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

TL;DR

This paper characterizes when words in the support of a free Lie algebra are twin or anti-twin, revealing that these relations depend on word reversal and length parity, with implications for combinatorics on words.

Contribution

It provides a complete characterization of twin and anti-twin words in the support of free Lie algebras over characteristic zero fields, based on word reversal and length parity.

Findings

Twin words are equal or reverses with odd length.

Anti-twin words are reverses with even length.

Palindromes of even length are excluded from support.

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).