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The Pure Virtual Braid Group Is Quadratic

Peter Lee

出版	Library and Archives Canada = Bibliothèque et Archives Canada, 2012
ISBN	04949741259780494974124
URL	http://books.google.com.hk/books?id=dDjLtwEACAAJ&hl=&source=gbs_api

註釋

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 thesis 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 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.