On Real Simple Left-symmetric Algebras

Abstract

In this article, we commence to study the real (simple) left-symmetric algebras. From the known classification of certain complex (semi)simple left-symmetric algebras, we classify their corresponding real forms. We not only obtain the classification of real simple left-symmetric algebras in low dimensions, but also find certain examples of real simple left-symmetric algebras in higher dimensions. In particular, there exists a complex simple left-symmetric algebra without any real form. We also give a geometric construction for a class of real simple left-symmetric algebras. At last, we apply the classification results to study some structures related to geometry.

Key Words:

• Real form
• Real left-symmetric algebra
• Semi-involution
• Simple left-symmetric algebra

2010 Mathematics Subject Classification:

• 17A30
• 17B60
• 53C15
• 53D05

ACKNOWLEDGMENTS

The authors thank D. Burde and D. P. Hou for valuable discussions. This work was supported in part by NSFC (10571091, 10621101, 10921061), NKBRPC (2006CB805905), and SRFDP (200800550015), Program for New Century Excellent Talents in University.

Notes

Communicated by A. Elduque.