An Introduction to Lie Theory Through Matrix Groups (original) (raw)
An Invitation to Lie Groups and Lie Algebras
A Lie group G is a space which possesses two structures: 1) structure of a group; 2) structure of a smooth manifold. These structures are compatible in the sense that the group operations (" multiplication " a, b → a · b and " taking the inverse element " a → a −1) are both smooth. Main examples of Lie groups are matrix groups:
Advances in Linear Algebra & Matrix Theory
This paper is made out of necessity as a doctoral student taking the exam from Lie groups. Using the literature suggested to me by the professor, I felt the need to, in addition to that literature, and since there was more superficial in that book with some remarks about the examples given in relation to the left group. I decided to try a little harder and collect as much literature as possible, both for the needs of me and the others who will take after me. Searching for literature in my mother tongue I could not find anything, in English as someone who comes from a small country like Montenegro, all I could find was through the internet. I decided to gather what I could find from the literature in my own way and to my observation and make this kind of work. The main content of this paper is to present the Lie group in the simplest way. Before and before I started writing or collecting about Lie groups, it was necessary to say something about groups and subgroups that are taught in basic studies in algebra. In them I cited several deficits and an example. The following content of the paper is related to Lie groups primarily concerning the definition of examples such as The General Linear Group GL(n, R), The Complex General Linear Group GL(n, C), The Special Linear Group
Differentiable Manifolds, 2001
The subject of Lie groups is one that slips by many a mathematician. Many claim that the topic is not accessible to undergraduate research. The book Lie Groups by Harriet Pollatsek came out a few years ago, and it was meant to be a new way to be introduced to the topic. However, the book does not quite get far enough to give a formal definition of a Lie group. The goal of this project is to "bridge the gap." The objective of this thesis is to include all the introductory material required to get to where the definition of a Lie group is no longer something so complicated. We will illustrate the major concepts by examples. Many matrix groups are Lie groups. Matrix groups are well-known, and they are an ideal place to start learning about what a Lie group can do. We then look at tangent spaces of the matrix groups, or the Lie algebra that is associated with each Lie group. After some motivation behind Lie algebras, we finally get to the feature presentation: a group and a differentiable manifold, put together into one super structure known as a Lie group. iii ACKNOWLEDGMENTS First of all, I would like to thank Dr. Goldthwait for all his help, his insight, and most importantly his patience. For the past year, he listened patiently to everything I had to say, right or wrong. All year long he asked all the right questions to get me thinking, and sometimes just to keep me on track. I knew from the start we would work well together. I'd also like to thank my committee member, Dr. Wingler, who throughout the entire last year took question after question I had, and not only had an answer but was excited to help-and sometimes have something interesting to work on while one of his classes took an exam. Next, I'd like to thank Dr. Wakefield, not only for being on my committee, but for always being supportive, attending my talks early this year, and for just always having such a positive attitude. Dr. Spalsbury deserves my thanks as well. She pushed me right when I needed a push. Not only that, but it was her book that I decided to borrow which got me started on this topic. I'd also like to thank my many friends in the mathematics department. I don't believe I could ever have done something like this without such a supportive, friendly environment and friends who can listen to you talk for extended periods of time despite not following anything that's being said-that's what they claim anyway, but I haven't seen a proof. Finally, I'd like to thank my family, who has been extremely supportive of every decision I've ever made throughout my life. I hear all the time how proud my parents are of me. I am truly blessed. iv Contents 0 Preliminaries vi 1 Tangent Spaces vii 2 The Exponential Map xii 3 Lie Algebras xv 4 Adjoints xviii 5 A Lie Group (Finally) xx 6 References xxv v 0 Preliminaries Definition 0.1. A set G along along with a binary operation (written in multiplicative form) is called a group if the following conditions are satisfied: 1. Closure: ab ∈ G for all a, b ∈ G. 2. Associativity: (ab)c = a(bc) for all a, b, c ∈ G. 3. Identity: there exists an element e ∈ G such that ae = ea = a for all a ∈ G. 4. Inverse: for all a ∈ G there exists a b ∈ G such that ab = e = ba. Definition 0.2. A mapping φ of a group G into a group G (φ : G → G) is called a homomorphism if it preserves the group operation. Symbolically, φ(ab) = φ(a)φ(b) for all a, b ∈ G. Definition 0.3. The set of all n by n matrices with real-valued entries (which is a vector space over R under matrix addition and scalar multiplication) is denoted M(n, R). Definition 0.4. The general linear group is the