## 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.

### MSC:

 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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