On twin and anti-twin words in the support of the free Lie algebra
Michos, Ioannis C.
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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 ũ 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 brack-eting and λ be its adjoint map with respect to the canonical scalar product on the free associative algebra K〈A〉. Studying the kernel of λ and using several techniques from combinatorics on words and the shuffle algebra (K〈A〉,+,), we show that, when K is of characteristic zero, two words u and ν of common length n that lie in the support of L K(A)-i.e., they are neither powers a n of letters a ∈ A with exponent n>1 nor palindromes of even length-are twin (resp. anti-twin) if and only if u = ν or u = ν̃ and n is odd (resp. u = ν̃ and n is even).