group of all invertible, n by n matrices with real entries under the group operation matrix multiplication. It is denoted GL(n, R) = {A ∈ M(n, R) : det A = 0}. Definition 0.5. "The general linear group" also describes the set of all invertible linear transformations from R n to R n and is denoted GL(R n) = {T : R n → R n : T is linear and invertible}. Take special note of the fact that the general linear group of matrices and the general linear group of transformations are essentially "the same." That is, the groups are isomorphic, which we will define later. Definition 0.6. The special linear group is the group of all n by n matrices with determinant 1, denoted SL(n, R) = {A ∈ M(n, R) : det A = 1}. Definition 0.7. The orthogonal group of n by n matrices (in the Euclidean case) is O(n, R) = {A ∈ M(n, R) : AA T = I n }. Definition 0.8. The special orthogonal group of n by n matrices is SO(n, R) = {A ∈ O(n, R) : det A = 1}. A few other comments worthy of note: 1. ≤ will be used as a subgroup symbol. 2. I n will be used to denote the n by n identity matrix. 3. 0 n will be used to denote the n by n zero matrix. vi
Seminar on Lie groups, Lie algebras and their representations
2020
Lie groups are important to describe symmetries, both in mathematics and in applications (physics, chemistry, engineering,. . . ). The classical Lie groups are for example the orthogonal groups O(n), the unitary groups U(n), but mathematicians and physicists are also fascinated by more exotic examples such as the symmetry group of the octonions which is discussed a lot in modern mathematical physics. Many of these Lie groups can be represented as subgroups of Gl(k,R) for some sufficiently large k, but there are also Lie groups which cannot. Lie groups are manifolds G together with a multiplication μ : G×G→ G which is a smooth map, such that (G,μ) is a group.1 Lie groups and their representation is a mighty theory which allows effect calculations both for problems inside mathematics and also for applications outside.
Algorithmic Lie Theory for Solving Ordinary Differential Equations, 2007
Suppose G is a Lie group and M is a manifold (G and M are not necessarily finite dimensional). Let D(M) denote the group of diffeomorphisms on M and V(M) denote the Lie algebra of vector fields on M. If X: isv a. complete-vector: field-then; Exp tX will denote the one-parameter group of X. A local action <£ of G oh M gives rise to a Lie algebra homomorphism + from L(G) into V(M). In particular if G is a subgroup-of D(M> and <|> : G x M-> M is the natural global action (g»p)-> g(p). then G is called a Lie transformation group of M. If M is a Hausdorff manifold and G is a Lie transformation group of M we show that is an isomorphism of L(G) onto (L(G)) and L = + (L(G)) satisfies the following conditions : (A) L consists of complete vector fields. (B) L has a Banach Lie algebra structure satisfying the following two conditions : (BI) the evaluation map ev : (X,p)-> X(p) is a vector bundle morphism from the trivial bundle L x M into T(M), (B2) there exists an open ball B r (0) of radius r at 0 such that Exp : L-> D(M) is infective on B r (0). Conversely, if L is a suba-lgebra of V(M) (M Hausdorff) satisfying conditions (A) and (B) we show there exists a unique connected Lie transfor-+ mation group with natural action : G x M-> M such that tj) is a Banach Lie algebra isomorphism of L(G) onto L .
New Mexico Tech, 2000
The Poincaré and Lorentz groups are typical examples of continuous groups. That is why we give below a very brief description of the theory of continuous groups.
Differential Geometry and Lie Groups A Computational Perspective
Differential Geometry and Lie Groups A Computational Perspective, 2020
To my daughter Mia, my wife Anne, my son Philippe, and my daughter Sylvie. To my parents Howard and Jane. discuss three results, one of which being the Hadamard and Cartan theorem about complete manifolds of non-positive curvature. The goal of Chapter 18 is to understand the behavior of isometries and local isometries, in particular their action on geodesics. We also intoduce Riemannian covering maps and Riemannian submersions. If π : M → B is a submersion between two Riemannian manifolds, then for every b ∈ B and every p ∈ π -1 (b), the tangent space T p M to M at p splits into two orthogonal components, its vertical component V p = Ker dπ p , and its horizontal component H p (the orthogonal complement of V p ). If the map dπ p is an isometry between H p and T b B, then most of the differential geometry of B can be studied by lifting B to M , and then projecting down to B again. We also introduce Killing vector fields, which play a technical role in the study of reductive homogeneous spaces. In Chapter 19, we return to Lie groups. Not every Lie group is a matrix group, so in order to study general Lie groups it is necessary to introduce left-invariant (and rightinvariant) vector fields on Lie groups. It turns out that the space of left-invariant vector fields is isomorphic to the tangent space g = T I G to G at the identity, which is a Lie algebra. By considering integral curves of left-invariant vector fields, we define the generalization of the exponential map exp : g → G to an arbitrary Lie group. The notion of immersed Lie subgroup is introduced, and the correspondence between Lie groups and Lie algebra is explored. We also consider the special classes of semidirect products of Lie algebras and Lie groups, the universal covering of a Lie group, and the Lie algebra of Killing vector fields on a Riemannian manifold. Chapter 20 deals with two topics: 1. A formula for the derivative of the exponential map for a general Lie group (not necessarily a matrix group). 