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)
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 \(A=A_++A_-\).
scientific article; zbMATH DE number 4122176
scientific article