Basic Lie products form a freely generating set of graded component of free Lie algebra

For free Lie ring
Suppose $$n$$ and $$r$$ are (unrelated) positive integers. Consider the free Lie ring on $$n$$ variables $$x_1,x_2,\dot,x_n$$. Consider the weight $$r$$ graded component, i.e., the abelian group generated by all iterated Lie brackets of length $$r$$. This is a finitely generated free abelian group and the basic Lie products form a freely generating set for it.

For free $$\mathbb{Q}$$-Lie algebra
Consider the free $$\mathbb{Q}$$-Lie algebra on variables $$x_1,x_2,\dot,x_n$$. Consider the weight $$r$$ graded component, i.e., the abelian group generated by all iterated Lie brackets of length $$r$$. This is a finite-dimensional vector space over $$\mathbb{Q}$$ and the basic Lie products form a basis for it.

This basis is called the Lyndon basis or Shirshov basis.

Applications

 * Formula for dimension of graded component of free Lie algebra