On the classification of unitary matrices (original) (raw)

On the Classifications of Unitary Matrices

We classify the dynamical action of matrices in SU(p, q) using the coefficients of their characteristic polynomial. This generalises earlier work of Goldman for SU(2,1) and the classical result for SU(1, 1), which is conjugate to SL(2, Ê). As geometrical applications, we show how this enables us to classify automorphisms of real and complex hyperbolic space and anti de Sitter space.

The automorphism groups of complex homogeneous spaces

Mathematische Annalen, 1997

If G is a (connected) complex Lie Group and Z is a generalized flag manifold for G, then the open orbits D of a (connected) real form G 0 of G form an interesting class of complex homogeneous spaces, which play an important role in the representation theory of G 0 . We find that the group of automorphisms, i.e., the holomorphic diffeomorphisms, is a finite-dimensional Lie group, except for a small number of open orbits, where it is infinite dimensional. In the finite-dimensional case, we determine its structure. Our results have some consequences in representation theory. §1. We determine the automorphism groups for a certain interesting class of complex homogeneous spaces. Denote by Z a generalized flag manifold for a connected complex semisimple Lie group G. A real form G 0 (which we assume to be connected) of G acts on Z with a finite number of orbits, thus there are always open orbits (cf. [22]). These open orbits play a key role in the representation theory of G 0 . An open G 0 -orbit D in Z has a G 0 -invariant complex structure. The identity component of the group of holomorphic diffeomorphisms of D will be denoted by Hol (D). In the main theorem below we determine Hol (D) for each measurable open orbit (see Definition 2.1). In the case where D is measurable, D carries a G 0 -invariant (usually) indefinite hermitian metric and we determine its group of hermitian isometries. Generally the open orbits D are non-compact, however, our results include the cases where G 0 is a compact real form and D is compact, so D = Z. The compact case is contained in [12], [21], [3] and [2], from various points of view.

The Quaternions and the Spaces S 3, SU(2), SO(3), and ℝ ℙ3

Texts in Applied Mathematics, 2011

One of the main goals of these notes is to explain how rotations in R n are induced by the action of a certain group, Spin(n), on R n , in a way that generalizes the action of the unit complex numbers, U(1), on R 2 , and the action of the unit quaternions, SU(2), on R 3

Matrix exponentials, SU(N) group elements, and real polynomial roots

The exponential of an N × N matrix can always be expressed as a matrix polynomial of order N − 1. In particular, a general group element for the fundamental representation of SU(N) can be expressed as a matrix polynomial of order N−1 in a traceless N×N hermitian generating matrix, with polynomial coefficients consisting of elementary trigonometric functions dependent on N − 2 invariants in addition to the group parameter. These invariants are just angles determined by the direction of a real N-vector whose components are the eigenvalues of the hermitian matrix. Equivalently, the eigenvalues are given by projecting the vertices of an (N − 1)-simplex onto a particular axis passing through the center of the simplex. The orientation of the simplex relative to this axis determines the angular invariants and hence the real eigenvalues of the matrix.

SELF-ADJOINT ELEMENTS IN THE PSEUDO-UNITARY GROUP U (p, p)

The pseudo-unitary group U (p, q) of signature (p, q) is the group of matrices that preserve the indefinite pseudo-Euclidean metric on the vector space C p,q. The goal of this paper is to describe the set Us (p, p) of Hermitian, or, self-adjoint elements in U (p, p). Mathematics Subject Classification (2010): 15B57, 15Axx, 20G20.

On the transposition anti-involution in real Clifford algebras I: the transposition map

Linear and Multilinear Algebra, 2011

A signature ε = (p, q) dependent transposition anti-involution T ε˜o f real Clifford algebras Cℓ p,q for non-degenerate quadratic forms was introduced in [1]. In [2] we showed that, depending on the value of (p − q) mod 8, the map T ε˜g ives rise to transposition, complex Hermitian, or quaternionic Hermitian conjugation of representation matrices in spinor representation. The resulting scalar product is in general different from the two known standard scalar products [12]. We provide a full signature (p, q) dependent classification of the invariance groups G ε p,q of this product for p + q ≤ 9. The map T εĩ s identified as the "star" map known [14] from the theory of (twisted) group algebras, where the Clifford algebra Cℓ p,q is seen as a twisted group ring R t [(Z 2) n ], n = p + q. We discuss and list important subgroups of stabilizer groups G p,q (f) and their transversals in relation to generators of spinor spaces.

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[.

Matrix Functions in SU(N)

We explicitly calculate the Riemannian metric on SU(2)SU(2)SU(2) and derive the associated Haar measure. For arbitrary elements of SU(N)SU(N)SU(N) we derive a polar representation in terms of the eigenvalues and polynomials of the Lie algebra elements. The results can be used to discuss the reducibility of gauge fields.

Automorphisms of sl(2) and dynamical r-matrices

Journal of Mathematical Physics, 1998

Two outer automorphisms of infinite-dimensional representations of sl(2) algebra are considered. The similar constructions for the loop algebras and yangians are presented. The corresponding linear and quadratic R-brackets include the dynamical r-matrices.