scholarly journals Some Topological and Algebraic Features of Symmetric Spaces

2020 ◽  
pp. 144-155
Author(s):  
Um Salama ◽  
Ahmed Abd Alla ◽  
A. Elemam
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Lie Groups ◽  
Lie Algebras ◽  
Symmetric Spaces ◽  
Algebraic Approach ◽  
Root Systems ◽  
Algebraic Properties ◽  
Important Field

In this study, we introduce some approaches, geometrical and algebraic, which help to give further understanding of symmetric spaces. Symmetric space is a very important field for understanding abstract and applied features of spaces. We have introduced Riemannian Manifold, Lie groups and Lie algebras, and some of their topological and algebraic properties, with some concentration on Lie algebras and root systems , which help classification and many applications of symmetric spaces. The paper is an attempt to explain some algebraic features of symmetric spaces and how to get some of their properties using algebraic approach, concluded with some results.

scholarly journals Symmetric Spaces in Riemannian and Semi-Riemannian Geometry

2020 ◽  
pp. 47-59
Author(s):  
Ehsan Hashempour ◽  
Mir Mohammad Seyedvalilo
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Lie Groups ◽  
Riemannian Geometry ◽  
Symmetric Spaces ◽  
Sufficient Conditions ◽  
Local Symmetry ◽  
Symmetric Pair ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

In this paper, we will obtain the necessary and sufficient conditions for the analysis of the position of local symmetry on an arbitrary Riemannian manifold. These conditions are devoid of the aspects of Lie groups, and thus can be used in calculations of procedures, without interfering with the concepts of Lie groups, and improve intuitive attitudes. Also, we will study and create equivalent conditions for a situation where a two-metric homogeneous Riemannian manifold is located symmetrically. In addition, in this paper it is stated that the symmetric space (M, g) can be seen as a homogeneous space G/K. Also, one-to-one correspondence between the symmetric space and the symmetric pair is shown, and curvature is studied on a symmetric space.

Corwin–Greenleaf multiplicity functions for complex semisimple symmetric spaces

2015 ◽  
Vol 26 (05) ◽  
pp. 1550039
Author(s):  
Salma Nasrin
Keyword(s):  
Symmetric Space ◽  
Lie Algebras ◽  
Lie Group ◽  
Symmetric Spaces ◽  
Coadjoint Orbit ◽  
Natural Projection ◽  
Real Form ◽  
Single Orbit ◽  
Complex Semisimple ◽  
Complex Symmetric

Let Gℂ be a complex simple Lie group, GU a compact real form, and [Formula: see text] the natural projection between the dual of the Lie algebras. We prove that, for any coadjoint orbit [Formula: see text] of GU, the intersection of [Formula: see text] with a coadjoint orbit [Formula: see text] of Gℂ is either an empty set or a single orbit of GU if [Formula: see text] is isomorphic to a complex symmetric space.

ON AMINO ACID AND CODON ASSIGNMENT IN ALGEBRAIC MODELS FOR THE GENETIC CODE

2010 ◽  
Vol 24 (04) ◽  
pp. 435-463 ◽  
Author(s):  
FERNANDO ANTONELI ◽  
MICHAEL FORGER ◽  
PAOLA A. GAVIRIA ◽  
JOSÉ EDUARDO M. HORNOS
Keyword(s):  
Amino Acid ◽  
Genetic Code ◽  
Lie Groups ◽  
Lie Algebras ◽  
Finite Groups ◽  
Algebraic Approach ◽  
Lie Superalgebras ◽  
Algebraic Models ◽  
Symmetry Principles

We give a list of all possible schemes for performing amino acid and codon assignments in algebraic models for the genetic code, which are consistent with a few simple symmetry principles, in accordance with the spirit of the algebraic approach to the evolution of the genetic code proposed by Hornos and Hornos. Our results are complete in the sense of covering all the algebraic models that arise within this approach, whether based on Lie groups/Lie algebras, on Lie superalgebras or on finite groups.

