IJPAM: Volume 4, No. 1 (2003)

A NOTE ON THE TAFT'S PROBLEM

Hao Zhifeng$^1$, Feng Lianggui$^2$
$^1$Department of Applied Mathematics
College of Science
South China University of Technology
Guangzhou 510641, P.R. CHINA,
e-mail: mazfhao@scut.edu.cn
$^2$Department of Mathematics and System Science
National University of Defense Technology
Changsha 410073, P.R. CHINA
e-mail: flg2000@yahoo.com


Abstract.Let $F$ be a field of characteristic zero. In this paper we work out the linearly recursive relation on Lie multiplication $[f, g]$ in Witt algebras $(W_1^{(i)} )^o$(resp. $(W^{(i)})^o$). This is an open problem proposed by Earl J. Taft. We show that if the characteric polynomial $p(x)$(resp. $q(x)$) of $f$(resp. $g$) $\in (W^{(i)} )^o$ or $(W_1^{(i)} )^o$ satisfy $p(x)\vert(x^i-a^i)$ and $q(x) \vert (x^i-a^i)$ for $a$ in the algebraically closure of $F$, then $[f, g]$ satisfies $LCM(p(x), q(x)), $the least common multiple of $p(x)$ and $q(x)$. Some examples illustrate the results.

Received: November 10, 2002

AMS Subject Classification: 16W30

Key Words and Phrases: Lie multiplication, linearly recursive sequences, recursive relation, Lie coalgebras

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2003
Volume: 4
Issue: 1