Algebraic Bol loops

2011 ◽  
Vol 23 (3) ◽  
Author(s):  
Alexander Grishkov ◽  
Gábor P. Nagy
Keyword(s):  
Bol Loops

scholarly journals Bol loops

1966 ◽  
Vol 123 (2) ◽  
pp. 341-341 ◽  
Author(s):  
D. A. Robinson
Keyword(s):  
Bol Loops

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 Commutants of Bol loops of odd order

2003 ◽  
Vol 132 (3) ◽  
pp. 617-619 ◽  
Author(s):  
Michael K. Kinyon ◽  
J. D. Phillips
Keyword(s):  
Odd Order ◽  
Bol Loops

On isomorphisms and isotopisms of bol loops of order 3p

1994 ◽  
Vol 22 (1) ◽  
pp. 345-347 ◽  
Author(s):  
Harald Niederreiter ◽  
Karl H. Robinson
Keyword(s):  
Bol Loops

Bol Loops of Nilpotence Class Two

2007 ◽  
Vol 59 (2) ◽  
pp. 296-310 ◽  
Author(s):  
Orin Chein ◽  
Edgar G. Goodaire
Keyword(s):  
Nilpotence Class ◽  
Bol Loops ◽  
Bol Loop

AbstractCall a non-Moufang Bol loop minimally non-Moufang if every proper subloop is Moufang and minimally nonassociative if every proper subloop is associative. We prove that these concepts are the same for Bol loops which are nilpotent of class two and in which certain associators square to 1. In the process, we derive many commutator and associator identities which hold in such loops.

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