Hall basis of twisted Lie algebras
Marc Aubry
DOI: 10.1007/s10801-010-0214-z
Abstract
In this paper we define a minimal generating system for the free twisted Lie algebra. This gives a correct formulation and a proof to an old statement of Barratt. To this aim we use properties of the Lyndon words and of the Klyachko idempotent which generalize to twisted Hopf algebras some similar results well known in the classical case.
Pages: 267–286
Keywords: keywords twisted Hopf algebras; twisted Lie algebras; klyachko idempotent; Hall basis; Dynkin word
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References
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2. Barratt, M.G.: Twisted Lie algebras. Geometric applications of homotopy theory. In: Proc. Conf., Evanston, Ill., 1977, II. Lecture Notes in Math., vol. 658, pp. 9-15. Springer, Berlin (1978)
3. Fresse, B.: Koszul duality of operads and homology of partition posets. In: Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K -theory. Contemp. Math., vol. 346, pp. 115-215. Am. Math. Soc., Providence (2004)
4. Garsia, A.M.: Combinatorics of the free Lie algebra and the symmetric group. In: Analysis, et Cetera, pp. 309-382. Academic Press, San Diego (1990)
5. Gelfand, I.M., Krob, D., Lascoux, A., Leclerc, B., Retakh, V.S., Thibon, J.-Y.: Noncommutative symmetric functions. Adv. Math. 112(2), 218-348 (1995)
6. Joyal, A.: Foncteurs analytiques et espèces de structures. In: Combinatoire Énumérative, Montréal, Qué., 1985/Québec, Que.,
1985. Lecture Notes in Math., vol. 1234, pp. 126-159. Springer, Berlin (1986)
7. Livernet, M., Patras, F.: Lie theory for Hopf operads. J. Algebra 319(12), 4899-4920 (2008)
8. Markl, M., Shnider, S., Stasheff, J.: Operads in Algebra, Topology and Physics. Mathematical Surveys and Monographs, vol.
96. Am. Math. Soc., Providence (2002). x+349 pp.
9. Patras, F., Reutenauer, C.: On Dynkin and Klyachko idempotents in graded bialgebras. Special issue in memory of Rodica Simion. Adv. Appl. Math. 28(3-4), 560-579 (2002)
10. Patras, F., Reutenauer, C.: On descent algebras and twisted bialgebras. Mosc. Math. J. 4 311(1), 199- 216 (2004)
11. Reutenauer, C.: Free Lie Algebras. London Mathematical Society Monographs. New Series, vol.
7. Oxford Science Publications, The Clarendon Press, Oxford University Press, London (1993). xviii+269 pp.
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