The Pure Virtual Braid Group Is Quadratic

TL;DR

This paper introduces the PVH Criterion, a sufficient condition for the associated graded algebra of an augmented algebra to be quadratic, and applies it to the pure virtual braid group, establishing the existence of a universal finite type invariant.

Contribution

The paper presents the PVH Criterion for quadraticity of associated graded algebras and demonstrates its application to the pure virtual braid group, confirming the existence of a universal finite type invariant.

Findings

The associated graded algebra of the pure virtual braid group is quadratic.

The PVH Criterion effectively determines quadraticity of graded algebras.

Existence of a universal finite type invariant for the pure virtual braid group.

Abstract

If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grK need not in general be quadratic: although it is generated in degree 1, its relations may not be generated by homogeneous relations of degree 2. In this paper we give a sufficient criterion (called the PVH Criterion) for grK to be quadratic. When K is the group algebra of a group G, quadraticity is known to be equivalent to the existence of a (not necessarily homomorphic) universal finite type invariant for G. Thus the PVH Criterion also implies the existence of such a universal finite type invariant for the group G. We apply the PVH Criterion to the group algebra of the pure virtual braid group (also known as the quasi-triangular group), and show that the corresponding associated graded algebra is quadratic, and hence that these groups have a (not necessarily homomorphic) universal finite type invariant.