The pure virtual braid group is quadratic

Abstract

If an augmented algebra K over Q is filtered by powers of its augmentation ideal I, the associated graded algebra grI K 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 grI K 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 universal finite type invariant.

The Pure Braid Groups and Their Relatives

“…A finite presentation for vP n was given by Bardakov [5]. The virtual pure braid group vP n and its 'upper' subgroup, vP + n , were both studied in depth (under different names) by Bartholdi et al and Lee in [11,50]. These groups are generated by elements x i j for i = j (respectively, for i < j), subject to the relations…”

“…Yes [11,50] No (for n ≥ 4) [70] Thus, the groups P n , wP n , and wP + n also enjoy this property. The fact that the pure braid groups P n are residually torsion-free nilpotent also follows from the work of Falk and Randell [35,36].…”

“…The groups vP n and vP + n were also independently studied by Bartholdi, Enriquez, Etingof, and Rains [11] and P. Lee [50] as groups arising from the Yang-Baxter equations. Classifying spaces for these groups (also known as the quasi-triangular groups and the triangular groups, respectively) can be constructed by taking quotients of permutahedra by suitable actions of the symmetric groups.…”

Koszulity of cohomology = K(π,1)-ness + quasi-formality

“…In this sense, Theorem 1.3 can be viewed as an analogue for relations of the type (2) of the Poincaré-Birkhoff-Witt theorem for relations of the type (1) [31,34] (cf. [23]). …”

“…Its subject can be roughly described as cohomological characterization of the coalgebras C defined by the relations (2) with the quadratic principal parts (3) q 2 (x) = 0 of the relations defining a Koszul graded coalgebra. In fact, according to the main theorem of [32] (see also [23]) a conilpotent coalgebra C is defined by a self-consistent system of relations (2) with Koszul quadratic principal part (3) if and only if its cohomology algebra H * (C) = Ext * C (k, k) is Koszul. Moreover, a certain seemingly weaker set of conditions on the algebra H * = H * (C) is sufficient, and implies Koszulity of algebras of the form H * (C).…”