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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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1615-7168
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Volume 16, Issue 2

Issues

On the axioms for Sabinin algebras

Gregor Weingart
Published Online: 2016-04-16 | DOI: https://doi.org/10.1515/advgeom-2016-0003

Abstract

The tangent space of a Lie loop, a non-associative Lie group, carries the structure of a Sabinin algebra, an algebraic concept generalizing Lie algebras. Alternatively a Sabinin algebra can be interpreted as the universal local covariant of a flat affine connection. Combining this interpretation of Sabinin algebras with a version of the parallel transport equation we prove that every Sabinin algebra structure determines and is determined by the one-sided linear terms in the Baker-Campbell-Hausdorff formula for the multiplication in the underlying loop. Based on this resultwe prove that every abstract Sabinin algebra over ℝ is the tangent algebra of a global Lie loop.

Keywords: Sabinin algebras; Lie loops

About the article

Received: 2014-07-26

Published Online: 2016-04-16

Published in Print: 2016-04-01


Citation Information: Advances in Geometry, Volume 16, Issue 2, Pages 205–229, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2016-0003.

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[1]
A. Grishkov and J.M. Pérez-Izquierdo
Linear Algebra and its Applications, 2018, Volume 544, Page 460

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