Cohomology of Hopf algebras and Massey products

V. M. Buchstaber ; F. Yu. Popelenskii

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 79 (2024) no. 4, pp. 567-648

Résumé

The theory of the trigraded Buchstaber spectral sequence $\operatorname{Bss}$ for graded Hopf algebras is developed. It is shown that the differentials of $\operatorname{Bss}$ define an increasing exhaustive filtration as a new structure in the cohomology of Hopf algebras. This structure is described explicitly for a number of known Hopf algebras. For the tensor algebra $T(s \operatorname{Ext}^{1,*}_{A}(\Bbbk,\Bbbk))$ of the suspension of the one-dimensional cohomology of a Hopf algebra $A$ over a field $\Bbbk$, the construction of partial multivalued operations $\operatorname{Bss}_p$, $p\geqslant 1$, is presented. This construction is used to describe the differentials in the spectral sequence $\operatorname{Bss}$ and the exhaustive filtration in $\operatorname{Ext}_{A}^{*,*}(\Bbbk,\Bbbk)$. It is shown that the structure introduced is an effective tool for solving several well-known problems: (1) realising cohomology classes of Hopf algebras by Massey products; (2) interpreting differentials in $\operatorname{Bss}$ as Massey operations; (3) effective construction of a certain class of Massey products in the form of differentials in $\operatorname{Bss}$. Bibliography: 74 titles.

Détail

Keywords: Hopf algebras, Landweber–Novikov algebra, Buchstaber spectral sequence, Eilenberg–Moore spectral sequence, $\operatorname{Bss}$-operations, cohomology of nilmanifolds.