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

• Authors: Ioannis C. Michos
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2012

Let ${\mathcal{L}}_{K}(A)$ 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 驴 let $\tilde{u}$ denote its reversal. Two words in A 驴 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 驴 be its adjoint map with respect to the canonical scalar product on the free associative algebra K驴A驴. Studying the kernel of 驴 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驴A with exponent n1 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).