Leibniz homology and the Hilton-Milnor theorem
Leibniz algebras and Leibniz homology were introduced by Loday as a non-commutative analogue of Lie algebras and Lie homology. In a previous paper [Math. Nachr. 175, 209-229 (1995; Zbl 0843.55014)], the author showed that a direct sum decomposition of the chain complex for the Leibniz homology of the infinite matrices over a group ring \(M(\mathbb{Q}[G])\) corresponds to geometry -- more precisely, to the standard filtration of the James model \(J(BG)\simeq\Omega\Sigma(BG)\) for the classifying space of the group \(G\). In this paper, he shows how Loday's Künneth formula for Leibniz homology [\textit{J.-L. Loday}, Künneth-style formula for the homology of Leibniz algebras, Math. Z. 221, No. 1, 41-47 (1996)] corresponds to the Hilton-Milnor theorem. ``The Hilton-Milnor theorem establishes a homotopy equivalence, via Samelson products, between \(\Omega\Sigma(X\vee Y)\) and the Cartesian product of other loop spaces. We show how a type of `Samelson product' can be defined on the chain level for Leibniz homology, which upon iteration becomes \(\alpha\), the inverse of Loday's Künneth theorem isomorphism. Moreover, we trace \(\alpha\) through the geometric construction of the author's previous paper (loc. cit.) and show that on Leibniz homology, \(\alpha\) covers the map on singular homology induced by the Hilton-Milnor homotopy equivalence. In this sense, the Künneth theorem for Leibniz homology is an algebraic version of the Hilton-Milnor theorem in homotopy theory''.
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