Bases des algèbres de Lie et série de Hausdorff. (French) Zbl 0332.17004

Michel, Jean

Semin. P. Dubreil, 27e annee 1973/74, Algebre, Fasc. 1, Expose 6, 9 p. (1975).

This paper first describes the Širšov basis which belongs to a new family of bases of lie algebras obtained by dropping the condition that longer elements are greater than shorter ones in axioms for the Hall basis [author, Bases des algebres de Lie libres; application à l’étude de la formule de Campbell-Hausdorff, Thèse Be cycle, Math., Univ. Paris-Sud (1974); G. V i e n n o t, Factorisations des monoides libres et algebres de Lie libres, Thèse Sc. math. Univ. Paris (1974), see also C. r. Acad. Sci. Paris, Sér. A 276, 511-514 (1973; ZbI. 252.17001), Sémin. P. Dubreil, 27e année 1973/74, Algère, Fasc. 1, Exposé 5 (1975; Zbl. 309.00019)]. It has computational advantages over classical Hall basis. Then this and other means are applied to the study of the convergence of Hausdorff series \( H(x, y) \) (defined by \( e^{x} e^{y}=e^{H}(x, y) \) ) in p-adic and real Banachic Lie algebras. In the p-adic case, the results of \( M \). I a \( z \) a \( r d \) [Bull. Soc. math. France 91, 435-451 (1963; Zbl. 117, 20)] are improved and this improved version is shown to be nearly optimal. In the real case the strength of the previous result of M. M é \( r \) i ot [Domaine de convergence de la série de Cámpbell-Hausdorff, Conférence faite à l’Université de Nice (multigraphie interne)] is estimated by studying the worst case in \( \mathrm{SI}_{2}(R) \).

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MSC:

17B99	Lie algebras and Lie superalgebras
17B05	Structure theory for Lie algebras and superalgebras