Lie *-Nilpotence of Group Rings

Abstract

Let KG be the group ring of a group G over a field K. Let * be an involution of a group G extended linearly to the group ring KG. Suppose that G is a torsion group without 2-elements and K is a field with characteristic different from 2. We prove that KG is Lie *-nilpotent if and only if KG is Lie nilpotent.

Key Words:

• Group ring
• Lie nilpotent
• Lie *-nilpotent

2010 Mathematics Subject Classifications:

• 16S34
• 16W10
• 16R50

ACKNOWLEDGMENTS

The author would like to thank the referee for comments and suggestions that helped to improve the final version of this article. This research is supported in part by the National Natural Science Foundation of China (11201064), the Specialized Research Fund for the Doctoral Program of Higher Education (20120092110020), and the Nature Science Foundation of Jiangsu Province (BK20130599). The author thanks Dr. Sudarshan Sehgal for all of his help with this article during her visit in University of Alberta.

Notes

Remark: Clearly, there are even numbers of x's in the equality. But it is easy to see that adding another x term does not hurt anything.

Communicated by S. Sehgal.