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In this paper, we consider the Landau-Lifshitz-Gilbert systems with spin-polarized transport from a bounded domain in \begin{document}$ \mathbb{R}^3 $\end{document} into \begin{document}$ S^2 $\end{document} and show the existence of global weak solutions to the Cauchy problems of such Landau-Lifshtiz systems. In particular, we show that the Cauchy problem to Landau-Lifshitz equation without damping but with diffusion process of the spin accumulation admits a global weak solution. The Landau-Lifshtiz system with spin-polarized transport into a compact Lie algebra is also discussed and some similar results are proved. The key ingredients of this article consist of the choices of test functions and approximate equations.

We give a complete description of the bounded (i.e. norm continuous) unitary representations of the Fr\'echet-Lie algebra of all smooth sections, as well as of the LF-Lie algebra of compactly supported smooth sections, of a smooth Lie algebra bundle whose typical fiber is a compact Lie algebra. For the Lie algebra of all sections, bounded unitary irreducible representations are finite tensor products of so-called evaluation representations, hence in particular finite-dimensional. For the Lie algebra of compactly supported sections, bounded unitary irreducible (factor) representations are possibly infinite tensor products of evaluation representations, which reduces the classification problem to results of Glimm and Powers on irreducible (factor) representations of UHF C*-algebras. The key part in our proof is the classification of irreducible bounded unitary representations of Lie algebras that are the tensor product of a compact Lie algebra and a unital real continuous inverse algebra: every such representation is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component of the group of smooth gauge transformations of a principal fiber bundle with compact base and structure group, and the connected component of the group of special unitary n times n matrices with values in an involutive commutative continuous inverse algebra., Comment: 44 pages

We highlight the general notion of a relative quantum field theory, which occurs in several contexts. One is in gauge theory based on a compact Lie algebra, rather than a compact Lie group. This is relevant to the maximal superconformal theory in six dimensions., 19 pages, 4 figures; v2 small changes for publication; v3 small final changes for publication

This paper deals with the striking fact that there is an essentially canonical path from the $i$-th Lie algebra cohomology cocycle, $i=1,2,... l$, of a simple compact Lie algebra $\g$ of rank $l$ to the definition of its primitive Casimir operators $C^{(i)}$ of order $m_i$. Thus one obtains a complete set of Racah-Casimir operators $C^{(i)}$ for each $\g$ and nothing else. The paper then goes on to develop a general formula for the eigenvalue $c^{(i)}$ of each $C^{(i)}$ valid for any representation of $\g$, and thereby to relate $c^{(i)}$ to a suitably defined generalised Dynkin index. The form of the formula for $c^{(i)}$ for $su(n)$ is known sufficiently explicitly to make clear some interesting and important features. For the purposes of illustration, detailed results are displayed for some classes of representation of $su(n)$, including all the fundamental ones and the adjoint representation., Latex, 16 pages

The sourceless Yang–Mills equation reduces to an algebraic problem if the potential A has constant components Aα in a Lie algebra L with respect to some gauge and choice of coordinates. For some types of L there exist nontrivial solutions, for other types they do not. It is decided this in the Euclidean case for all Lie algebras with a dimension not greater than 5. A constant Yang–Mills potential on a Euclidean or Minkowski space with values in a compact Lie algebra is shown to be trivial.

We study how tensor products of representations decompose when restricted from a compact Lie algebra to one of its subalgebras. In particular, we are interested in tensor squares which are tensor products of a representation with itself. We show in a classification-free manner that the sum of multiplicities and the sum of squares of multiplicities in the corresponding decomposition of a tensor square into irreducible representations has to strictly grow when restricted from a compact semisimple Lie algebra to a proper subalgebra. For this purpose, relevant details on tensor products of representations are compiled from the literature. Since the sum of squares of multiplicities is equal to the dimension of the commutant of the tensor-square representation, it can be determined by linear-algebra computations in a scenario where an a priori unknown Lie algebra is given by a set of generators which might not be a linear basis. Hence, our results offer a test to decide if a subalgebra of a compact semisimple Lie algebra is a proper one without calculating the relevant Lie closures, which can be naturally applied in the field of controlled quantum systems.

The Wigner–Eckart theorem is a well known result for tensor operators of 𝔰𝔲(2) and, more generally, any compact Lie algebra. In this paper, the theorem will be generalized to the particular non-compact case of 𝔰𝔩(2, ℝ). In order to do so, recoupling theory between representations that are not necessarily unitary will be studied, namely, between finite-dimensional and infinite-dimensional representations. As an application, the Wigner–Eckart theorem will be used to construct an analogue of the Jordan–Schwinger representation, previously known only for representations in the discrete class, which also covers the continuous class.

It is shown, under rather general smoothness conditions on the gauge potential, which takes values in an arbitrary semi-simple compact Lie algebra ${\bf g}$, that if a (${\bf g}$-valued) solution to the gauge covariant Laplace equation exists, which vanishes at spatial infinity, in the cases of 1,2,3,... space dimensions, then the solution is identically zero. This result is also valid if the Lie algebra is merely compact. Consequently, a solution to the gauge covariant Poisson equation is uniquely determined by its asymptotic radial limit at spatial infinity. In the cases of one or two space dimensions a related result is proved, namely that if a solution to the gauge covariant Laplace equation exists, which is unbounded at spatial infinity, but with a certain dimension-dependent condition on the asymptotic growth of its norm, then the solution in question is a covariant constant.

In this article we study perturbed Hamiltonian dynamics in the class of collective Hamiltonian systems associated to Lie algebra Hamiltonian actions. For perturbations of collective systems generated by contractions of a compact Lie algebra into a limiting algebra 0, we give general conditions to transform such perturbations coming from deformations of the action into perturbations produced by corrections in the energy function of the collective system associated to the action by the limiting Lie algebra g0. We give an application to perturbations of collective systems generated by contraction of so(4) into e(3).

Irreducible representations of the real compact Lie algebra g2 are given in g2⊇so(4) bases. All missing labels are accounted for by the explicit construction of a g2⊇so(4) basis of vector holomorphic functions. The general problem of missing internal labels is also briefly discussed.

Two independent algorithms are presented, which together allow the determination of branching rules from an irreducible representation of a compact Lie algebra to those of a subalgebra (or subjoined algebra). The first gives the subalgebra Weyl orbits contained in an algebra orbit. The second gives the irreducible representations of an algebra contained in an orbit, and by inversion of a triangular matrix, the orbits contained in an irreducible representation.

Irreducible representations of the real compact Lie algebra g2 are given in g2⊇ su(3) bases. A missing label is accounted for by the explicit construction of a g2⊇ su(3) basis of vector holomorphic functions. Analytical results are given for two multiplicity‐free classes of irreps. It is also shown how vector coherent state (VCS) theory accommodates the decomposition of the nilpotent raising operator subalgebra of an arbitrary Lie algebra into a finite but arbitrary number of irreducible tensorial sets under transformations generated by a stability algebra.

The classical limit of operators X belonging to any compact Lie algebra g is computed. If X∈g, the classical limit in the representation ΓΛ, whose highest weight is Λ, is lim ΓΛ(X/N) =Σsig (fi,X,Ω), where the limit is taken as N→∞, the sum runs from i=1 to r=rank g, Λ=Σμifi,fi are the highest weights of the r fundamental representations of g,si=lim μi/N, and g (fi,X,Ω) is the expectation value of X with respect to the coherent states ‖fi, Ω〉 in the representation Γfi. Examples and applications are given.