An Introduction to Lie Theory Through Matrix Groups

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 <J> + 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 <j> is an isomorphism of L(G) onto <f> (L(G)) and L = <j> + (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 <j> : 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, 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

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[.

2020

In this research article, Lie Groups and Lie Algebras are projected in a distinct direction and with innovative proofs. We develop the necessary and sufficient condition for a topological group to be Hausdorff. The criteria of topological group and Haussdorff to be connected is also derived. This research article mainly explores the concept of left in variant field and presents an important hypothesis namely ―any Lie group is parallelizable‖ with a very simple accurate logical and analytical reasoning. Moreover this paper covers the impervious of another property namely every one parameter subgroup is an integral curve of some left-invariant vector field.

American Journal of Computational Mathematics, 2020

As recounted in this paper, the idea of groups is one that has evolved from some very intuitive concepts. We can do binary operations like adding or multiplying two elements and also binary operations like taking the square root of an element (in this case the result is not always in the set). In this paper , we aim to find the operations and actions of Lie groups on manifolds. These actions can be applied to the matrix group and Bi-invariant forms of Lie groups and to generalize the eigenvalues and eigenfunctions of differential operators on n . A Lie group is a group as well as differentiable manifold, with the property that the group operations are compatible with the smooth structure on which group manipulations, product and inverse, are distinct. It plays an extremely important role in the theory of fiber bundles and also finds vast applications in physics. It represents the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. Here we did work flat out to represent the mathematical aspects of Lie groups on manifolds.

Journal of Computational and Applied Mathematics, 2010

In this paper we explore the computation of the matrix exponential in a manner that is consistent with Lie group structure. Our point of departure is the decomposition of Lie algebra as the semidirect product of two Lie subspaces and an application of the Baker-Campbell-Hausdorff formula. Our results extend the results in Iserles and Zanna (2005) [2], Zanna and Munthe-Kaas(2001/02) [4] to a range of Lie groups: the Lie group of all solid motions in Euclidean space, the Lorentz Lie group of all solid motions in Minkowski space and the group of all invertible (upper) triangular matrices. In our method, the matrix exponential group can be computed by a less computational cost and is more accurate than the current methods. In addition, by this method the approximated matrix exponential belongs to the corresponding Lie group.

It is shown that every abelian Lie group with smooth exponential mapping is a quotient of its Lie algebra via the exponential mapping.

2010