Non-abelian tensor and exterior products of multiplicative Lie rings (Q524703)

The main goal of the paper is the introduction of the notion of non-abelian tensor product of multiplicative Lie rings and to show that this concept generalizes the corresponding notions of non-abelian tensor product of groups in [\textit{R. Brown} and \textit{J.-L. Loday}, Topology 26, 311--335 (1987; Zbl 0622.55009)] and for Lie algebras [\textit{G. J. Ellis}, Glasg. Math. J. 33, No. 1, 101--120 (1991; Zbl 0724.17016)], together with the properties of these tensor products. After that, an analogue of Miller's theorem for free multiplicative Lie rings is proved and a relation between the second homology and the non-abelian exterior square of multiplicative Lie rings is obtained. The paper concludes with a question about the description of the third homology group of a multiplicative Lie ring in terms of non-abelian exterior product. The authors provide the following partial answer to this question: Let \(1 \to \mathfrak{r} \to \mathfrak{f} \to \mathfrak{g} \to 1\) be a free presentation of a multiplicative Lie ring \(\mathfrak{g}\). Assume that \(\mathrm{Ker}(\mathfrak{r}' \wedge \mathfrak{f}' \to \mathfrak{f}' \wedge \mathfrak{f}') = 1\) for all free presentations \(1 \to \mathfrak{r}' \to \mathfrak{f}' \to \mathfrak{f} \to 1\). Then there exists an isomorphism \(HS_3^{\mathrm{mlr}}( \mathfrak{g} ) \cong \text{Ker} (\mathfrak{r} \wedge \mathfrak{f} \to \mathfrak{f} \wedge \mathfrak{f})\), where \(HS_3^{\mathrm{mlr}}( \mathfrak{g} )\) denotes the third strong homology group of \(\mathfrak{g}\).