Loops as Invariant Sections in Groups, and their Geometry

1994 ◽  
Vol 46 (5) ◽  
pp. 1027-1056 ◽  
Author(s):  
Péter T. Nagy ◽  
Karl Strambach
Keyword(s):  
Isotopy Class ◽  
Closed Loops ◽  
Connected Manifold ◽  
Moufang Loops ◽  
Configurational Condition ◽  
Bol Loops

AbstractWe investigate left conjugacy closed loops which can be given by invariant sections in the group generated by their left translations. These loops are generalizations of the conjugacy closed loops introduced in [13] just as Bol loops generalize Moufang loops. The relations of these loops to common classes of loops are studied. For instance on a connected manifold we construct proper topological left conjugacy closed loops satisfying the left Bol condition but show that any differentiable such loop must be a group. We show that the configurational condition in the 3-net corresponding to an isotopy class of left conjugacy closed loops has the same importance in the geometry of 3-nets as the Reidemeister or the Bol condition.

Differential equations of smooth loops related to some space-time models: Integrability conditions and geometry

2018 ◽  
Vol 27 (07) ◽  
pp. 1841004
Author(s):  
L. Sbitneva
Keyword(s):  
Differential Equations ◽  
Homogeneous Spaces ◽  
Space Time ◽  
Lie Triple System ◽  
Lie Group Theory ◽  
Moufang Loops ◽  
Integrability Conditions ◽  
Curvature And Torsion ◽  
The Right ◽  
Bol Loops

The original approach of Lie to the theory of transformation groups acting on smooth manifolds, on the basis of differential equations, being applied to smooth loops, has permitted the development of the infinitesimal theory of smooth loops generalizing the Lie group theory. A loop with the law of associativity verified for its binary operation is a group. It has been shown that the system of differential equations characterizing a smooth loop with the right Bol identity and the integrability conditions lead to the binary-ternary algebra as a proper infinitesimal object, which turns out to be the Bol algebra (i.e. a Lie triple system with an additional bilinear skew-symmetric operation). There exist the analogous considerations for Moufang loops. We will consider the differential equations of smooth loops, generalizing smooth left Bol loops, with the identities that are the characteristic identities for the algebraic description of some relativistic space-time models. Further examinations of the integrability conditions for the differential equations allow us to introduce the proper infinitesimal object for some subclass of loops under consideration. The geometry of corresponding homogeneous spaces can be described in terms of tensors of curvature and torsion.

scholarly journals Exceptional smooth Bol loops

2002 ◽  
Vol 30 (9) ◽  
pp. 575-579
Author(s):  
Larissa V. Sbitneva
Keyword(s):  
New Class ◽  
Moufang Loops ◽  
Bol Loops

One new class of smooth Bol loops, exceptional Bol loops, is introduced and studied. The approach to the Campbell-Hausdorff formula is outlined. Bol-Bruck loops and Moufang loops are exceptional which justifies our consideration.

An isotopically invariant property of automorphic Moufang loops

2019 ◽  
Vol 58 (4) ◽  
pp. 458-466
Author(s):  
A. N. Grishkov ◽  
M. N. Rasskazova ◽  
L. L. Sabinina
Keyword(s):  
Invariant Property ◽  
Moufang Loops

scholarly journals Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

2020 ◽  
pp. 1-17
Author(s):  
THOMAS BARTHELMÉ ◽  
SERGIO R. FENLEY ◽  
STEVEN FRANKEL ◽  
RAFAEL POTRIE
Keyword(s):  
Large Class ◽  
Universal Cover ◽  
Isotopy Class ◽  
Seifert Manifold ◽  
Hyperbolic Diffeomorphisms ◽  
Surface Homeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

Abstract We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

A multiple closed‐loops robotic calibration for accurate surgical puncture

2021 ◽  
Author(s):  
Lingxiang Zheng ◽  
Zhesi Zhang ◽  
Zhigang Wang ◽  
Kaiyang Bao ◽  
Lin Yang ◽  
...  
Keyword(s):  
Closed Loops

scholarly journals The restricted Burnside problem for Moufang loops

2021 ◽  
pp. 1-11
Author(s):  
ALEXANDER GRISHKOV ◽  
LIUDMILA SABININA ◽  
EFIM ZELMANOV
Keyword(s):  
Prime Number ◽  
Burnside Problem ◽  
Moufang Loops ◽  
Restricted Burnside Problem ◽  
Positive Integers

Abstract We prove that for positive integers $m \geq 1, n \geq 1$ and a prime number $p \neq 2,3$ there are finitely many finite m-generated Moufang loops of exponent $p^n$ .

scholarly journals Diassociativity in Conjugacy Closed Loops

2004 ◽  
Vol 32 (2) ◽  
pp. 767-786 ◽  
Author(s):  
Michael K. Kinyon ◽  
Kenneth Kunen ◽  
J. D. Phillips
Keyword(s):  
Closed Loops
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