On Anti-Commutative Algebras and Analytic Loops

1965 ◽  
Vol 17 ◽  
pp. 550-558 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Binary Operation ◽  
Identity Element ◽  
Unique Element ◽  
Commutative Algebras ◽  
Algebraic Properties

In (4) Malcev generalizes the notion of the Lie algebra of a Lie group to that of an anti-commutative "tangent algebra" of an analytic loop. In this paper we shall discuss these concepts briefly and modify them to the situation where the cancellation laws in the loop are replaced by a unique two-sided inverse. Thus we shall have a set H with a binary operation xy defined on it having the algebraic properties(1.1) H contains a two-sided identity element e;(1.2) for every x ∊ H, there exists a unique element x-1 ∊ H such that xx-1 = x-1x = e;

GENERALIZED COMPOSITIONS OF FUZZY RELATIONS

2002 ◽  
Vol 10 (supp01) ◽  
pp. 149-163 ◽  
Author(s):  
JÓZEF DREWNIAK ◽  
KRZYSZTOF KULA
Keyword(s):  
Binary Operation ◽  
Identity Element ◽  
Fuzzy Relations ◽  
Triangular Norms ◽  
Algebraic Properties

We examine compositions of fuzzy relations based on a binary operation *. We discuss the dependences between algebraic properties of the operation * and the induced sup –* composition. It is examined independently for monotone operations, for operations with idempotent, zero or identity element, for distributive and associative operations. Finally, we present consequences of these results for compositions based on triangular norms, triangular conorms and uninorms.

scholarly journals A Note on Simple Anti-Commutative Algebras Obtained from Reductive Homogeneous Spaces

1968 ◽  
Vol 31 ◽  
pp. 105-124 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Homogeneous Space ◽  
Lie Algebra ◽  
Lie Group ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Commutative Algebras ◽  
Direct Sum ◽  
Reductive Homogeneous Spaces ◽  
Connected Lie Group

LetGbe a connected Lie group andHa closed subgroup, then the homogeneous spaceM = G/His calledreductiveif there exists a decomposition(subspace direct sum) withwhereg(resp.) is the Lie algebra ofG(resp.H); in this case the pair (g,) is called areductive pair.

ON QUOTIENT STRUCTURE OF TAKASAKI QUANDLES

2013 ◽  
Vol 22 (12) ◽  
pp. 1341001 ◽  
Author(s):  
YONGJU BAE ◽  
SEONGJEONG KIM
Keyword(s):  
Abelian Group ◽  
Binary Operation ◽  
Identity Element ◽  
Algebraic Properties

A Takasaki quandle is defined by the binary operation a * b = 2b - a on an abelian group G. A Takasaki quandle depends on the algebraic properties of the underlying abelian group. In this paper, we will study the quotient structure of a Takasaki quandle in terms of its subquandle. If a subquandle X of a quandle Q is a subgroup of the underlying group Q, then we can define the quandle structure on the set {X * g | g ∈ Q}, which is called the quotient quandle of Q. Also we will study conditions for a subquandle X to be a subgroup of the underlying group when it contains the identity element.

ε‎-Invariance

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Lie Algebra ◽  
Lie Group

The discussions up to Chapter 4 have been concerned with the Lie group. In this chapter, the Lie algebra is constructed by defining the operators ∇ and ∇̃.

scholarly journals COMMUTATORS OF SKEW-SYMMETRIC MATRICES

2005 ◽  
Vol 15 (03) ◽  
pp. 793-801 ◽  
Author(s):  
ANTHONY M. BLOCH ◽  
ARIEH ISERLES
Keyword(s):  
Optimal Control ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Geometric Integration ◽  
Symmetric Matrices ◽  
Frobenius Norm

In this paper we develop a theory for analysing the "radius" of the Lie algebra of a matrix Lie group, which is a measure of the size of its commutators. Complete details are given for the Lie algebra 𝔰𝔬(n) of skew symmetric matrices where we prove [Formula: see text], X, Y ∈ 𝔰𝔬(n), for the Frobenius norm. We indicate how these ideas might be extended to other matrix Lie algebras. We discuss why these ideas are of interest in applications such as geometric integration and optimal control.

scholarly journals A Drift-Free Left Invariant Control System on the Lie Group SO(3)×R3×R3

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Camelia Pop
Keyword(s):  
Control System ◽  
Numerical Integration ◽  
Lie Algebra ◽  
Lie Group ◽  
Poisson Structure ◽  
Dual Space ◽  
Free System ◽  
Geometrical Properties ◽  
Free Left ◽  
Invariant Control

A controllable drift-free system on the Lie group G=SO(3)×R3×R3 is considered. The dynamics and geometrical properties of the corresponding reduced Hamilton’s equations on g∗,·,·- are studied, where ·,·- is the minus Lie-Poisson structure on the dual space g∗ of the Lie algebra g=so(3)×R3×R3 of G. The numerical integration of this system is also discussed.

