Homology of Free Nilpotent Lie Rings

Abstract

Some results obtained by the calculations of the integral homology of free nilpotent Lie algebras Hi (L(x1, x2,. . . , xr)/γN+1) in the system for computational discrete algebra GAP are presented. The attention is focused on the occurrence of unexpected torsion in this homology, similar to one that arises for a 4-generated free nilpotent group of class 2. The main result is that even for the case of two generators, the torsion occurs in the fourth integral homology when the nilpotency class is 5. Moreover, only 7-torsion occurs, and no others. Namely, there is an isomorphism H4(L(x1, x2)/γ6) ≅ ℤ85 ⨁ ℤ /7ℤ. We also give an explicit mathematical proof of the fact that H4(L(x1, x2, x3, x4)/γ3) has nontrivial 3-torsion. Bibliography: 6 titles.