Note on the replicas of matrices
YOZÔ MATSUSHIMA · Proceedings of the Japan Academy Series A Mathematical Sciences · 1947
The concept o the replicas o matrices was introduced by C. Chevalley).and very interesting application o it to the study o al- gebraic Lie groups was gven in a joint paper by hmsel and H. F.Tuan.Chevallcy determined the replicas of matrices over a field o characteristic zero and H. F. Tuan ) gave an elementary proof to the same result and in fact in a somewhat general form.In the present notC we shall prove Chevalley's results in a somewhat different way and obtain some properties o the replicas which shall be used in a orthcoming paper ). 1) C. Chevalley, On a kind of new relationship between matrices, Amer, I. Math.Vol.65 (1943), I have not yet access to this paper and the results obtained in this note may perhaps have much contact with Chevalley's paper.2) C. Chevalley and H. F. Tuan, On algebraic Lie algebras, Proc.Nat.Acad.Sci.U. S. A. (1946).3) H. F. Tuan, A note on the replicas of hi]potent matrices, Bull.Amer.Math.Soc.(1945).4) I express my hearty thanks to M. Gotb for his kind remarks during the pre- paration of this note.5) Y. Matsushima, On algebraic Lie groups and algebras, Journ. of the Math.Soc. of Japan, Vol. 1 No. 1 (1948).6) In the following we assume for simplicity that the field K is algebraically closed.No. 5.