On twin and anti-twin words in the support of the free Lie algebra

Abstract

Let $L_{K}(A)$ be the free Lie algebra on a finite alphabet $A$ over a commutative ring $K$ with unity. For a word $u$ in the free monoid $A^{*}$ let $\tilde{u}$ denote its reversal. Two words in $A^{*}$ are called twin (resp. anti-twin) if they appear with equal (resp. opposite) coefficients in each Lie polynomial. Let $l$ denote the left-normed Lie bracketing and $\lambda$ be its adjoint map with respect to the canonical scalar product on the corresponding free associative algebra. Studying the kernel of $\lambda$ and using several techniques from combinatorics on words and the shuffle algebra, we show that when $K$ is of characteristic zero two words $u$ and $v$ of common length $n$ that lie in the support of ${\mathcal L}_{K}(A)$ - i.e., they are neither powers $a^{n}$ of letters $a \in A$ with exponent $n > 1$ nor palindromes of even length - are twin (resp. anti-twin) if and only if $u = v$ or $u = \tilde{v}$ and $n$ is odd (resp. $u = \tilde{v}$ and $n$ is even).