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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 consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra $$\mathfrak{g}$$, which can be viewed as the extension of Landau-Lifshitz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the work by the second author to show the existence of global weak solutions to the Cauchy problems of such LL equations from an n-dimensional closed Riemannian manifold $$\mathbb{T}$$ or a bounded domain in ℝn into a unit sphere $$S_\mathfrak{g}$$(1) in $$\mathfrak{g}$$. In particular, we consider the Hamiltonian system associated with the non-local energy—micromagnetic energy defined on a bounded domain of ℝ3 and show the initial-boundary value problem to such LL equation without damping terms admits a global weak solution. The key ingredient of this article consists of the choices of test functions and approximate equations.

In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) Condition), which means that in each tangent plane, we can find a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset spaces admitting non-negatively curved homogeneous Finsler metrics satisfying the (FP) Condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove that any Lie group whose Lie algebra is a rank $2$ non-Abelian compact Lie algebra admits a left invariant Finsler metric satisfying the (FP) condition. As by-products, we find the first example of non-compact coset space $S^3\times \mathbb{R}$ which admits homogeneous flag-wise positively curved Finsler metrics. Moreover, we find some non-negatively curved Finsler metrics on $S^2\times S^3$ and $S^6\times S^7$ which satisfy the (FP) condition, as well as some flag-wise positively curved Finsler metrics on $S^3\times S^3$, shedding some light on the long standing general Hopf conjecture., 23 pages. The newest version has strengthened the main results in the paper, and provides more examples. We add a short survey on the most recent progress inspired by this paper in the introduction section

The realification of the ( 2 n + 1 ) -dimensional complex Heisenberg Lie algebra is a ( 4 n + 2 ) -dimensional real nilpotent Lie algebra with a 2-dimensional commutator ideal coinciding with the centre, and admitting the compact algebra sp ( n ) of derivations. We investigate, in general, whether a real nilpotent Lie algebra with 2-dimensional commutator ideal coinciding with the centre admits a compact Lie algebra of derivations. This also gives us the occasion to revisit a series of classic results, with the expressed aim of attracting the interest of a broader audience.

Results from the forthcoming papers [4] and [8] are announced. We introduce a singular current construction, or base change, for pseudoalgebras which may be used to obtain a primitive Lie pseudoalgebra of type H from a suitable one of type K. When applied to representations, it derives the pseudo de Rham complex of type H from that of type K—which is related to Rumin’s construction from [15]—both with standard coefficients and with nontrivial Galois coefficients. In the latter case, the construction yields exact complexes of modules for the Poisson linearly compact Lie algebra \(P_{2N}\) exhibiting a nontrivial central action.

In this paper, we study invariant Einstein metrics on Ledger-Obata spaces $F^m/\operatorname{diag}(F)$. In particular, we classify invariant Einstein metrics on $F^4/\operatorname{diag}(F)$ and estimate the number of invariant Einstein metrics on general Ledger-Obata spaces $F^{m}/\operatorname{diag}(F)$., Comment: 17 pages, comments are welcome

A generalization of the cyclic Jacobi algorithm is proposed that works in an arbitrary compact Lie algebra. This allows, in particular, a unified treatment of Jacobi algorithms on different classes of matrices, e.g., skew-symmetric or skew-Hermitian Hamiltonian matrices. Wildberger has established global, linear convergence of the algorithm for the classical Jacobi method on compact Lie algebras. Here we prove local quadratic convergence for general cyclic Jacobi schemes.

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

Let 𝓰 be a compact simple Lie algebra of dimension d. It is a classical result that the convolution of any d non-trivial, G-invariant, orbitalmeasures is absolutely continuous with respect to Lebesgue measure on 𝓰, and the sum of any d non-trivial orbits has non-empty interior. The number d was later reduced to the rank of the Lie algebra (or rank +1 in the case of type An). More recently, the minimal integer k = k(X) such that the k-fold convolution of the orbital measure supported on the orbit generated by X is an absolutely continuous measure was calculated for each X ∈ 𝓰.In this paper 𝓰 is any of the classical, compact, simple Lie algebras. We characterize the tuples (X1 , . . . , XL), with Xi ∊ 𝓰, which have the property that the convolution of the L-orbital measures supported on the orbits generated by the Xi is absolutely continuous, and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of 𝓰 and the structure of the annihilating roots of the Xi. Such a characterization was previously known only for type An.