2. A formula for the Taylor expansion of µ(X, Y ) = log(exp(X) exp(Y )) near the origin. The second problem is solved by a formula known as the Campbell-Baker-Hausdorff formula. An explicit formula was derived by Dynkin (1947), and we present this formula. Chapter 21 is devoted to the study of metrics, connections, geodesics, and curvature, on Lie groups. Since a Lie group G is a smooth manifold, we can endow G with a Riemannian metric. Among all the Riemannian metrics on a Lie groups, those for which the left translations (or the right translations) are isometries are of particular interest because they take the group structure of G into account. As a consequence, it is possible to find explicit formulae for the Levi-Civita connection and the various curvatures, especially in the case of metrics which are both left and right-invariant. In Section 21.2 we give four characterizations of bi-invariant metrics. The first one refines the criterion of the existence of a left-invariant metric and states that every bi-invariant metric on a Lie group G arises from some Ad-invariant inner product on the Lie algebra g. In Section 21.3 we show that if G is a Lie group equipped with a left-invariant metric, then it is possible to express the Levi-Civita connection and the sectional curvature in terms Recall that a real symmetric matrix is called positive (or positive semidefinite) if its eigenvalues are all positive or null, and positive definite if its eigenvalues are all strictly positive. We denote the vector space of real symmetric n × n matrices by S(n), the set of symmetric positive matrices by SP(n), and the set of symmetric positive definite matrices by SPD(n). The next proposition shows that every symmetric positive definite matrix A is of the form e B for some unique symmetric matrix B. The set of symmetric matrices is a vector space, but it is not a Lie algebra because the Lie bracket [A, B] is not symmetric unless A and B commute, and the set of symmetric (positive) definite matrices is not a multiplicative group, so this result is of a different flavor as Theorem 2.6. Proposition 2.8. For every symmetric matrix B, the matrix e B is symmetric positive definite. For every symmetric positive definite matrix A, there is a unique symmetric matrix B such that A = e B . Proof. We showed earlier that e B = e B . If B is a symmetric matrix, then since B = B, we get e B = e B = e B , and e B is also symmetric. Since the eigenvalues λ 1 , . . . , λ n of the symmetric matrix B are real and the eigenvalues of e B are e λ 1 , . . . , e λn , and since e λ > 0 if λ ∈ R, e B is positive definite. To show the surjectivity of the exponential map, note that if A is symmetric positive definite, then by Theorem 12.3 from Chapter 12 of Gallier [50], there is an orthogonal matrix P such that A = P D P , where D is a diagonal matrix CHAPTER 2. THE MATRIX EXPONENTIAL; SOME MATRIX LIE GROUPS
Notes on group actions, manifolds, lie groups, and lie algebras
2005
CHAPTER 2. REVIEW OF GROUPS AND GROUP ACTIONS 3. Similarly, the sets R of real numbers and C of complex numbers are groups under addition (with identity element 0), and R * = R − {0} and C * = C − {0} are groups under multiplication (with identity element 1). 4. The sets R n and C n of n-tuples of real or complex numbers are groups under componentwise addition: (x 1 ,. .. , x n) + (y 1 , • • • , y n) = (x 1 + y n ,. .. , x n + y n), with identity element (0,. .. , 0). All these groups are abelian. 5. Given any nonempty set S, the set of bijections f : S → S, also called permutations of S, is a group under function composition (i.e., the multiplication of f and g is the composition g • f), with identity element the identity function id S. This group is not abelian as soon as S has more than two elements. 6. The set of n × n matrices with real (or complex) coefficients is a group under addition of matrices, with identity element the null matrix. It is denoted by M n (R) (or M n (C)). 7. The set R[X] of polynomials in one variable with real coefficients is a group under addition of polynomials. 8. The set of n × n invertible matrices with real (or complex) coefficients is a group under matrix multiplication, with identity element the identity matrix I n. This group is called the general linear group and is usually denoted by GL(n, R) (or GL(n, C)). 9. The set of n × n invertible matrices with real (or complex) coefficients and determinant +1 is a group under matrix multiplication, with identity element the identity matrix I n. This group is called the special linear group and is usually denoted by SL(n, R) (or SL(n, C)). 10. The set of n × n invertible matrices with real coefficients such that RR = I n and of determinant +1 is a group called the orthogonal group and is usually denoted by SO(n) (where R is the transpose of the matrix R, i.e., the rows of R are the columns of R). It corresponds to the rotations in R n. 11. Given an open interval ]a, b[, the set C(]a, b[) of continuous functions f : ]a, b[ → R is a group under the operation f + g defined such that (f + g)(x) = f (x) + g(x) for all x ∈]a, b[.