Theory of K-Loops

Kiechle, Hubert

doi:10.1007/b83276

Cited by 86 publications

(87 citation statements)

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“…Further readings on Bol loops can be found in (Robinson, 1966;Pflugfelder, 1990;Kiechle 2002). Next we define some groups acting on L, namely right multiplication group and left multiplication group.…”

Section: Preliminariesmentioning

confidence: 97%

A Family of Left Lie Bol Loops

Bulut¹

2017

JMR

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In this paper the left Bol split extension method is used to build left Bol Lie loops from the Lie groups H and K such that H is a Lie subgroup of Aut(K). Furthermore, we investigated some of the properties of those loops constructed in this way. Examples are given for finite and infinite dimensional left Bol Lie loops. Moreover, we showed that the twisted semidirect product of Lie algebras is an Akivis algebra.

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Section: Preliminariesmentioning

confidence: 97%

A Family of Left Lie Bol Loops

Bulut¹

2017

JMR

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“…Kiechle demonstra em [9] que um laço à esquerda (� , ⊕) é um grupo se, e somente se, os giradores g y r [a, b ] = I d � para quaisquer que sejam a, b ∈ � . Posteriormente, usaremos esse fato.…”

Section: Girogruposunclassified

Gyrogroups and Gyrovectors Spaces

Aguitoni

Semensato

2013

Revista Ciencias Exatas E Naturais

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Resumo: Para um grupo arbitrário G e um subgrupo normal H ⊂ G, o espaço G H herda a estrutura do grupo G. Já no caso em que H não é normal em G, isso não ocorre. Neste trabalho, quando H não é normal em G, vamos introduzir uma estrutura de girogrupo em G H , o que é possível já que a maioria das propriedades características dos girogrupos são propriedades de laços homogêneos com a propriedade inversa do automorfismo.Também definiremos espaços girovetoriais, os quais dão suporte à geometria hiperbólica assim como espaços vetoriais para a geometria euclidiana.Palavras-chave: espaços girovetoriais; girogrupos; laços.Abstract: For an arbitrary group G and a normal subgroup H ⊂ G, the space G H inherits the structure of the group G. Now in the case that H is not normal in G, this does not happen. In this work, when H is not normal in G we introduce a structure of gyrogroup in G H , what is possible since most of the characteristic properties of gyrogroups are properties of homogeneous loops with the automorphism inverse property. We will also define gyrovectors spaces, what provides supports for hyperbolic geometry just as vectors spaces for euclidean geometry.Recebido em 18/07/2013 -Aceito em 28/08/2013. RECEN15 (1)

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“…This allows one to define a multiplication on L as follows: for a, b ∈ L, we write ab = xh ∈ LH for unique x ∈ L, h ∈ H and define a * b := x. Then as an algebraic structure (L, * ) is a left loop, that is, it has an identity, namely the identity e of the group, and there is a unique solution of the equation a * x = b for all a, b ∈ L (see, for example, [11,Theorem 2.7]). …”

Section: Dyadic Symmetric Sets and B-loopsmentioning

confidence: 99%

“…The connection of symmetric spaces with involutive groups is well-established; indeed Helgason [6] develops the theory of Riemannian symmetric spaces in the context of Lie groups with involution. More recent work has made connections between twisted subgroups, involutive groups, and certain classes of loops; see for example, [10], [1], [3], [11]. In this paper we tie all of these together by making precise these connections for the specific class of symmetric sets that we have in mind.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Sets With Midpoints and Algebraically Equivalent Theories

Lawson

Lim

2004

Results. Math.

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Abstract. In this paper we consider an algebraic generalization of symmetric spaces of noncompact type to a more general class of symmetric structures equipped with midpoints. These symmetric structures are shown to have close relationships to and even categorical equivalences with a variety of other algebraic structures:transversal twisted subgroups of involutive groups, a special class of loops called B-loops, and gyrocommutative gyrogroups.

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Theory of K-Loops

Cited by 86 publications

References 0 publications

A Family of Left Lie Bol Loops

A Family of Left Lie Bol Loops

Gyrogroups and Gyrovectors Spaces

Symmetric Sets With Midpoints and Algebraically Equivalent Theories

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