The Akivis algebra of a homogeneous loop.

*(English)*Zbl 0601.22002The authors define an Akivis algebra to be a vector space equipped with an anticommutative bilinear operation [\(\cdot,\cdot]\) and a trilinear operation \(<\cdot,\cdot,\cdot >\) which measures how badly [\(\cdot,\cdot]\) fails to satisfy the Jacobi identity. The authors have shown in a previous work [Pac. J. Math. 123, 301-327 (1986; Zbl 0596.22002)] that Akivis algebras occur as the tangent algebras of analytic loops (i.e., non-associative groups).

In this paper they consider a Lie group G with Lie algebra L(G) and a closed subgroup H such that L(H) has a vector space complement K in L(G) with [L(H),K]\(\subseteq K\). (In this case the homogeneous space \(M=G/H\) is called reductive.) M admits the structure of a local loop by taking a local cross-section \(\sigma\) in a neighborhood of eH, and using it to transport the multiplication to a neighborhood of eH in M. The purpose of this paper is to calculate the Akivis algebra operations of the local Lie loop M in terms of the Lie algebra structure on L(G). It is shown that L(M) can be identified with K and that the Akivis algebra operations are given by \([X,Y]_ M=\pi_ K[X,Y]\) and \(<X,Y,Z>_ M=-(1/2)\) \([\pi_ H[X,Y],Z]\), where \(\pi_ K\) and \(\pi_ H\) are projections for L(G) onto K and L(H), respectively.

In this paper they consider a Lie group G with Lie algebra L(G) and a closed subgroup H such that L(H) has a vector space complement K in L(G) with [L(H),K]\(\subseteq K\). (In this case the homogeneous space \(M=G/H\) is called reductive.) M admits the structure of a local loop by taking a local cross-section \(\sigma\) in a neighborhood of eH, and using it to transport the multiplication to a neighborhood of eH in M. The purpose of this paper is to calculate the Akivis algebra operations of the local Lie loop M in terms of the Lie algebra structure on L(G). It is shown that L(M) can be identified with K and that the Akivis algebra operations are given by \([X,Y]_ M=\pi_ K[X,Y]\) and \(<X,Y,Z>_ M=-(1/2)\) \([\pi_ H[X,Y],Z]\), where \(\pi_ K\) and \(\pi_ H\) are projections for L(G) onto K and L(H), respectively.

Reviewer: J.D.Lawson

##### MSC:

22A99 | Topological and differentiable algebraic systems |

17D99 | Other nonassociative rings and algebras |

17B99 | Lie algebras and Lie superalgebras |

##### Keywords:

reductive subalgebras; reductive homogeneous space; Akivis algebra; analytic loops; Lie group; local Lie loop
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\textit{K. H. Hofmann} and \textit{K. Strambach}, Mathematika 33, 87--95 (1986; Zbl 0601.22002)

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##### References:

[1] | DOI: 10.2307/2372398 · Zbl 0059.15805 · doi:10.2307/2372398 |

[2] | Kikkawa, Hiroshima Math. J. 5 pp 141– (1975) |

[3] | Sagle, Pacific J. Math. 48 pp 247– (1973) · Zbl 0285.17003 · doi:10.2140/pjm.1973.48.247 |

[4] | Akivis, Sib. Mat. Zh. 17 pp 5– (1976) · Zbl 0337.53018 · doi:10.1007/BF00969285 |

[5] | Hofmann, The Theory and Applications of Quasigroups and Loop |

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