Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-24T17:09:14.323Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Akivis, M. A.. Local algebras of a multidimensional web. Sib. Mat. Zh., 17 (1976), 5–11. Translated in Sib. Mat. J., 17 (1976), 3–8.CrossRefGoogle Scholar
2.Hofmann, K. H. and Strambach, K.. Lie's fundamental theorems for local analytical loops. Pacific J. Math., 123 (1986). To appear.CrossRefGoogle Scholar
3.Hofmann, K. H. and Strambach, K.. Topological and Analytical Loops, Chapter IX in: The Theory and Applications of Quasigroups and Loop, Chein, O., Pflugfelder, H., and Smith, J. H., Eds. (Heldermann Verlag, Berlin). To appear.Google Scholar
4.Kikkawa, M.. Geometry of homogeneous Lie loops. Hiroshima Math. J., 5 (1975), 141–179.CrossRefGoogle Scholar
5.Nomizu, K.. Invariant affine connections on homogeneous spaces. Atner. J. Math., 76 (1954), 33–65.CrossRefGoogle Scholar
6.Sagle, A. A. and Schumi, J. R.. Multiplication on homogeneous spaces, non-associative algebras, and connections. Pacific J. Math., 48 (1973), 247–266.CrossRefGoogle Scholar