Journal of Algebraic Combinatorics (EMIS Electronic Version)

Home > Vol 36, No 3 (2012)

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

Ioannis C. Michos

General Department (Mathematics-Physics), Frederick University, Cyprus, 1036, Nicosia, Cyprus

DOI: 10.1007/s10801-011-0339-8

Abstract

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 λ be its adjoint map with respect to the canonical scalar product on the free associative algebra K\langle A\rangle . Studying the kernel of λ 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).

Pages: 355–388

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

Full Text: PDF

References

Duchamp, G., Krob, D.: Combinatorics in trace monoids II. In: Diekert V., Rozenberg G. (eds.) The Book of Traces, pp. 83-129. World Scientific, Singapore (1995) CrossRef Duchamp, G., Laugerotte, É., Luque, J.-G.: On the support of graph Lie algebras. Theoret. Comput. Sci. 273, 283-294 (2002) CrossRef Duchamp, G., Thibon, J.-Y.: Le support de l'algèbre de Lie libre. Discrete Math. 76, 123-132 (1989) CrossRef Johnson, M., Stöhr, R.: Lie powers and pseudo-idempotents. Canad. Math. Bull. 54, 297-301 (2011) CrossRef Katsura, M., Kobayashi, Y.: The shuffle algebra and its derivations. Theoret. Comput. Sci. 115, 359-369 (1993) CrossRef Lothaire, M.: Combinatorics on Words. Encyclopedia of Mathematics and Its Applications, vol.
17. Addison-Wesley, Reading (1983) Michos, I.: On the support of the free Lie algebra: the Schützenberger problems. Discrete Math. Theor. Comput. Sci. $12(3)$, 1-28 (2010) Minh, H.N., Petitot, M.: Lyndon words, polylogarithms and the Riemann $ζ$function. Discrete Math. 217, 273-292 (2000) CrossRef Ree, R.: Lie elements and algebra associated with shuffles. Ann. Math. 68, 210-220 (1958) CrossRef Reutenauer, C.: Free Lie Algebras. London Mathematical Society Monographs. New Series, vol.
7. Oxford Science Publications, The Clarendon Press, Oxford University Press, London (1993)

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	JOURNAL OF ALGEBRAIC COMBINATORICS
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	Home > Vol 36, No 3 (2012) On twin and anti-twin words in the support of the free Lie algebra Ioannis C. Michos General Department (Mathematics-Physics), Frederick University, Cyprus, 1036, Nicosia, Cyprus DOI: 10.1007/s10801-011-0339-8 Abstract 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 λ be its adjoint map with respect to the canonical scalar product on the free associative algebra K\langle A\rangle . Studying the kernel of λ 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). Pages: 355–388 Keywords: free Lie algebras; combinatorics on words; shuffle algebra Full Text: PDF References Duchamp, G., Krob, D.: Combinatorics in trace monoids II. In: Diekert V., Rozenberg G. (eds.) The Book of Traces, pp. 83-129. World Scientific, Singapore (1995) CrossRef Duchamp, G., Laugerotte, É., Luque, J.-G.: On the support of graph Lie algebras. Theoret. Comput. Sci. 273, 283-294 (2002) CrossRef Duchamp, G., Thibon, J.-Y.: Le support de l'algèbre de Lie libre. Discrete Math. 76, 123-132 (1989) CrossRef Johnson, M., Stöhr, R.: Lie powers and pseudo-idempotents. Canad. Math. Bull. 54, 297-301 (2011) CrossRef Katsura, M., Kobayashi, Y.: The shuffle algebra and its derivations. Theoret. Comput. Sci. 115, 359-369 (1993) CrossRef Lothaire, M.: Combinatorics on Words. Encyclopedia of Mathematics and Its Applications, vol. 17. Addison-Wesley, Reading (1983) Michos, I.: On the support of the free Lie algebra: the Schützenberger problems. Discrete Math. Theor. Comput. Sci. $12(3)$, 1-28 (2010) Minh, H.N., Petitot, M.: Lyndon words, polylogarithms and the Riemann $ζ$function. Discrete Math. 217, 273-292 (2000) CrossRef Ree, R.: Lie elements and algebra associated with shuffles. Ann. Math. 68, 210-220 (1958) CrossRef Reutenauer, C.: Free Lie Algebras. London Mathematical Society Monographs. New Series, vol. 7. Oxford Science Publications, The Clarendon Press, Oxford University Press, London (1993) © 1992–2009 Journal of Algebraic Combinatorics © 2012 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition

JOURNAL OF ALGEBRAIC COMBINATORICS

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

Abstract

References

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