Generalized chord diagrams and weight systems

The paper is devoted to a description of the recent progress in understanding the extension of Lie algebra weight systems to permutations. Lie algebra weight systems are functions on chord diagrams arising naturally in Vassiliev's theory of finite-type knot invariants. These functions satisfy certain linear restrictions known as Vassiliev's 4-term relations. Chord diagrams can be interpreted as fixed-point-free involutions in symmetric groups, and an extension of Lie algebra weight systems to arbitrary permutations was aimed at finding an efficient way to compute their values. We show that this extension is of interest on its own, which suggests introducing the notion of weight system on permutations. To this end we define generalized Vassiliev's relations for permutations, which reduce to conventional ones for chord diagrams. We also describe the corresponding Hopf algebra structures on spaces of permutations that match the classical Hopf algebra structure on the space of chord diagrams modulo 4-term relations. Among main results of the paper is an explicit formula for the average value of the universal $\mathfrak{gl}$-weight system on permutations. This formula implies, in particular, that this average value is a tau-function for the Kadomtsev-Petviashvili hierarchy of partial differential equations. Its proof is based on an analysis of a quantum version of the universal $\mathfrak{gl}$-weight system.