Let $A = \{a,b,c,d\}$ be a set of totally ordered alphabets, a Lyndon word over $A$ is a word $w$ in $A^*$ such that if $w=uv$ is a factorization of $w$ into non-empty subwords, then $u<v$ in lexicographic order. This is equivalent to being the unique minimum word (in lexicographic order) among all its rotations.
Let $\mathcal{L}(A)$ be the free Lie algebra on the set $A$. It is well known that Lyndon words in $A$ indexes a basis for $\mathcal(L)(A)$.
Assume $a$ is the least alphabet and let $w=a_1a_2\cdots a_k \in A^*$ (free monoid over $A$) define $L(w) = [a_1,[a_2,[\cdots[a_{k-1},a_k]\cdots]$ to be the right associative Lie monomial associated to $w$ in $\mathcal{L}(A)$. By rotation, I always mean another word $w^{'}$ which is obtained from $w$ by rotating the alphabets cyclically and further we also want this word to start with $a$.
Let $w = abcad$ and let $w^{'}=adabc$ be a rotation of $w$. My question is $L(w)$ and $L(w^{'})$ are linearly dependent as Lyndon words indexes a basis and $w$ and $w^{'}$ are rotationally equivalent.
More generally, I want to understand the linear dependence relation between the Lie monomials $L(w),L(w_1),\dots,L(w_k)$ where $w,w_1,w_2,\dots,w_k$ are all possible rotations of $w$ in the above sense.
How to understand the linear dependence relation between general right-associative Lie monomials and what makes the Lyndon words the indexing set for the basis of free Lie algebras? How the rotation of words affects the Lyndon words.
Kindly share your thoughts.
Thank you.
