A and B of the book. 1 Introduction The concept of symmetry, and especially gauge symmetry, is central to this course. Now what is a symmetry: you have something, e.g. a vase, and you do something to it, e.g. turn it by 29 degrees, and it still looks the same then we call the operation performed, i.e. the rotation, a symmetryoperation on the object. But if we first rotate it by 29 degrees and then by 13 degrees and it still looks the same then also the combined operation of a rotation by 42 degrees is a symmetry operation as well. The mathematics that is involved here is that of groups and their representations. The symmetry operations form the group and the objects the operations work on are in representations of the group. 2 Groups A group is a set of elements g where there exist an operation ∗ that combines two group elements and the results is a third: ∃∗ : 8g1; g2 2 G : g1 ∗ g2 = g3 2 G (1) This operation must be associative: 8g1; g2; g3 2 G :(g1 ∗ g2) ∗ g3 = g1 ∗ (g2 ∗ g3) (2) and there exists an element unity: 91 2 G : 8g 2 G : g ∗ 1 = 1 ∗ g = g (3) and for every element in G there exists an inverse: 8g 2 G : 9g−1 2 G : g ∗ g−1 = g−1 ∗ g = 1 (4) This is the general definition of a group. Now if all elements of a group commute, i.e. -
Lie Algebras from Lie Groups
Preprint typeset in JHEP style - HYPER VERSION Lecture 8: Lie Algebras from Lie Groups Gregory W. Moore Abstract: Not updated since November 2009. March 27, 2018 -TOC- Contents 1. Introduction 2 2. Geometrical approach to the Lie algebra associated to a Lie group 2 2.1 Lie's approach 2 2.2 Left-invariant vector fields and the Lie algebra 4 2.2.1 Review of some definitions from differential geometry 4 2.2.2 The geometrical definition of a Lie algebra 5 3. The exponential map 8 4. Baker-Campbell-Hausdorff formula 11 4.1 Statement and derivation 11 4.2 Two Important Special Cases 17 4.2.1 The Heisenberg algebra 17 4.2.2 All orders in B, first order in A 18 4.3 Region of convergence 19 5. Abstract Lie Algebras 19 5.1 Basic Definitions 19 5.2 Examples: Lie algebras of dimensions 1; 2; 3 23 5.3 Structure constants 25 5.4 Representations of Lie algebras and Ado's Theorem 26 6. Lie's theorem 28 7. Lie Algebras for the Classical Groups 34 7.1 A useful identity 35 7.2 GL(n; k) and SL(n; k) 35 7.3 O(n; k) 38 7.4 More general orthogonal groups 38 7.4.1 Lie algebra of SO∗(2n) 39 7.5 U(n) 39 7.5.1 U(p; q) 42 7.5.2 Lie algebra of SU ∗(2n) 42 7.6 Sp(2n) 42 8. Central extensions of Lie algebras and Lie algebra cohomology 46 8.1 Example: The Heisenberg Lie algebra and the Lie group associated to a symplectic vector space 47 8.2 Lie algebra cohomology 48 { 1 { 9. -
Lie Algebras by Shlomo Sternberg
Lie algebras Shlomo Sternberg April 23, 2004 2 Contents 1 The Campbell Baker Hausdorff Formula 7 1.1 The problem. 7 1.2 The geometric version of the CBH formula. 8 1.3 The Maurer-Cartan equations. 11 1.4 Proof of CBH from Maurer-Cartan. 14 1.5 The differential of the exponential and its inverse. 15 1.6 The averaging method. 16 1.7 The Euler MacLaurin Formula. 18 1.8 The universal enveloping algebra. 19 1.8.1 Tensor product of vector spaces. 20 1.8.2 The tensor product of two algebras. 21 1.8.3 The tensor algebra of a vector space. 21 1.8.4 Construction of the universal enveloping algebra. 22 1.8.5 Extension of a Lie algebra homomorphism to its universal enveloping algebra. 22 1.8.6 Universal enveloping algebra of a direct sum. 22 1.8.7 Bialgebra structure. 23 1.9 The Poincar´e-Birkhoff-Witt Theorem. 24 1.10 Primitives. 28 1.11 Free Lie algebras . 29 1.11.1 Magmas and free magmas on a set . 29 1.11.2 The Free Lie Algebra LX ................... 30 1.11.3 The free associative algebra Ass(X). 31 1.12 Algebraic proof of CBH and explicit formulas. 32 1.12.1 Abstract version of CBH and its algebraic proof. 32 1.12.2 Explicit formula for CBH. 32 2 sl(2) and its Representations. 35 2.1 Low dimensional Lie algebras. 35 2.2 sl(2) and its irreducible representations. 36 2.3 The Casimir element. 39 2.4 sl(2) is simple. -
