Homology of free nilpotent Lie rings

V. R. Romanovskiǐ

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 34, Tome 478 (2019), pp. 202-210

Résumé

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 $H_4(L(x_1,x_2)/\gamma_{6})\cong \mathbb Z^{85}\oplus \mathbb Z/7$.