Enveloping algebras of preLie algebras, Solomon idempotents and the Magnus formula (Q2842024)

If \(L\) is a Lie algebra, the Poincaré-Birkhoff-Witt theorem shows that the enveloping algebra \(\mathcal{U}(L)\) and the symmetric algebra \(S(L)\) are linearly isomorphic. The decomposition of \(S(L)\) into homogeneous components defines projections on \(\mathcal{U}(L)\), called the Solomon idempotents (also known as the Eulerian, the Barr, or the canonical idempotents).NEWLINENEWLINEWhen \(L\) is pre-Lie, the Guin-Oudom construction [\textit{J.-M. Oudom} and \textit{D. J. Guin}, K-Theory 2, No. 1, 147--167 (2008; Zbl 1178.17011)] allows to equip \(S(L)\) with a product \(*\), making the isomorphism with \(\mathcal{U}(L)\) explicit. The Solomon idempotents are here explicitly described, with the help of this product. It is shown that the first Solomon idempotent is related to the Magnus expansion formula used for differential equations.