Non-abelian unipotent periods and monodromy of iterated integrals. (English) Zbl 1044.11057

The author studies the Lie algebras associated to non-abelian unipotent periods on \(P^1_{Q(\mu_n)}\setminus\{0, \mu_n,\infty\}\). Let \(n\) be a prime number. For any \(m\geq 1\), the numbers \(\text{Li}_{m+1}(\xi^k_n)\) for \(1\leq k\leq (n- 1)/2\) are assumed linearly independent over \(\mathbb{Q}\) in \(\mathbb{C}/(2\pi i)^{m+1}\mathbb{Q}\). Let \(S= \{k_1,\dots, k_q\}\) be a subset of \(\{1,\dots, p-1\}\) such that if \(k\in S\), then \(p-k\in S\) and \((S+S)\cap S= \emptyset\) (the sum of two elements of \(S\) is calculated \(\text{mod\,}p\)). Then the author shows that in the Lie algebra associated to non-abelian unipotent periods on \(P_{Q(\mu_n)}\setminus \{0,\mu_n, \infty\}\) there are derivations \(D^{k_1}_{m+1},\dots, D^{k_q}_{m+1}\) in each degree \(m+1\) and these derivations are free generators of a free Lie subalgebra of this Lie algebra.


11G55 Polylogarithms and relations with \(K\)-theory
14F35 Homotopy theory and fundamental groups in algebraic geometry
11M41 Other Dirichlet series and zeta functions
17B56 Cohomology of Lie (super)algebras
19E20 Relations of \(K\)-theory with cohomology theories
19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
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