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

We discuss the appearance of additional, hidden supersymmetries for simple 0+1 $Ad(G)$-invariant supersymmetric models and analyse some geometrical mechanisms that lead to them. It is shown that their existence depends crucially on the availability of odd order invariant skewsymmetric tensors on the (generic) compact Lie algebra $\cal G$, and hence on the cohomology properties of the Lie algebra considered., Misprints corrected, two refs. added. To appear in NPB

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

For each compact Lie algebra g and each real representationV of g we construct a two-step nilpotent Lie groupN(g, V), endowed with a natural left-invariant riemannian metric. The main goal of this paper is to show that this construction produces many new Gelfand pairs associated with nilpotent Lie groups. Indeed, we will give a full classification of the manifoldsN(g, V) which are commutative spaces, using a characterization in terms of multiplicity-free actions.

We study the representation ${\cal D}$ of a simple compact Lie algebra $\g$ of rank l constructed with the aid of the hermitian Dirac matrices of a (${\rm dim} \g$)-dimensional euclidean space. The irreducible representations of $\g$ contained in ${\cal D}$ are found by providing a general construction on suitable fermionic Fock spaces. We give full details not only for the simplest odd and even cases, namely su(2) and su(3), but also for the next (${dim} \g$)-even case of su(5). Our results are far reaching: they apply to any $\g$-invariant quantum mechanical system containing ${\rm dim} \g$ fermions. Another reason for undertaking this study is to examine the role of the $\g$-invariant fermionic operators that naturally arise. These are given in terms of products of an odd number of gamma matrices, and include, besides a cubic operator, (l-1) fermionic scalars of higher order. The latter are constructed from the Lie algebra cohomology cocycles, and must be considered to be of theoretical significance similar to the cubic operator. In the (${\rm dim} \g$)-even case, the product of all l operators turns out to be the chirality operator $\gamma_q, q=({{\rm dim} \g+1})$.

For each compact Lie algebra ? and each real representation V of ? we consider a two-step nilpotent Lie group N(?,V), endowed with a natural left-invariant riemannian metric. The homogeneous nilmanifolds so obtained are precisely those which are naturally reductive. We study some geometric aspects of these manifolds, finding many parallels with H-type groups. We also obtain, within the class of manifolds N(?,V), the first examples of non-weakly symmetric, naturally reductive spaces and new examples of non-commutative naturally reductive spaces.

Let $$\mathfrak{g}$$ u be a compact Lie algebra and let $$\mathfrak{g}$$ u be its complexification. Let ζ−1/2 be the inverse on the set of regular elements of $$\mathfrak{g}$$ u of a square root of the discriminant of $$\mathfrak{g}$$ . Generalizing a result of W. Lichtenstein in the case $$\mathfrak{g}$$ u = $$\mathfrak{s}\mathfrak{u}$$ (n, ℂ) or $$\mathfrak{s}\mathfrak{o}$$ (nℝ), we prove that ∂(q).ζ1/2 is non zero for all harmonic polynomialsq ∈S( $$\mathfrak{g}$$ ) \ {0}. This fact is deduced from results about equivariantD-modules supported on the nilpotent cone of $$\mathfrak{g}$$ .

We describe a family of non-nilpotent Riemannian solvable Lie groups whose isotropy group has a prescribed compact Lie algebra.

We describe a family of homogeneous nilmanifolds whose isotropy group has a prescribed compact Lie algebra.

