Lie Algebras and their corresponding linear groups

Global Journal of Science Frontier Research, 2018

This paper begins the study of bicomplex matrices. In this paper, we have defined bicomplex matrices, determinant of a bicomplex square matrix and singular and non-singular matrices in C 2. We have proved that the set of all bicomplex square matrices of order n is an algebra. We have given some definitions and results regarding adjoint and inverse of a matrix in C 2. We have defined three types of conjugates and three types of tranjugates of a bicomplex matrix. With the help of these conjugates and tranjugates, we have also defined symmetric and skew-symmetric matrices, Hermitian and Skew-Hermitian matrices in C 2 .

Mathematical Sciences and Applications E-Notes, 2018

We present a new study on the square roots of real 2 × 2 matrices with a special view towards examples, some of them inspired by geometry. We begin with the following general matrix A = a c b d ∈ M 2 (R) and ask: is there a matrix B ∈ M 2 (R) such that B 2 = A? Such a matrix B is called square root of A. We point out that the more complicated case of a real matrix of order 3 is discussed in [4]. Although the case we consider is also well studied (according to the bibliography of [3]) we add several examples and facts concerning this notion as well as a series of geometrical applications. The Euclidean example We recall the n-orthogonal group: O(n) = {A ∈ M n (R) : A t · A = I n }; is the invariant group of the Euclidean inner product < ·, · > (yielding the usual Euclidean norm ·). If A ∈ O(n) then (det A t) · (det A) = detI n = 1 implies that det A = ±1. Hence, the orthogonal group splits into two components: O(n) = SO(n) O − (n) where SO(n) contains the matrices from O(n) having detA = 1 and O − (n) those with detA = −1; represents the disjoint reunion of sets. SO(n) is a subgroup in O(n) and is called n-special orthogonal group. O − (n) is not closed under product: A 1 , A 2 ∈ O − (n) implies that A 1 A 2 ∈ SO(n). Since M 1 (R) = R we have that O(1) = {±1} with SO(1) = {1} and O − (1) = {−1}; we remark that O(1) contains the integer unit roots! We know O(2) as well: R(t) = cos t − sin t sin t cos t ∈ SO(2), S(t) = cos t sin t sin t − cos t ∈ O − (2), t ∈ R. Hence, we have that: S(t) 2 = cos t sin t sin t − cos t cos t sin t sin t − cos t = I 2 which means that any S(t) is a root of the unit matrix I 2. We recall that from a geometrical point of view a square root of the unit matrix is called almost product structure, see for example [6]. Geometrical significance: R(t) is the matrix of rotation of angle t in trigonometrical sense (i.e anticlockwise) around the origin and S(t) is the matrix of axial symmetry with respect to d t/2 =line from plane R 2 which contains the origin O and makes the oriented angle t/2 with Ox. We have that S(t 2) · S(t 1) = A(t 2 − t 1) = S(t 1) · S(t 2). 2

2015

We explore properties and representations of first and second order spectral shift functions evaluated for pairs of self-adjoint matrices guided by examples. We examine the conditions sufficient for the first and second order spectral shift functions to be identically 0 in 3.11, 4.10, and 4.13.

Documenta Mathematica

Chapter 1. Generalities Definitions of group, isomorphism, representation, vector space and algebra. Biographical notes on Galois, Abel and Jacobi are given. Chapter 2. Lie groups and Lie algebras Lie Groups, infinitesimal generators, structure constants, Cartan's metric tensor, simple and semisimple groups and algebras, compact and non-compact groups. Biographical notes on Euler, Lie and Cartan are given. Chapter 3. Rotations: SO(3) and SU(2) Rotations and reflections, connectivity, center, universal covering group. Chapter 4. Representations of SU(2) Irreducible representations, Casimir operators, addition of angular momenta, Clebsch-Gordan coefficients, the Wigner-Eckart theorem, multiplicity. Biographical notes on Casimir, Weyl, Clebsch, Gordan and Wigner are given. Chapter 5. The so(n) algebra and Clifford numbers Spin(n), spinors and semispinors, Schur's lemma. Biographical notes on Clifford and Schur are given. Chapter 6. Reality properties of spinors Conjugate, orthogonal and symplectic representations. Chapter 7. Clebsch-Gordan series for spinors Antisymmetrie tensors, duality. Chapter 8. The center and outer automorphisms of Spin(n) Inversion, 12, 14 and 12 x 12 centers. A biographical note on Dynkin is given.

SIAM Journal on Matrix Analysis and Applications, 2004

Given an n-by-n Hermitian matrix A and a real number λ, index i is said to be Parter (resp. neutral, downer) if the multiplicity of λ as an eigenvalue of A(i) is one more (resp. the same, one less) than that in A. In case the multiplicity of λ in A is at least 2 and the graph of A is a tree, there are always Parter vertices. Our purpose here is to advance the classification of vertices and, in particular, to relate classification to the combinatorial structure of eigenspaces. Some general results are given and then used to deduce some rather specific facts, not otherwise easily observed. Examples are given.