Lie Elements and the Matrix-Tree Theorem

For a finite-dimensional representation V of a group G we introduce and study the notion of a Lie element in the group algebra kof Lie elements is a Lie algebra and a G-module acting on the original representation V .Lie elements often exhibit nice combinatorial properties.Thus, for a G = Sn and V , a permutation representation, we prove a formula for the characteristic polynomial of a Lie element similar to the classical matrix-tree theorem.