Dual-depth Adapted Irreducible Formal Multizeta Values - Schneps, Leila (Mathematica Scandinavica, 2013)
Let ds denote the double shuffle Lie algebra, equipped with the standard weight grading and depth filtration; we write ds=_n3 ds_n and denote the filtration by ds^1ds^2…. The double shuffle Lie algebra is dual to the new formal multizeta space nfz=_n3 nfz_n, which is equipped with the dual depth filtration nfz^1nfz^2… Via an explicit canonical isomorphism ds→nfz, we define the "dual" in nfz of an element in ds. For each weight n3 and depth d1, we then define the vector subspace ds_n,d of ds as the space of elements in ds_n^d-ds_n^d+1 whose duals lie in nfz_n^d. We prove the direct sum decomposition \[ ds=_n3_d1 ds_n,d, \] oindent which yields a canonical vector space isomorphism between ds and its associated graded for the depth filtration, ds_n,dds_n^d/ ds_n^d+1. A basis of ds respecting this decomposition is dual-depth adapted, which means that it is adapted to the depth filtration on ds, and the basis of dual elements is adapted to the dual depth filtration on nfz. We use this notion to give a canonical depth 1 generator f_n for ds in each odd weight n3, namely the dual of the new formal single zeta value ζ(n)nfz_n. At the end, we also apply the result to give canonical irreducibles for the formal multizeta algebra in weights up to 12.