The space of Lie algebra 2‐cocycles of an affine transformation group

1995 ◽  
Vol 36 (10) ◽  
pp. 5949-5967
Author(s):  
G. Harnett
Keyword(s):  
Lie Algebra ◽  
Affine Transformation ◽  
Transformation Group ◽  
Algebra 2

scholarly journals Generalized Symmetries and mCK Method Analysis of the (2+1)-Dimensional Coupled Burgers Equations

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1473 ◽  
Author(s):  
Gangwei Wang ◽  
Yixing Liu ◽  
Shuxin Han ◽  
Hua Wang ◽  
Xing Su
Keyword(s):  
Symmetry Group ◽  
Lie Algebra ◽  
Transformation Group ◽  
Symmetry Transformation ◽  
Burgers Equations ◽  
Generalized Symmetries ◽  
Method Analysis

In this paper, generalized symmetries and mCK method are employed to analyze the (2+1)-dimensional coupled Burgers equations. Firstly, based on the generalized symmetries method, the corresponding symmetries of the (2+1)-dimensional coupled Burgers equations are derived. And then, using the mCK method, symmetry transformation group theorem is presented. From symmetry transformation group theorem, a great many of new solutions can be derived. Lastly, Lie algebra for given symmetry group are considered.

scholarly journals A Theorem on the Affine Transformation Group of a Riemannian Manifold

1955 ◽  
Vol 9 ◽  
pp. 39-41 ◽  
Author(s):  
Shoshichi Kobayashi
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Affine Transformation ◽  
Transformation Group ◽  
Affine Connection

Every Riemannian manifold has a unique affine connection without torsion, which is necessarily invariant by any isometrical transformation of the manifold. However, an affine transformation (i.e., transformation leaving invariant the affine connection) is not necessarily an isometrical transformation. (Consider, for example, the ordinary Euclidean space).

On the compact real form of the Lie algebra 2

2009 ◽  
Vol 148 (1) ◽  
pp. 87-91 ◽  
Author(s):  
ROBERT A. WILSON
Keyword(s):  
Lie Algebra ◽  
Real Form ◽  
Group Of Automorphisms ◽  
Elementary Construction ◽  
Algebra 2 ◽  
Group 2

AbstractWe give an elementary construction of the compact real form of the Lie algebra 2. This construction exhibits the group 23·L3(2) as a group of automorphisms. We also show that there is a unique 14-dimensional real Lie algebra invariant under the action of this group.

On Connections Of Cartan

1956 ◽  
Vol 8 ◽  
pp. 145-156 ◽  
Author(s):  
Shôshichi Kobayashi
Keyword(s):  
Tangent Bundle ◽  
Affine Transformation ◽  
Affine Space ◽  
General Linear Group ◽  
Transformation Group ◽  
Fibre Bundle ◽  
Differentiable Manifold ◽  
Isotropic Subgroup ◽  
Principal Fibre ◽  
Principal Fibre Bundle

Introduction. Consider a differentiable manifold M and the tangent bundle T(M) over M, the structure group of which is usually the general linear group G'. Let P' be the principal fibre bundle associated with T(M). Consider the fibre F of T(M) as an affine space, then we have acting on F the affine transformation group G, which contains G' as the isotropic subgroup.

scholarly journals Generalized affine transformation monoids on Galois rings

2006 ◽  
Vol 2006 ◽  
pp. 1-6
Author(s):  
Yonglin Cao
Keyword(s):  
Algebraic Structure ◽  
Affine Transformation ◽  
Transformation Group ◽  
Galois Ring ◽  
Explicit Description ◽  
Green’S Relations ◽  
Galois Rings ◽  
Transformation Monoid ◽  
Green's Relations ◽  
Transformation Monoids

LetAbe a ring with identity. The generalized affine transformation monoidGaff(A)is defined as the set of all transformations onAof the formx↦xu+a(for allx∈A), whereu,a∈A. We study the algebraic structure of the monoidGaff(A)on a finite Galois ringA. The following results are obtained: an explicit description of Green's relations onGaff(A); and an explicit description of the Schützenberger group of every-class, which is shown to be isomorphic to the affine transformation group for a smaller Galois ring.

The Standard Metric

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Affine Transformation ◽  
Convex Sets ◽  
Finite Dimension ◽  
Affine Subspace ◽  
Affine Transformations ◽  
Real Vector ◽  
Euclidean Metric ◽  
Affine Hyperplane ◽  
Tits Building ◽  
Convex Closure

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

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