Geodesic
Homology of free nilpotent Lie rings
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 34, Tome 478 (2019), pp. 202-210 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper presents the results of calculations of integer homology of free nilpotent Lie algebras $H_i(L(x_1,\dots,x_r)/\gamma_{N+1})$ in the system of computational algebra GAP. Our attention was focused on the occurrence of unexpected torsion in these homology, similar to the one that arises for $4$-generated free nilpotent groups of class $2$. The main result is that even for two generators torsion occurs in the fourth integer homology when the nilpotency class is $5$. Moreover, only a $7$-torsion occurs, and no others. Namely, there is an isomorphism $H_4(L(x_1,x_2)/\gamma_{6})\cong \mathbb Z^{85}\oplus \mathbb Z/7$. 
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     author = {V. R. Romanovskiǐ},
     title = {Homology of free nilpotent {Lie} rings},
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}
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V. R. Romanovskiǐ. Homology of free nilpotent Lie rings. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 34, Tome 478 (2019), pp. 202-210. http://geodesic.mathdoc.fr/item/ZNSL_2019_478_a9/

[1] J. Huebschmann, “Cohomology of nilpotent groups of class 2”, J. Algebra, 126 (1989), 400–450 | DOI | MR | Zbl

[2] K. Igusa, K. E. Orr, “Links, pictures and the homology of nilpotent groups”, Topology, 40 (2001), 1125–1166 | DOI | MR | Zbl

[3] L. Lambe, “Cohomology of principal G-bundles over a torus when H(BG; R) is polynomial”, Bull. Soc. Math. de Belg., 38 (1986), 247–264 | MR | Zbl

[4] C. Reutenauer, Free Lie algebras, London Mathematical Society Monographs, 7, Clarendon Press, Oxford, 1993 | MR | Zbl

[5] A. Kartan, S. Eilenberg, Gomologicheskaya algebra, 1960

[6] Yu. V. Kuzmin, Yu. S. Semenov, “O gomologiyakh svobodnoi nilpotentnoi gruppy klassa 2”, Matem. sb., 189:4 (1998), 49–82 | DOI | Zbl