an:02011916
Zbl 1044.11057
Wojtkowiak, ZdzisĹ‚aw
Non-abelian unipotent periods and monodromy of iterated integrals.
EN
J. Inst. Math. Jussieu 2, No. 1, 145-168 (2003).
1474-7480 1475-3030
2003
j
11G55 14F35 11M41 17B56 19E20 19F27
polylogarithms; iterated integrals; monodromy; Lie algebras; periods; fundamental group
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.
Mina Teicher (Ramat Gan)