Harish-Chandra Modules over ℤ

2019 ◽  
pp. 126-167
Author(s):  
Günter Harder
Keyword(s):  
Fixed Points ◽  
Lie Algebra ◽  
Lie Group ◽  
Maximal Torus ◽  
Reductive Group ◽  
Group Scheme ◽  
Finitely Generated ◽  
Cartan Involution ◽  
Split Maximal Torus ◽  
Definition Of

This chapter shows that certain classes of Harish-Chandra modules have in a natural way a structure over ℤ. The Lie group is replaced by a split reductive group scheme G/ℤ, its Lie algebra is denoted by 𝖌ℤ. On the group scheme G/ℤ there is a Cartan involution 𝚯 that acts by t ↦ t −1 on the split maximal torus. The fixed points of G/ℤ under 𝚯 is a flat group scheme 𝒦/ℤ. A Harish-Chandra module over ℤ is a ℤ-module 𝒱 that comes with an action of the Lie algebra 𝖌ℤ, an action of the group scheme 𝒦, and some compatibility conditions is required between these two actions. Finally, 𝒦-finiteness is also required, which is that 𝒱 is a union of finitely generated ℤ modules 𝒱I that are 𝒦-invariant. The definitions imitate the definition of a Harish-Chandra modules over ℝ or over ℂ.

Summary of Elements of Algebraic Theory

1995 ◽  
Author(s):  
F. Iachello ◽  
R. D. Levine
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Algebraic Structure ◽  
Binary Operation ◽  
Algebraic Theory ◽  
Quantum Mechanical ◽  
Ordinary Quantum

Algebraic theory makes use of an algebraic structure. The structure appropriate to ordinary quantum mechanical problems is that of a Lie algebra. We begin this chapter with a brief review of the essential concepts of Lie algebras. The binary operation (“multiplication”) in the Lie algebra is that of taking the commutator. As usual, we denote the commutator by square brackets, [A, B] = AB - BA. A set of operators {X} is a Lie algebra when it is closed under commutation.

scholarly journals On Formality of Some Homogeneous Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1011
Author(s):  
Aleksy Tralle
Keyword(s):  
Homogeneous Space ◽  
Root System ◽  
Lie Algebra ◽  
Lie Group ◽  
Maximal Torus ◽  
Homogeneous Spaces ◽  
Direct Proof ◽  
Classical Type ◽  
Sasakian Manifolds ◽  
Full Analysis

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

Characteristic functions of semigroups in semi-simple Lie groups

2019 ◽  
Vol 31 (4) ◽  
pp. 815-842
Author(s):  
Luiz A. B. San Martin ◽  
Laercio J. Santos
Keyword(s):  
Laplace Transform ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Convex Cone ◽  
Convex Set ◽  
Flag Manifold ◽  
Riemannian Metric ◽  
Iwasawa Decomposition ◽  
Characteristic Functions

Abstract Let G be a noncompact semi-simple Lie group with Iwasawa decomposition {G=KAN} . For a semigroup {S\subset G} with nonempty interior we find a domain of convergence of the Helgason–Laplace transform {I_{S}(\lambda,u)=\int_{S}e^{\lambda(\mathsf{a}(g,u))}\,dg} , where dg is the Haar measure of G, {u\in K} , {\lambda\in\mathfrak{a}^{\ast}} , {\mathfrak{a}} is the Lie algebra of A and {gu=ke^{\mathsf{a}(g,u)}n\in KAN} . The domain is given in terms of a flag manifold of G written {\mathbb{F}_{\Theta(S)}} called the flag type of S, where {\Theta(S)} is a subset of the simple system of roots. It is proved that {I_{S}(\lambda,u)<\infty} if λ belongs to a convex cone defined from {\Theta(S)} and {u\in\pi^{-1}(\mathcal{D}_{\Theta(S)}(S))} , where {\mathcal{D}_{\Theta(S)}(S)\subset\mathbb{F}_{\Theta(S)}} is a B-convex set and {\pi:K\rightarrow\mathbb{F}_{\Theta(S)}} is the natural projection. We prove differentiability of {I_{S}(\lambda,u)} and apply the results to construct of a Riemannian metric in {\mathcal{D}_{\Theta(S)}(S)} invariant by the group {S\cap S^{-1}} of units of S.

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