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We introduce a new Lie-algebraic approach to explicitly construct the motiviccoaction and single-valued map of multiple polylogarithms in any number ofvariables. In both cases, the appearance of multiple zeta values is controlledby conjugating generating series of polylogarithms with Lie-algebra generatorsassociated with odd zeta values. Our reformulation of earlier constructions ofcoactions and single-valued polylogarithms preserves choices of fibrationbases, exposes the correlation between multiple zeta values of different depthsand paves the way for generalizations beyond genus zero.

Motivic coaction and single-valued map of polylogarithms from zeta generators