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The Akivis algebra of a homogeneous loop

Part of: Locally compact groups and their algebras

Published online by Cambridge University Press:  26 February 2010

Karl Heinrich Hofmann  and
Karl Strambach
Karl Heinrich Hofmann
Affiliation:
Fachbereich Mathematik, Technische Hochschule Darmstadt, Schlossgartenstr. 7, D-6100 Darmstadt, Federal Republic of Germany.
Karl Strambach
Affiliation:
Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstr. l½, D-8520 Erlangen, Federal Republic of Germany.
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Extract

A (local) Lie loop is a real analytic manifold M with a base point e and three analytic functions (x, y) → x° y, x\y, x/y: M × M → M (respectively, U × U → M for an open neighbourhood U of e in M) such that the following conditions are satisfied: (i) x ° e = e ° x = e, (ii) x ° (x\y) = y, and (iii) (x/y)° y = x for all x, y ε M (respectively, U). If the multiplication ° is associative, then M is a (local) Lie group. The tangent vector space L(M) in e is equipped with an anticommutative bilinear operation (X, Y) →[X, Y] and a trilinear operation (X, Y, Z) →〈X, Y, Z〉. These are defined as follows: Let B be a convex symmetric open neighbourhood of 0 in L(M) such that the exponential function maps B diffeomorphically onto an open neighbourhood V of e in M and transport the operation ° into L(M) by defining X ° Y = (exp|B)−1((exp X)° (exp Y)) for X and Y in a neighbourhood C of 0 in B such that (exp C) ° (exp C) ⊂ V. Similarly, we transport / and \.

MSC classification

Secondary: 22D99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 33 , Issue 1 , June 1986 , pp. 87 - 95
Copyright
Copyright © University College London 1986

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References

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