Série de Hausdorff d'une algèbre de Lie et projections canoniques dans l'algèbre enveloppante. (Hausdorff series of a Lie algebra and canonical projections in the enveloping algebra)
type: Item
instance of: scholarly article
title: Série de Hausdorff d'une algèbre de Lie et projections canoniques dans l'algèbre enveloppante. (Hausdorff series of a Lie algebra and canonical projections in the enveloping algebra)
zbMATH Open document ID: 0685.17007
DOI: 10.1016/0021-8693(89)90194-4
author: Jacques Helmstetter
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let A be a Lie algebra over a field of characteristic zero, UA its universal enveloping algebra, \(U^ kA\) the subspace of A spanned by elements of the form \((1/k!)\sum_{w}a_{w(1)}a_{w(2)}...a_{w(k)}\) with \(a_ i\in A\), sum over all permutations. Then \(UA=\oplus_{k\geq 0}U^ kA\). The canonical projections mentioned in the title are the projections \(UA\to U^ kA\), and the paper is concerned with their explicit calculation. In a first step, due to \textit{L. Solomon} [J. Comb. Theory 4, 363-375 (1968; Zbl 0174.066)], the problem is reduced to the case \(k=1\), i.e. to P: UA\(\to U^ 1A=A\) (which is essentially equivalent to finding a formula for the homogeneous components of the Hausdorff series mentioned in the title). It suffices to take for A the free Lie algebra generated by \(a_ 1,...,a_ m\). The author proves the following formula. \[ P(\frac{a_ 1^{p_ 1}}{p_ 1!}... \frac{a_ m^{p_ m}}{p_ m!})=\sum \quad \lambda (\alpha,\beta,q_ 1,...,q_ n) a^{q_ 1}_{i_ 1}... a^{q_ n}_{i_ n}; \] the coefficient \(\lambda (\alpha,\beta,q_ 1,...,q_ n)\) of \(a^{q_ 1}_{i_ 1}... a^{q_ n}_{i_ n}\) depends on the exponents \(q_ 1,...,q_ n\) and on the number \(\alpha\) (resp. \(\beta)\) of successive index pairs \(i_ k\), \(i_{k+1}\) with \(i_ k<i_{k+1}\) (resp. \(i_ k>i_{k+1})\). They are determined by the generating function \[ \int^{1}_{0}u^{\alpha}(u-1)^{\beta} F(u,x_ 1)... F(u,x_ n)=\sum \lambda (\alpha,b,q_ 1,...,q_ n) x_ 1^{q_ 1}... x_ n^{q_ n}, \] with \(F(u,x)=\frac{e^ x-1}{(1- u)e^ x+u}.\) A special case of this result is due to Solomon [loc. cit.]. The author proves in fact a generalization of this formula for \({\mathbb{Z}}_ 2\)-graded Lie algebras \(A=A_++A_-\).
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B05
zbMATH DE Number: 4122176
zbMATH Keywords: universal enveloping algebra
zbMATH Keywords: canonical projections
zbMATH Keywords: Hausdorff series
zbMATH Keywords: free Lie algebra
zbMATH Keywords: generating function
zbMATH Keywords: \({\mathbb{Z}}_ 2\)-graded Lie algebras
Wikidata QID: Q115366104
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90194-4
OpenAlex ID: W1976033809
cites work: Q3941610
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cites work: On the Poincaré-Birkhoff-Witt theorem
description: scientific article; zbMATH DE number 4122176