Universal Polynomial $\mathfrak{so}$ Weight System

TL;DR

This paper introduces a universal polynomial weight system that generalizes several Lie algebra and superalgebra weight systems, with an efficient recursive computation method.

Contribution

It extends the universal polynomial weight system framework to include $ ext{so}$, $ ext{sp}$, and $ ext{osp}$ Lie (super)algebras, building on previous work for $ ext{gl}$.

Findings

Unified polynomial weight system for multiple Lie (super)algebras

Efficient recursive algorithm for computation

Specializations recover known weight systems

Abstract

We introduce a universal weight system (a function on chord diagrams satisfying the $4$-term relation) taking values in the ring of polynomials in infinitely many variables whose particular specializations are weight systems associated with the Lie algebras $\mathfrak{so}(N)$, $\mathfrak{sp}(2M)$, as well as Lie superalgebras $\mathfrak{osp}(N|2M)$. We extend this weight system to permutations and provide an efficient recursion for its computation. The construction for this weight system extends a similar construction for the universal polynomial weight system responsible for the Lie algebras $\mathfrak{gl}(N)$ and superalgebras $\mathfrak{gl}(N|M)$ introduced earlier by the second named author.