Lyndon-Shirshov basis and anti-commutative algebras ∗
Lyndon-Shirshov basis and anti-commutative algebras ∗
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DOI：
10.1016/j.jalgebra.2012.12.017
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发表时间：
2011-10
影响因子：
0.9
通讯作者：
L. Bokut;L. Bokut;Yuqun Chen;Yu Li
中科院分区：
文献类型：
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作者：
L. Bokut;L. Bokut;Yuqun Chen;Yu Li
Chen，Fox，Lyndon（1958）[10]和Shirshov（1958）[29]引入了非结合的Lyndon-Shirshov词，并证明了它们独立地构成自由李代数的线性基。本文给出了Lyndon-Shirshov基定义的另一种方法，即我们找到了一个自由李代数的反交换Gröbner-Shirshov基S，使得Irr（S）是所有非结合Lyndon-Shirshov字的集合，其中Irr（S）是N（X）的所有单项式的集合，它是X上的自由反交换代数的一个基，不包含来自S的多项式的极大单项式.根据Shirshov的反交换Gröbner-Shirshov基理论（Shirshov，1962 [32]），集合Irr（S）是自由李代数的线性基。
Chen, Fox, Lyndon (1958) [10] and Shirshov (1958) [29] introduced non-associative Lyndon–Shirshov words and proved that they form a linear basis of a free Lie algebra, independently. In this paper we give another approach to definition of Lyndon–Shirshov basis, i.e., we find an anti-commutative Gröbner–Shirshov basis S of a free Lie algebra such that Irr(S) is the set of all non-associative Lyndon–Shirshov words, where Irr(S) is the set of all monomials of N(X), a basis of the free anti-commutative algebra on X, not containing maximal monomials of polynomials from S. Following from Shirshovʼs anti-commutative Gröbner–Shirshov bases theory (Shirshov, 1962 [32]), the set Irr(S) is a linear basis of a free Lie algebra.