Anti-commutative Grbner-Shirshov basis of a free Lie algebra  

作　　者：BOKUT L. A. 

机构地区：[1]School of Mathematical Sciences, South China Normal University [2]Sobolev Institute of Mathematics, Russian Academy of Sciences, Siberian Branch,Novosibirsk 630090, Russia

出　　处：《Science China Mathematics》2009年第2期244-253,共10页中国科学：数学（英文版）

基　　金：supported by the grant LSS (Grant No. 344.2008.1);the SB RAS Integration Grant (GrantNo. 2006.1.9) (Russia);National Natural Science Foundation of China (Grant No. 10771077);Natural Science Foundation of Guangdong Province (Grant No. 06025062)

摘　　要：The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie algebra by using direct construction, that is, first he started with a linear space spanned by Hall words, then defined the Lie product of Hall words and finally checked that the product yields the Lie identities. In this paper, we give a Grbner-Shirshov basis for a free Lie algebra. As an application, by using the Composition-Diamond lemma established by Shirshov in 1962 for free anti-commutative (non-associative) algebras, we provide another method different from that of M. Hall to construct a basis of a free Lie algebra.The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie algebra by using direct construction, that is, first he started with a linear space spanned by Hall words, then defined the Lie product of Hall words and finally checked that the product yields the Lie identities. In this paper, we give a Grbner-Shirshov basis for a free Lie algebra. As an application, by using the Composition-Diamond lemma established by Shirshov in 1962 for free anti-commutative (non-associative) algebras, we provide another method different from that of M. Hall to construct a basis of a free Lie algebra.

关 键 词：LIE ALGEBRA anti-commutative ALGEBRA HALL words Grbner-Shirshov BASIS 

分 类 号：O152.5[理学—数学]