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Mosc. Math. J., 2004, Volume 4, Number 1, Pages 199–216 (Mi mmj148)  

This article is cited in 12 scientific papers (total in 12 papers)

On descent algebras and twisted bialgebras

F. Patrasa, Ch. Reutenauerb

a Laboratoire de Mathématiques Jean Alexandre Dieudonné, Université de Nice Sophia Antipolis
b Université de Montréal

Abstract: Bialgebras in the category of tensor species (twisted bialgebras) deserve a particular attention, in particular in view of applications to algebraic combinatorics. In order to study these bialgebras, a new class of descent algebras is introduced. The fine structure of Barratt's permutation bi-ring (the direct sum of the symmetric group algebras) is investigated in detail from this point of view, leading to the definition of an enveloping algebra structure on it.

Key words and phrases: Descent algebra, tensor species, symmetric group, permutation bi-ring, free Lie algebra.

DOI: https://doi.org/10.17323/1609-4514-2004-4-1-199-216

Full text: http://www.ams.org/.../abst4-1-2004.html
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MSC: Primary 16W30, 05E10; Secondary 17A30, 17B01, 17B35
Received: February 4, 2003
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Citation: F. Patras, Ch. Reutenauer, “On descent algebras and twisted bialgebras”, Mosc. Math. J., 4:1 (2004), 199–216

Citation in format AMSBIB
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\by F.~Patras, Ch.~Reutenauer
\paper On descent algebras and twisted bialgebras
\jour Mosc. Math.~J.
\yr 2004
\vol 4
\issue 1
\pages 199--216
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    This publication is cited in the following articles:
    1. Kharchenko V.K., “Braided version of Shirshov-Witt theorem”, J. Algebra, 294:1 (2005), 196–225  crossref  mathscinet  zmath  isi  elib
    2. Patras F., Schocker M., “Twisted descent algebras and the Solomon-Tits algebra”, Adv. Math., 199:1 (2006), 151–184  crossref  mathscinet  zmath  isi  scopus
    3. Patras F., Schocker M., “Trees, set compositions and the twisted descent algebra”, Journal of Algebraic Combinatorics, 28:1 (2008), 3–23  crossref  mathscinet  zmath  isi  scopus
    4. Hivert F., Novelli J.-Ch., Thibon J.-Y., “Commutative combinatorial Hopf algebras”, Journal of Algebraic Combinatorics, 28:1 (2008), 65–95  crossref  mathscinet  zmath  isi  scopus
    5. Livernet M., Patras F., “Lie theory for Hopf operads”, Journal of Algebra, 319:12 (2008), 4899–4920  crossref  mathscinet  zmath  isi  scopus
    6. Schedler T., “Poisson algebras and Yang–Baxter equations”, Advances in Quantum Computation, Contemporary Mathematics Series, 482, 2009, 91–106  crossref  mathscinet  zmath  isi
    7. Aubry M., “Hall basis of twisted Lie algebras”, J Algebraic Combin, 32:2 (2010), 267–286  crossref  mathscinet  zmath  isi  scopus
    8. Brouder Ch., Patras F., “Nonlocal, noncommutative diagrammatics and the linked cluster theorems”, Journal of Mathematical Chemistry, 50:3 (2012), 552–576  crossref  mathscinet  zmath  isi  scopus
    9. Kharchenko V.K., Shestakov I.P., “Generalizations of Lie Algebras”, Adv. Appl. Clifford Algebr., 22:3, SI (2012), 721–743  crossref  mathscinet  zmath  isi  elib  scopus
    10. Aguiar M., Bergeron N., Thiem N., “Hopf Monoids From Class Functions on Unitriangular Matrices”, Algebr. Number Theory, 7:7 (2013), 1743–1779  crossref  mathscinet  zmath  isi  scopus
    11. Marberg E., “Strong Forms of Linearization For Hopf Monoids in Species”, J. Algebr. Comb., 42:2 (2015), 391–428  crossref  mathscinet  zmath  isi  scopus
    12. Marberg E., “Strong Forms of Self-Duality For Hopf Monoids in Species”, Trans. Am. Math. Soc., 368:8 (2016), 5433–5473  crossref  mathscinet  zmath  isi  scopus
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