Related papers: A Compact Formula for Rotations as Spin Matrix Pol…
Cayley rational forms for rotations are given as explicit spin matrix polynomials for any j. The results are compared to the Curtright-Fairlie-Zachos matrix polynomials for rotations represented as exponentials.
The concept of the Schwinger Representation of a finite or compact simple Lie group is set up as a multiplicity-free direct sum of all the unitary irreducible representations of the group. This is abstracted from the properties of the…
Recent results for rotations expressed as polynomials of spin matrices are derived here by elementary differential equation methods. Structural features of the results are then examined in the framework of biorthogonal systems, to obtain an…
There are two well-known ways of describing elements of the rotation group SO$(m)$. First, according to the Cartan-Dieudonn\'e theorem, every rotation matrix can be written as an even number of reflections. And second, they can also be…
Any nonsingular function of spin j matrices always reduces to a matrix polynomial of order 2j. The challenge is to find a convenient form for the coefficients of the matrix polynomial. The theory of biorthogonal systems is a useful…
A symmetry $SU(2,2)$ group in terms of ladder operators is presented for the Jacobi polynomials, $J_{n}^{(\alpha,\beta)}(x)$, and the Wigner $d_j$-matrices where the spins $j=n+(\alpha+\beta)/2$ integer and half-integer are considered…
Polynomial relations for generators of $su(2)$ Lie algebra in arbitrary representations are found. They generalize usual relation for Pauli operators in spin 1/2 case and permit to construct modified Holstein-Primakoff transformations in…
We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a,b,h;p,q,r)…
Matrix Lie groups provide a language for describing motion in such fields as robotics, computer vision, and graphics. When using these tools, we are often faced with turning infinite-series expressions into more compact finite series (e.g.,…
An algebraic interpretation of the one-variable quantum $q$-Krawtchouk polynomials is provided in the framework of the Schwinger realization of $\mathcal{U}_{q}(sl_{2})$ involving two independent $q$-oscillators. The polynomials are shown…
We show that a new unitary transform with characteristics almost similar to those of the finite Fourier transform can be defined in any finite-dimensional Hilbert space. It is defined by using the Kravchuk polynomials, and we call it…
We present a new approach to Hamilton's theory of turns for the groups SO(3) and SU(2) which renders their properties, in particular their composition law, nearly trivial and immediately evident upon inspection. We show that the entire…
We show that all spin groups of non-definite, quinary quadratic forms over a field with characteristic 0 can be represented as 2 by 2 matrices with entries in an associated quaternion algebra. Over local and global fields, we further study…
Hamilton in the course of his studies on quaternions came up with an elegant geometric picture for the group SU(2). In this picture the group elements are represented by ``turns'', which are equivalence classes of directed great circle arcs…
Remarkably simple closed-form expressions for the elements of the groups SU(n), SL(n,R), and SL(n,C) with n=2, 3, and 4 are obtained using linear functions of biquaternions instead of n x n matrices. These representations do not directly…
In the framework of the Schwinger boson representation for the su(2)-algebra, the closed form is derived for the total spin eigenstates which result from the coupling of n su(2)-spins. In order to demonstrate its usefulness, the orthogonal…
We derive the representation theory of $SU(2)$ from the expository theory of Lie groups and Lie algebras. Based on this, the mathematics of non-relativistic quantum mechanics of a spin $\frac{1}{2}$ particle are described from a…
Explicit forms are given of matrix elements of generalized coherent operators based on Lie algebras su(1,1) and su(2). We also give a kind of factorization formula of the associated Laguerre polynomials.
Individual spinors in a SU(2) spin network are described by their relations to the background spin network. A 'covariant' formulation of these relations yields the de Sitter group SO(3,2) as the fundamental symmetry group. Locally this…
Previously we showed that a loop in a simply connected compact Lie group K has a unique triangular factorization if and only if the loop has a unique root subgroup factorization (relative to a choice of a reduced sequence of simple…