On the primitive subspace of Lando framed graph bialgebra

TL;DR

This paper characterizes the primitive subspace of the Lando framed graph bialgebra, providing generators, relations, and a new algebraic operation, along with a polynomial module structure and a 4-invariant.

Contribution

It offers an explicit description of the primitive subspace, introduces leaf addition as a module operation, and constructs a novel 4-invariant within the algebra.

Findings

Explicit generators of the primitive subspace identified

Relations between generators explicitly described

A new 4-invariant satisfying a simple identity constructed

Abstract

Lando framed graph bialgebra is generated by framed graphs modulo 4-term relations. We provide an explicit set of generators of its primitive subspace and a description of the set of relations between the generators. We also define an operation of leaf addition that endows the primitive subspace of Lando algebra with a structure of a module over the ring of polynomials in one variable and construct a 4-invariant that satisfies a simple identity with respect to the vertex-multiplication.