Characterization of SU(N)
University of Rochester Group Theory for Physicists Professor Sarada Rajeev Characterization of SU(N) David Mayrhofer PHY 391 Independent Study Paper December 13th, 2019 1 Introduction At this point in the course, we have discussed SO(N) in detail. We have de- termined the Lie algebra associated with this group, various properties of the various reducible and irreducible representations, and dealt with the specific cases of SO(2) and SO(3). Now, we work to do the same for SU(N). We de- termine how to use tensors to create different representations for SU(N), what difficulties arise when moving from SO(N) to SU(N), and then delve into a few specific examples of useful representations. 2 Review of Orthogonal and Unitary Matrices 2.1 Orthogonal Matrices When initially working with orthogonal matrices, we defined a matrix O as orthogonal by the following relation OT O = 1 (1) This was done to ensure that the length of vectors would be preserved after a transformation. This can be seen by v ! v0 = Ov =) (v0)2 = (v0)T v0 = vT OT Ov = v2 (2) In this scenario, matrices then must transform as A ! A0 = OAOT , as then we will have (Av)2 ! (A0v0)2 = (OAOT Ov)2 = (OAOT Ov)T (OAOT Ov) (3) = vT OT OAT OT OAOT Ov = vT AT Av = (Av)2 Therefore, when moving to unitary matrices, we want to ensure similar condi- tions are met. 2.2 Unitary Matrices When working with quantum systems, we not longer can restrict ourselves to purely real numbers. Quite frequently, it is necessarily to extend the field we are with with to the complex numbers. -
Covariant Dirac Operators on Quantum Groups
Covariant Dirac Operators on Quantum Groups Antti J. Harju∗ Abstract We give a construction of a Dirac operator on a quantum group based on any simple Lie alge- bra of classical type. The Dirac operator is an element in the vector space Uq (g) ⊗ clq(g) where the second tensor factor is a q-deformation of the classical Clifford algebra. The tensor space Uq (g) ⊗ clq(g) is given a structure of the adjoint module of the quantum group and the Dirac operator is invariant under this action. The purpose of this approach is to construct equivariant Fredholm modules and K-homology cycles. This work generalizes the operator introduced by Bibikov and Kulish in [1]. MSC: 17B37, 17B10 Keywords: Quantum group; Dirac operator; Clifford algebra 1. The Dirac operator on a simple Lie group G is an element in the noncommutative Weyl algebra U(g) cl(g) where U(g) is the enveloping algebra for the Lie algebra of G. The vector space g generates a Clifford⊗ algebra cl(g) whose structure is determined by the Killing form of g. Since g acts on itself by the adjoint action, U(g) cl(g) is a g-module. The Dirac operator on G spans a one dimensional invariant submodule of U(g)⊗ cl(g) which is of the first order in cl(g) and in U(g). Kostant’s Dirac operator [10] has an additional⊗ cubical term in cl(g) which is constant in U(g). The noncommutative Weyl algebra is given an action on a Hilbert space which is also a g-module.