Structure Of Symmetric Spaces Research Articles

On the structure of symmetric spaces of semidihedral groups

On the structure of symmetric spaces of semidihedral groups

On geometric structure of symmetric spaces

In this article we discuss local approach to strict K-monotonicity and local uniform rotundity in symmetric spaces. We prove several general results on local structure of symmetric spaces E showing relation between strict monotonicity and strict K-monotonicity and the Kadec–Klee property for global convergence in measure. We also present the full criteria for points of upper K-monotonicity in Lorentz spaces Γp,w for degenerated weight function w. Next we characterize local uniform rotundity in symmetric spaces E proving several correspondences between x∈E a point of local uniform rotundity and its decreasing rearrangement x⁎ and absolute value |x|. Finally, we apply these results to find complete criteria for local uniform rotundity of Lorentz spaces Γp,w.

Local monotonicity structure of symmetric spaces with applications

In this paper we discuss several monotonicity properties in Banach lattices. We start with several general results on local structure of symmetric Banach function spaces discussing in particular whether a point x∈E has some local property if and only if its nonincreasing rearrangement x∗ has the same property (Section 2). In that section we also prove some general facts for nonincreasing rearrangements which may be of independent interest. Next, we apply these results to find complete criteria for local monotonicity structure of Lorentz spaces Γp,ω and Orlicz–Lorentz spaces Λφ,ω (Sections 3, 4.1 and 4.2). We conclude with the description of global monotonicity structure of Lorentz spaces Γp,ω (Section 4.3). We finish with the applications of upper monotonicity and lower monotonicity points to local best dominated approximation problems in Banach lattices (Section 5).

Dyadic sets, maximal functions and applications on ax + b-groups

Let S be the Lie group \({{\mathbb R}^n\ltimes {\mathbb R}}\), where \({{\mathbb R}}\) acts on \({{\mathbb R}^n}\) by dilations, endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure ρ, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245: 37–61, 2003] proved that any integrable function on (S, ρ) admits a Calderón–Zygmund decomposition which involves a particular family of sets, called Calderón–Zygmund sets. In this paper, we show the existence of a dyadic grid in the group S, which has nice properties similar to the classical Euclidean dyadic cubes. Using the properties of the dyadic grid, we prove a Fefferman–Stein type inequality, involving the dyadic Hardy–Littlewood maximal function and the dyadic sharp function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space H 1 and the space BMO introduced in [Collect. Math. 60: 277–295, 2009].

Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions

Einstein gravity in both 3 and 4 dimensions, as well as some interesting genera- lizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant models in 3 dimensions include Einstein gravity in Chern-Simons form, as well as a new formulation of topologically massive gravity, with arbitrary cosmological constant, as a single constrained Chern-Simons action. In 4 dimensions the main model of interest is MacDowell-Mansouri gravity, generalized to include the Immirzi parameter in a natural way. I formulate these theories in Cartan geometric language, emphasizing also the role played by the symmetric space structure of the model. I also explain how, from the perspective of these Cartan-geometric formulations, both the topological mass in 3d and the Immirzi parameter in 4d are the result of non- simplicity of the Lorentz Lie algebra so(3,1) and its relatives. Finally, I suggest how the language of Cartan geometry provides a guiding principle for elegantly reformulating any 'gauge theory of geometry'.

Algorithms for Computations in Local Symmetric Spaces

In the last two decades much of the algebraic/combinatorial structure of Lie groups, Lie algebras, and their representations has been implemented in several excellent computer algebra packages, including LiE, GAP4, Chevie, Magma, and Maple. The structure of reductive symmetric spaces or more generally symmetric k-varieties is very similar to that of the underlying Lie group, with a few additional complications. A computer algebra package enabling one to do computations related to these symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc. In this article we lay the groundwork for computing the fine structure of symmetric spaces over the real numbers and other base fields, give a complete set of algorithms for computing the fine structure of symmetric varieties and use this to compute nice bases for the local symmetric varieties.

Zero Curvature Formalism for Supersymmetric Integrable Hierarchies in Superspace

We generalize the Drinfeld–Sokolov formalism of bosonic integrable hierarchies to superspace, in a way which systematically leads to the zero curvature formulation for the supersymmetric integrable systems starting from the Lax equation in superspace. We use the method of symmetric space as well as the non-Abelian gauge technique to obtain the supersymmetric integrable hierarchies of the AKNS type from the zero curvature condition in superspace with the graded algebras, sl (n+1,n), providing the Hermitian symmetric space structure.

A new algebraic approach for calculating the heat kernel in quantum gravity

It is shown that the heat kernel operator for the Laplace operator on any covariantly constant curved background, i.e., in symmetric spaces, may be presented in the form of an averaging over the Lie group of isometries with some nontrivial measure. Using this representation, the heat kernel diagonal, i.e., the heat kernel in coinciding points is obtained. Related topics concerning the structure of symmetric spaces and the calculation of the effective action are discussed.

Linear dynamical systems, coherent state manifolds, flows, and matrix Riccati equation

The classical motion and the quantum evolution determined by linear Hamiltonians on coherent state manifolds of compact and noncompact Hermitian symmetric space structure are considered. A matrix Riccati equation, with opposite signs of the quadratic terms in the compact and noncompact cases, determines the classical motion and the quantum evolution. The geometric meaning of equations of motion as flow on a complex Grassmann manifold and his noncompact dual is emphasized. Possibilities of generalizing the results to flag manifolds are pointed out.

On equations of motion on compact Hermitian symmetric spaces

The classical equations of motion on compact coherent state manifolds that have Hermitian symmetric space structure are considered. In the case of Hamiltonians linear in the generators of the group that determines the structure of the manifold of coherent states, the classical motion and the quantum evolution are both described by the same first-order second degree differential equation. The explicit forms of the classical equations of motion as Matrix Riccati equations on the Grassmann manifold GA(Cn) and the coset manifold SO (2n)/U(n), attached to the Hartree–Fock and, respectively, Hartree–Fock–Bogoliubov problem are obtained.

Prolongations of tensor fields and connections to tangent bundles I -- General theory --

Publisher Summary Using the notion of complete lift it is shown that G structures as a pseudo-Riemannian structure, an almost complex structure and a symplectic structure on M induce similar structures on the tangent bundle T(M). An unexpected but perhaps more interesting result is that each pseudo-Riemannian (resp. affine) symmetric space structure on M induces a pseudo-Riemannian (resp. affine) symmetric space structure on T(M). This suggests a method of producing a large class of affine symmetric spaces. The chapter considers A be a local algebra of the form A=R+I where R is the field of real numbers and I is the maximal ideal of A such that dim I k =0 for some k. A successful generalization of the theory to A(M) furnishs a useful tool for the differential geometry of higher order contact and yield a large number of affine symmetric spaces.