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

Michos, Ioannis C.

Journal of Algebraic Combinatorics, Tome 36 (2012) no. 3, pp. 355-388.

Résumé

Summary: 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 $^{\ast }$ let $\tilde{u}$ denote its reversal. Two words in A $^{\ast }$ 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 free associative algebra K$\langle A\rangle $. 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).

Détail

Keywords: free Lie algebras, combinatorics on words, shuffle algebra