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The pure virtual braid group is quadratic

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Abstract

If an augmented algebra \(K\) over \(\mathbb Q \) is filtered by powers of its augmentation ideal \(I\), the associated graded algebra \(gr_I K\) 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 \(gr_I K\) 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.

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Notes

  1. See also the summary of 1-formality in Papadima [26].

  2. In fact, by a theorem in [1] (originally due to Morgan), 1-formality for a finitely presentable group is equivalent to the existence of a filtered isomorphism between \(M_G\) and any quadratic Lie algebra (with the filtration induced by the lower central series), although their proof uses real coefficients. When this theorem applies, the existence of a Hopf algebra expansion would always imply 1-formality.

  3. Quillen [31] and Quillen [32]. See also the summary of the construction of \(M_G\) and \(grM_G\) in Papadima [26].

  4. See [23], Sect. 5.7, for the Lie algebra structure on \(grG\).

  5. See [20]. This theorem extends (a part of) Theorem (8.1)’ of Sullivan [35]. See also [9].

  6. We recall that \(G^{(1)}\!/ G^{(2)}\cong H^{1*}\) (see [15]).

  7. Alternatively, whenever the result in footnote 2 applies, quadraticity is equivalent to graded 1-formality, whether or not \(hol(G) \cong \mathcal P (q({{gr_I} K}))\).

  8. This can be seen as follows. Essentially by definition, \(F=TX\), the tensor algebra over \(X\). But then it is easy to see that the algebra homomorphism which maps \(x \mapsto \bar{x}+1\) converts from the \(TX\) presentation to the \(T\tilde{X}\) presentation.

  9. Note the \(\pi _p^{Syz}: \mathfrak{R }^F\rightarrow [ker\ \partial _A]_p\) referred to in the Introduction are just the \(\pi ^{Syz}_p\) above extended by \(0\) outside of \(ker\ \partial _K|_{\mathfrak{R }_{\ge p}}\).

  10. The statement about Koszulness, however, relies on results about Koszul algebras whose graded components are finitely generated over the ground ring. Hence, for purposes of this part of the theorem, we assume the algebra \(K\) to be finitely generated, which is sufficient to ensure that \(A^m\) is a finite dimensional \(\mathbb Q \)-vector space for all \(m\).

  11. In fact, \(A\) need only be 2-Koszul, that is, its Koszul complex need only be exact up to homological degree 2 inclusive.

  12. The picture builds on xy-pic templates due to Aaron Lauda—see [21].

  13. As noted in footnote 10, we rely on results about Koszul algebras which have only been developed for graded algebras whose graded components are finitely generated over the ground ring. Hence, wherever we rely on Koszulness of \(A\), we assume the algebra \(K\) to be finitely generated. This is sufficient to ensure that \(A^m\) is finitely generated over \(\mathbb Q \).

  14. If we assume that \(\partial _A: \mathbb Q \mathcal{Y }\rightarrow R\) is injective (and hence the PVH Criterion is satisfied in degree 2) then level 3 syzygies must have at least degree 3 in the generators of \(A\). Given a level 3, degree 3 syzygy, we can also get level 3 syzygies of higher degree by pre- or postmultiplying all terms in the syzygy by monomials in the generators, although level 3 syzygies of higher degree need not all arise in this way (except when the algebra is Koszul).

  15. As per footnote 11, \(A\) need only be 2-Koszul, that is, its Koszul complex need only be exact up to homological degree 2 inclusive.

  16. Strictly speaking \(\mathfrak{pvb }_n^!\) is generated by dual generators \(\{r_{ij}^*\}\). To simplify the notation, we write \(r_{ij}\) instead of \(r_{ij}^*\).

  17. Instead of writing \(r_{ij}^{**}\) for generators of \(\mathfrak pvb _n^{!*}\), we write \(r_{ij}\).

  18. In fact, the reductions (a\(^{\prime }\)) and (b\(^{\prime }\)) may really involve two reductions, applicable to different terms.

  19. As per the previous footnote, note that reductions (a\(^{\prime }\)) and (b\(^{\prime }\)), indicated by the stars in the RHS of the first lines, actually involve two reductions, applicable to separate terms.

  20. See [25] for more on such bases.

  21. See the proof of Corollary 4.6, (iii). Our ordered 2-step partitions differ from their ‘2-step partitions’ in that our underlying sets \(S_i\) are cyclically ordered and theirs are unordered.

  22. The Down and Up graphs correspond respectively to red and black graphs in the terminology used in the definition of 2-step partition immediately prior to Proposition 4.5 of Bartholdi et al. [6].

  23. Strictly speaking \(\mathfrak{pvb }_n^!\) is generated by dual generators \(\{r_{ij}^*\}\). As per footnote 16, and to simplify the notation, we write \(r_{ij}\) instead of \(r_{ij}^*\).

  24. Theorem 6.16.

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Acknowledgments

Many thanks to D. Bar-Natan, A. Polishchuk, L. Positselski, P. Etingof and E. Rains for their feedback on the paper and/or very helpful discussions. Thanks also to the referee for a very thorough reading of the paper, which revealed a number of inaccuracies, and for making some helpful suggestions to improve the presentation.

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    Peter Lee

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Lee, P. The pure virtual braid group is quadratic. Sel. Math. New Ser. 19, 461–508 (2013). https://doi.org/10.1007/s00029-012-0107-1

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