As we have seen in Chapter 5, Levi’s Theorem 5.6.6 is a central result in the structure theory of Lie algebras. It often allows splitting problems: one separately considers solvable and semisimple Lie algebras, and one puts together the results for both types. Naturally, this strategy also works to some extent for Lie groups. After dealing with nilpotent and solvable Lie groups in Chapter 11, we turn to the other side of the spectrum, to groups with semisimple or reductive Lie algebras. Here an important subclass is the class of compact Lie groups and the slightly larger class of groups with compact Lie algebra. Many problems can be reduced to compact Lie groups, and they are much easier to deal with than noncompact ones. The prime reason for that is the existence of a finite Haar measure whose existence was shown in Section 10.4.

Let G be a Lie group and E be a locally convex topological G-module. If E is sequentially complete, then E and its space of smooth vectors are modules for the algebra D(G) of compactly supported smooth functions on G. However, the module multiplication need not be continuous. The pathology can be ruled out if E is (or embeds into) a projective limit of Banach G-modules. Moreover, in this case the space of analytic vectors is a module for the algebra A(G) of superdecaying analytic functions introduced by Gimperlein, Kroetz and Schlichtkrull. We prove that the space of analytic vectors is a topological A(G)-module if E is a Banach space or, more generally, if every countable set of continuous seminorms on E has an upper bound. The same conclusion is obtained if G has a compact Lie algebra. The question of whether D(G) and A(G) are topological algebras is also addressed., Comment: 33 pages, LaTeX; v3: update of references

In the preceding chapter, we studied groups with a compact Lie algebra. For these groups, we have seen how to split them into a direct product of a compact and a vector group, how to complement the commutator group by an abelian Lie group, and that all compact Lie groups are linear. We now proceed with our program to obtain similar results for arbitrary Lie groups with finitely many connected components. First, we turn to the important special case of semisimple Lie groups.

We exhibit a simple algorithmic procedure to show that any element of a compact Lie algebra is conjugate to an element of a fixed maximal abelian subalgebra. An estimate of the convergence of the algorithm is obtained. As an application, we provide a new proof of Kostant's theorem on the projection of orbits onto a maximal abelian subalgebra. 0 Let M E M(n, C) be a Hermitian matrix and consider the problem of diagonalizing M, that is, finding a unitary n x n matrix g such that g-1Mg is diagonal. This problem is essentially equivalent to that of finding the eigenvalues and eigenvectors of M. We propose an algorithm for solving this problem which utilizes the Lie algebra structure of 9, the n x n skew-Hermitian matrices, and the adjoint action of G, the n x n unitary group, on j. In fact our method applies generally to any compact connected Lie group G and its Lie algebra . Fix a maximal torus T C G with Lie algebra t C O and let (0.1) g = tE E Oa aEl+ be the decomposition of 9 into weight spaces under the adjoint action of T. Here 1+ is a set of positive roots and each space Oa is two-dimensional. Given Z E 9, we will write (0.2) z = zO + E za aET+ corresponding to (0.1). The idea is then to choose a E 1+ such that Za has maximum norm and then find g E G such that Ad(g)Z has no Oa component. This turns out to be essentially a problem in SU(2), which we can solve using only quadratic equations. If d(Z) denotes the distance from Z to the subspace Received by the editors November 29, 1990 and, in revised form, February 25, 1992. 1991 Mathematics Subject Classification. Primary 22E15; Secondary 58F05.

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 construct the BRST and anti-BRST operator for a compact Lie algebra which is a direct sum of abelian and simple ideals. Two different inner products are defined on the ghost space and the hermiticity properties of the ghost and BRST operators with respect to these inner products are discussed. A decomposition theorem for ghost states is derived and the cohomology of the BRST complex is shown to reduce to the standard Lie-algebra cohomology. We show that the cohomology classes of the Lie algebra are given by all invariant anti-symmetric tensors and explain how these can be obtained as zero modes of an invariant operator in the representation space of the ghosts. Explicit examples are given.

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.

We consider a certain polynomial F on the space of endomorphisms from a compact Lie algebra g to su(n) (regarded as vector spaces). The function F was first discovered in string theory and physical considerations suggest that it is minimized on the set of irreducible representations of g. We prove this conjecture for g = su(2).

