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Generalized Lie Elements

Published online by Cambridge University Press:  20 November 2018

Rimhak Ree
Rimhak Ree*
Affiliation:
University of British Columbia
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Let λ(ij), i,j = 1, 2, … , m, be m2 elements in a field K of characteristic zero such that λ(ij)λ(ji) = 1 for all i and j, and X1, x2, … , xm non-commutative associative indeterminates over K. Define the elements [xi1Xi2 … xin] inductively by [xi] = xi and

Any linear combination of the elements

with coefficients in K will be called a generalized Lie elememt. Generalized Lie elements reduce to ordinary Lie elements if λ(ij) = 1 for all i and j.

The purpose of this paper is to generalize to the generalized Lie elements the following: a theorem of Friedrichs, a theorem of Dynkin-Specht-Wever (2), and the Witt formula on the dimension of the space spanned by homogeneous Lie elements of a fixed degree. The set of all generalized Lie elements will be made into an algebra which generalizes the ordinary free Lie algebra. This algebra turns out to be free in a certain sense. We shall also generalize the algebra associated with shuffles in (2).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 493 - 502
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Chevalley, C., Fundamental concepts of algebra (New York: Academic Press Inc., 1956).Google Scholar
2. Ree, Rimhak, Lie elements and an algebra associated with shuffles, Ann. Math., 68 (1958), 210–220.Google Scholar
3. Séminaire, “Sophus Lie,” Théorie des algebres de Lie et topologie des groupes de Lie, 1954-5.Google Scholar
4. Witt, Ernst, Treue Darstellung Liescher Ringe, J. Reine Angew. Math., 177 (1937), 152–160.Google Scholar