Cohomology of the universal enveloping algebras of certain bigraded Lie algebras
type: Item
instance of: scholarly article
title: Cohomology of the universal enveloping algebras of certain bigraded Lie algebras
zbMATH Open document ID: 1409.55015
author: Linan Zhong
author: Wenhuai Shen
author: Hao Zhao
publication date: 2018-12-21T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 6996450
zbMATH Keywords: Steenrod algebra
zbMATH Keywords: Hopf algebra
zbMATH Keywords: Lie algebra
zbMATH Keywords: spectral sequence
zbMATH Keywords: stable homotopy groups of spheres
reviewed by: Branislav Prvulović
MaRDI profile type: Publication
full work available at URL: s10114-018-7273-9
OpenAlex ID: W2789442931
cites work: 3-dimensional cohomology of the mod $p$ Steenrod algebra.
cites work: The double suspension and exponents of the homotopy groups of spheres
cites work: The factorization of cyclic reduced powers by secondary cohomology operations
cites work: The cohomology of restricted Lie algebras and of Hopf algebras
cites work: On the structure of Hopf algebras
cites work: The Smith-Toda complex \(V((p + 1)/2)\) does not exist
cites work: Q3747338
cites work: On spectra realizing exterior parts of the Steenrod algebra
cites work: Convergence of the generators of \(H^{*,*} (U (L))\)
DOI: 10.1007/S10114-018-7273-9
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6996450