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On the primitive subspace of Lando framed graph bialgebra

Maksim Karev

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.

On the primitive subspace of Lando framed graph bialgebra

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.
Paper Structure (3 sections, 9 theorems, 18 equations)

This paper contains 3 sections, 9 theorems, 18 equations.

Table of Contents

  1. Bialgebra structures on the spaces generated by isomorphism classes of framed finite simple graphs
  2. Leaf attachment
  3. A 4-invariant related to the number of 3-colorings of a graph.

Key Result

Theorem 1.1

The inclusion $\iota\colon G_\mathcal{JR} \to G_\mathcal{C}$ gives rise to a bialgebra isomorphism $\phi\colon G_\mathcal{JR} \to G_\mathcal{C}/I_\mathcal{C},$ where the bialgebra structure on $G_\mathcal{C}/I_\mathcal{C}$ is induced from $G_\mathcal{C}$.

Theorems & Definitions (19)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.1
  • proof
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • proof
  • ...and 9 more