Lyndon-Shirshov basis and anti-commutative algebras ∗

Abstract 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 Grobner–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 Grobner–Shirshov bases theory (Shirshov, 1962 [32] ), the set Irr ( S ) is a linear basis of a free Lie algebra.

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