It is a classical theorem that the exponential function of a Lie group with compact Lie algebra is surjective. We recall that a Lie algebra is said to be compact, if there exists a positive definite bilinear form which is invariant under the adjoint action. One can prove this theorem by means of the Pontryagin maximum principle (PMP). The idea is to consider a certain optimal control problem and prove that the solutions are one-parameter semigroups. The latter is equivalent to the statement that the optimal controls are constant. Another theorem from Lorentzian geometry states that two points p and q of a Lorentzian manifold may be joined by a geodesic segment, provided that the order interval [p, q] is compact. These two results are not as unrelated as one might expect, for they can be deduced from a more general theorem on the exponential function of an ordered manifold with affine connection. >

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.

Nelson and Stinespring proved that in any unitary representation of a Lie group with compact Lie algebra the representation of Hermitian elements in the enveloping algebra are essentially self-adjoint. If the Lie algebra is noncompact, we construct in its enveloping algebra a Hermitian element u such that in any locally faithful unitary representation the representative of u has no self-adjoint extension.

In this paper we shall show that the equations of motion of a solid, and also Liouville's method of integration of Hamiltonian systems, appear in a natural manner when we study the geometry of level surfaces of a finite-dimensional space of integrals that are closed with respect to the Poisson bracket. If V is a finite-dimensional space of integrals on a symplectic manifold X which is closed with respect to the Poisson bracket, and if the integrals at thepoints of common position form linearly independent differentials, then we can refer to a level surface Mp of integrals which forms a manifold at the points of common position. The space V is a Lie algebra, and its associated group @ is a group of Hamiltonian transformations of the symplectic manifold X. The level surface Mp is not invariant under the action of the group ~, but the group ~ contains a subgroup ~ , that leaves invariant the manifold Mp. In the case of a compact Lie algebra V, the subgroup ~ will be commutative, and its algebra H is a Cartan subalgebr~.of V. In the general case the subalgebra H is an annihilator of the covector at the common position. If the action of the group ~ on Mp is free (or if it has a single orbit type), then the factor manifold Y =~fp/~ wili be a symplectic manifold, and the Hamiltonian vector field on the level surface Mp will be projected onto a Hamiltonian vector field on a manifold Y. It is natural tO refer to the latter as Euler's equation for the original Hamiltonian system on the manifold X. A particular case of this scheme is Euler's equation of motion of a solid, or, more generally, of motion of geodesic left-invariant metrics on the Lie group 6The algebra V of integrals consists of integrals of the moment of momentum. This algebra coincides with the Lie algebra of the group @, acting on a cotangent fibering with the aid of left shifts. The level surfaces are invariant under right shifts; therefore the factor manifold Y is homeomorphic to the orbit of a coadjoint representation, and the symplectic form on it coincides with Kirillov's form. Euler's equations of motion of a solid leave these orbits invariant, and they coincide with the projection of a Hamiltonian system onto a cotangent fibering of the group ~.

In this paper it is shown that for a connected semisimple Lie group with no nontrivial compact quotient any finite central measure is a discrete measure concentrated on the center of the group. More generally, the largest possible support set for a central measure on any semisimple Lie group is determined. From these results it follows that the center of the algebra L1(H) is trivial for any locally compact group H which has a noncompact connected simple Lie group as a homomorphic image. Let Iu be a finite complex measure on a locally compact group G. The measure Iu is called central if 4u(xSx-1)=#u(S) for all Borel sets S and all xeG. Central measures are precisely the measures in the center of the measure algebra M(G) (see [2, p. 269]). We shall show that for a connected semisimple Lie group with no nontrivial compact quotient any central measure is a discrete measure supported on the center of the group. To state our main result in complete generality we need the concept of the maximal connected compact normal subgroup, C, of a connected semisimple Lie group G. C is determined as follows. Let g be the Lie algebra of G and suppose g = g+ + * * + g& where each gi is a simple ideal. (We refer to [1] for this (p. 122) and in general for all Lie theory facts we use.) We may assume that for i

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.