scholarly journals Submaximally symmetric quaternion Hermitian structures

2020 ◽  
Vol 31 (11) ◽  
pp. 2050084
Author(s):  
Boris Kruglikov ◽  
Henrik Winther
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Symmetric Spaces ◽  
Geometric Properties ◽  
Hermitian Manifolds ◽  
Kähler Structures ◽  
Hermitian Structures

We consider and resolve the gap problem for almost quaternion-Hermitian structures, i.e. we determine the maximal and submaximal symmetry dimensions, both for Lie algebras and Lie groups, in the class of almost quaternion-Hermitian manifolds. We classify all structures with such symmetry dimensions. Geometric properties of the submaximally symmetric spaces are studied, in particular, we identify locally conformally quaternion-Kähler structures as well as quaternion-Kähler with torsion.

scholarly journals Reflections and symmetries in compact symmetric spaces

1988 ◽  
Vol 38 (3) ◽  
pp. 377-386 ◽  
Author(s):  
Bang-Yen Chen ◽  
Lieven Vanhecke
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Symmetric Spaces ◽  
Compact Symmetric Space ◽  
Compact Symmetric Spaces

Point symmetries and reflections are two important transformations on a Riemannian manifold. In this article we study the interactions between point symmetries and reflections in a compact symmetric space when the reflections are global isometries.

scholarly journals Helgason spheres of compact symmetric spaces and immersions of finite type

2001 ◽  
Vol 63 (2) ◽  
pp. 243-255
Author(s):  
Bang-Yen Chen
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Finite Type ◽  
Symmetric Spaces ◽  
Unit Vector ◽  
Geodesic Sphere ◽  
Compact Type ◽  
Totally Geodesic ◽  
Compact Symmetric Space ◽  
Unit Speed

A unit speed curve γ = γ(s) in a Riemannian manifold N is called a circle if there exists a unit vector field Y(s) along γ and a positive constant k such that ∇sγ′(s) = kY(s), ∇sY(s) = −kγ′(s). A maximal totally geodesic sphere with maximal sectional curvature in a compact irreducible symmetric space M is called a Helgason sphere. A circle which lies in a Helgason sphere of a compact symmetric space is called a Helgason circle. In this article we establish some fundamental relationships between Helgason circles, Helgason spheres of irreducible symmetric spaces of compact type and the theory of immersions of finite type.

scholarly journals THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

2021 ◽  
pp. 1-20
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Continuous Measure ◽  
Convolution Product ◽  
Absolutely Continuous ◽  
Regular Elements ◽  
Connected Subgroup ◽  
Decay Properties ◽  
Absolutely Continuous Measure ◽  
Special Case

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

An algebraic approach to N-soft sets with application in decision-making using TOPSIS

2021 ◽  
pp. 1-21
Author(s):  
Muhammad Shabir ◽  
Rimsha Mushtaq ◽  
Munazza Naz
Keyword(s):  
Decision Making ◽  
Mathematical Models ◽  
Real World ◽  
Algebraic Approach ◽  
Multi Criteria Decision Making ◽  
Commutative Monoids ◽  
Soft Sets ◽  
Algebraic Properties ◽  
Different Types ◽  
Selection Of

In this paper, we focus on two main objectives. Firstly, we define some binary and unary operations on N-soft sets and study their algebraic properties. In unary operations, three different types of complements are studied. We prove De Morgan’s laws concerning top complements and for bottom complements for N-soft sets where N is fixed and provide a counterexample to show that De Morgan’s laws do not hold if we take different N. Then, we study different collections of N-soft sets which become idempotent commutative monoids and consequently show, that, these monoids give rise to hemirings of N-soft sets. Some of these hemirings are turned out as lattices. Finally, we show that the collection of all N-soft sets with full parameter set E and collection of all N-soft sets with parameter subset A are Stone Algebras. The second objective is to integrate the well-known technique of TOPSIS and N-soft set-based mathematical models from the real world. We discuss a hybrid model of multi-criteria decision-making combining the TOPSIS and N-soft sets and present an algorithm with implementation on the selection of the best model of laptop.

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