On automorphisms of graded quasi-Lie algebras. (English) Zbl 1499.17019
Summary: Let \(\mathbb Z\) be the ring of integers and let \(K(\mathbb Z,2n)\) denote the Eilenberg-MacLane space of type \((\mathbb Z,2n)\) for \(n\ge 1\). In this article, we prove that the graded group
\[
A_m:= \operatorname{Aut}(\pi_{\le 2mn+1}(\Sigma K(\mathbb Z,2n))/\text{torsions})
\]
of automorphisms of the graded quasi-Lie algebras \(\pi_{\le 2mn+1}(\Sigma K(\pi_{\le 2mn+1},2n))\) modulo torsions that preserve the Whitehead products is a finite group for \(m\le 2\) and an infinite group for \(m\ge 3\), and that the group \(\operatorname{Aut}(\pi_*(\Sigma K(\mathbb Z,2n))/\text{torsions})\) is non-abelian. We extend and apply those results to techniques in localization (or rationalization) theory.
MSC:
| 17B70 | Graded Lie (super)algebras |
| 17B01 | Identities, free Lie (super)algebras |
| 55P20 | Eilenberg-Mac Lane spaces |
| 55P60 | Localization and completion in homotopy theory |
| 55S37 | Classification of mappings in algebraic topology |
Keywords:
automorphisms; commutator; Eilenberg-MacLane space; rational homotopy Lie algebra; localization; rationalization; graded quasi-Lie algebraReferences:
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