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Lie Dimension Subrings

We compare, for L a Lie ring over the integers, its lower central series (γ_n(L))_{n>0} and its dimension series defined by δ_n(L):=L\cap \varpi^n(L) in the universal enveloping algebra of L. We show that γ_n(L)=δ_n(L) for all n<4, but give an example showing that they may differ if n=4. We introduce simplicial methods to describe these results, and to serve as a possible tool for further study of the dimension series.