Eulerian idempotent, pre-Lie logarithm and combinatorics of trees
Speaker: Ruggero Bandiera, Penn State
Abstract: The aim of this talk is to bring together the three objects in the title. Recall that, given a Lie algebra g, the Eulerian idempotent is a canonical projection from the enveloping algebra U(g) to g. The Baker-Campbell-Hausdorff product and the Magnus expansion can both be expressed in terms of the Eulerian idempotent, which makes it interesting to look for explicit formulas for the latter. We show how to reduce the computation of the Eulerian idempotent to the computation of a logarithm in a certain pre-Lie algebra of planar, binary, rooted trees. The problem of finding formulas for the pre-Lie logarithm, which is interesting in its own right, with connections to operad theory, numerical analysis and renormalization, is addressed using techniques inspired by umbral calculus. As a consequence of our analysis, we find formulas both for the Eulerian idempotent and the pre-Lie logarithm in terms of combinatorics of trees. Based on joint work with Florian Schaetz.
Room Reservation Information
Room Number: 104 Osmond Building
Time: 2:35pm - 3:30pm