Cohomology of the universal enveloping algebras of certain bigraded Lie algebras
The calculation of \(\mathrm{Ext}_{\mathcal A}^{s,t}(\mathbb Z_p,\mathbb Z_p)\), where \(p\) is an odd prime and \(\mathcal{A}\) the mod \(p\) Steenrod algebra, is of great importance for determining the stable homotopy groups of spheres (via the Adams spectral sequence). Following [\textit{J. P. May}, J. Algebra 3, 123--146 (1966; Zbl 0163.03102)], the authors consider an easier problem of determining \(\mathrm{Ext}_{\mathbb P}^{s,t}(\mathbb Z_p,\mathbb Z_p)\), where \(\mathbb P\) is the Hopf subalgebra of \(\mathcal A\) generated by the reduced power operations \(\mathcal P^i\) (\(i\geq0\)). The augmentation ideal of \(\mathbb P\) is used to define a filtration on \(\mathbb P\), which gives a graded Hopf algebra \(E^0\mathbb P\). Let \(L\) be the restricted Lie algebra consisting of the primitive elements of \(E^0\mathbb P\). The main result of this paper concerns the cohomology \(H^{*,*}(U(L))\) of the universal enveloping algebra of \(L\). The authors present a list of multiplicative generators of \(H^{s,t}(U(L))\), as well as the nontrivial products among them, up to total degree \(t-s<\max\{(5p^3+6p^2+6p+4)q-10,p^4q\}\). They use this, along with some spectral sequences, to obtain an upper bound for \(\mathrm{rank}(\mathrm{Ext}_{\mathbb P}^{s,t}(\mathbb Z_p,\mathbb Z_p))\). They also point out that this cohomology satisfies a Poincaré duality property.
scientific article; zbMATH DE number 6996450
scientific article