Remarks on one-parameter subsemigroups of the affine group and their homo- and isomorphisms

1970 ◽  
Vol 4 (1-2) ◽  
pp. 1-10 ◽  
Author(s):  
J. Aczél ◽  
St. Gołąb
Keyword(s):  
Affine Group

The complements in an affine group

2013 ◽  
Vol 50 (2) ◽  
pp. 258-265
Author(s):  
Pál Hegedűs
Keyword(s):  
Permutation Group ◽  
Structural Similarity ◽  
Affine Group ◽  
Permutation Module ◽  
Regular Module ◽  
Brauer Characters

In this paper we analyse the natural permutation module of an affine permutation group. For this the regular module of an elementary Abelian p-group is described in detail. We consider the inequivalent permutation modules coming from nonconjugate complements. We prove their strong structural similarity well exceeding the fact that they have equal Brauer characters.

scholarly journals Solitons of the midpoint mapping and affine curvature

2021 ◽  
Vol 112 (1) ◽  
Author(s):  
Christine Rademacher ◽  
Hans-Bert Rademacher
Keyword(s):  
Differential Equation ◽  
Large Class ◽  
Affine Group ◽  
Arc Length ◽  
Smooth Curves ◽  
Affine Map ◽  
Affine Curvature

AbstractFor a polygon $$x=(x_j)_{j\in \mathbb {Z}}$$ x = ( x j ) j ∈ Z in $$\mathbb {R}^n$$ R n we consider the midpoints polygon $$(M(x))_j=\left( x_j+x_{j+1}\right) /2.$$ ( M ( x ) ) j = x j + x j + 1 / 2 . We call a polygon a soliton of the midpoints mapping M if its midpoints polygon is the image of the polygon under an invertible affine map. We show that a large class of these polygons lie on an orbit of a one-parameter subgroup of the affine group acting on $$\mathbb {R}^n.$$ R n . These smooth curves are also characterized as solutions of the differential equation $$\dot{c}(t)=Bc (t)+d$$ c ˙ ( t ) = B c ( t ) + d for a matrix B and a vector d. For $$n=2$$ n = 2 these curves are curves of constant generalized-affine curvature $$k_{ga}=k_{ga}(B)$$ k ga = k ga ( B ) depending on B parametrized by generalized-affine arc length unless they are parametrizations of a parabola, an ellipse, or a hyperbola.

IMBEDDING OF TRANSFORMATION GROUPS IN AFFINE ONES AND RENORMALIZATION OF TWO-DIMENSIONAL HOMOGENEOUS σ-MODELS

1993 ◽  
Vol 08 (31) ◽  
pp. 2937-2942
Author(s):  
A. V. BRATCHIKOV
Keyword(s):  
Homogeneous Space ◽  
Transformation Group ◽  
Affine Group ◽  
Coupling Constants ◽  
Transformation Groups ◽  
Two Dimensional

The BLZ method for the analysis of renormalizability of the O(N)/O(N − 1) model is extended to the σ-model built on an arbitrary homogeneous space G/H and in arbitrary coordinates. For deriving Ward-Takahashi (WT) identities an imbedding of the transformation group G in an affine group is used. The structure of the renormalized action is found. All the infinities can be absorbed in a coupling constants renormalization and in a renormalization of auxiliary constants which are related to the imbedding.

scholarly journals Gauss-Bonnet theorems in the generalized affine group and the generalized BCV spaces

2021 ◽  
Vol 6 (11) ◽  
pp. 11655-11685
Author(s):  
Tong Wu ◽  
◽  
Yong Wang
Keyword(s):  
Gaussian Curvature ◽  
Smooth Surface ◽  
Geodesic Curvature ◽  
Affine Group ◽  
Smooth Curves ◽  
Characteristic Points ◽  
Curves On Surfaces

<abstract><p>In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean $ C^2 $-smooth surface in the generalized affine group and the generalized BCV spaces away from characteristic points and signed geodesic curvature for Euclidean $ C^2 $-smooth curves on surfaces. We get Gauss-Bonnet theorems in the generalized affine group and the generalized BCV spaces.</p></abstract>

Automorphisms, isomorphisms, and Tits classification

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Affine Group ◽  
Classification Theorem ◽  
Isomorphism Theorem ◽  
Reductive Groups ◽  
Semisimple Groups ◽  
Dynkin Diagrams ◽  
Weil Restriction ◽  
Characteristic 2 ◽  
Smooth Affine

This chapter considers automorphisms, isomorphisms, and Tits classification. It begins by establishing a version of the Isomorphism Theorem for pseudo-split pseudo-reductive groups, along with a pseudo-reductive variant of the Isogeny Theorem for split connected semisimple groups. The key to both proofs is a technique to construct pseudo-reductive subgroups of an ambient smooth affine group. Some instructive examples over imperfect fields k of characteristic 2 are given. The chapter goes on to discuss the behavior of the k-group ZG,C with respect to Weil restriction in the pseudoreductive case. It also describes automorphism schemes for pseudo-reductive groups, focusing only on the pseudo-semisimple case because commutative pseudo-reductive groups that are not tori generally have a non-representable automorphism functor. Finally, it examines Tits-style classification, using Dynkin diagrams to express the classification theorem.

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