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Title: Reductive Lie-admissible algebras applied to H-spaces and connections

Abstract

An algebra A with multiplication xy is Lie-admissible if the vector space A with new multiplication (x,y) = xy-yx is a Lie algebra; we denote this Lie algebra by A/sup -/. Thus, an associative algebra is Lie-admissible but a Cayley algebra is not Lie-admissible. In this paper we show how Lie-admissible algebras arise from Lie groups and their application to differential geometry on Lie groups via the following theorem. Let A be an n-dimensional Lie-admissible algebra over the reals. Let G be a Lie group with multiplication function ..mu.. and with Lie algebra g which is isomorphic to A/sup -/. Then there exiss a corrdinate system at the identify e in G which represents ..mu.. by a function F:gxg..-->..g defined locally at the origin, such that the second derivative, F/sup 2/, at the origin defines on the vector space g the structure of a nonassociative algebra (g, F/sup 2/). Furthermore this algebra is isomorphic to A and (g, F/sup 2/)/sup -/ is isomorphic to A/sup -/. Thus roughly, any Lie-admissible algebra is isomorphic to an algebra obtained from a Lie algebra via a change of coordinates in the Lie group. Lie algebras arise by using canonical coordinates and the Campbell-Hausdorffmore » formula. Applications of this show that any G-invariant psuedo-Riemannian connection on G is completely determined by a suitable Lie-admissible algebra. These results extend to H-spaces, reductive Lie-admissible algebras and connections on homogeneous H-spaces. Thus, alternative and other non-Lie-admissible algebras can be utilized.« less

Authors:
Publication Date:
Research Org.:
Inst. for Basic Research, Cambridge, MA
OSTI Identifier:
6369925
Report Number(s):
CONF-820136-
Journal ID: CODEN: HAJOD
Resource Type:
Conference
Journal Name:
Hadronic J.; (United States)
Additional Journal Information:
Journal Volume: 5:4; Conference: 1. international conference on non-potential interactions and their Lie-admissible treatment, Orleans, France, 5 Jan 1982
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; LIE GROUPS; MATHEMATICAL SPACE; MATHEMATICS; SPACE; SYMMETRY GROUPS; 658000* - Mathematical Physics- (-1987)

Citation Formats

Sagle, A A. Reductive Lie-admissible algebras applied to H-spaces and connections. United States: N. p., 1982. Web.
Sagle, A A. Reductive Lie-admissible algebras applied to H-spaces and connections. United States.
Sagle, A A. 1982. "Reductive Lie-admissible algebras applied to H-spaces and connections". United States.
@article{osti_6369925,
title = {Reductive Lie-admissible algebras applied to H-spaces and connections},
author = {Sagle, A A},
abstractNote = {An algebra A with multiplication xy is Lie-admissible if the vector space A with new multiplication (x,y) = xy-yx is a Lie algebra; we denote this Lie algebra by A/sup -/. Thus, an associative algebra is Lie-admissible but a Cayley algebra is not Lie-admissible. In this paper we show how Lie-admissible algebras arise from Lie groups and their application to differential geometry on Lie groups via the following theorem. Let A be an n-dimensional Lie-admissible algebra over the reals. Let G be a Lie group with multiplication function ..mu.. and with Lie algebra g which is isomorphic to A/sup -/. Then there exiss a corrdinate system at the identify e in G which represents ..mu.. by a function F:gxg..-->..g defined locally at the origin, such that the second derivative, F/sup 2/, at the origin defines on the vector space g the structure of a nonassociative algebra (g, F/sup 2/). Furthermore this algebra is isomorphic to A and (g, F/sup 2/)/sup -/ is isomorphic to A/sup -/. Thus roughly, any Lie-admissible algebra is isomorphic to an algebra obtained from a Lie algebra via a change of coordinates in the Lie group. Lie algebras arise by using canonical coordinates and the Campbell-Hausdorff formula. Applications of this show that any G-invariant psuedo-Riemannian connection on G is completely determined by a suitable Lie-admissible algebra. These results extend to H-spaces, reductive Lie-admissible algebras and connections on homogeneous H-spaces. Thus, alternative and other non-Lie-admissible algebras can be utilized.},
doi = {},
url = {https://www.osti.gov/biblio/6369925}, journal = {Hadronic J.; (United States)},
number = ,
volume = 5:4,
place = {United States},
year = {1982},
month = {6}
}

Conference:
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