# 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 »

- 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}

}