Zinbiel algebras and multiple zeta values

Abstract

Multiple zeta values are the convergent iterated integrals from 0 to 1 of the differential forms ω0 = dt/t and ω1 = dt/(1− t). They form an algebra over Q, which has many interesting connections with different domains, including knot theory and perturbative quantum field theory [18, 11]. This algebra is expected to be graded by the weight, and a famous conjecture of Zagier [19] states that the dimensions of homogeneous components are given by the Padovan numbers. The algebra AMZV of motivic multiple zeta values is a more subtle construction, in the setting of periods and mixed motives [5, 6, 11]. It can be defined as the quotient of the commutative algebra A1,0, whose elements are seen as formal iterated integrals of ω0 and ω1, by the non-explicit ideal of all relations that can be proved using algebraic geometry. This algebra is known to be graded by the weight and its dimensions are given by the Padovan sequence, by results of Brown [5]. There is a surjective morphism, called the period map, from the motivic algebra AMZV to the usual algebra of multiple zeta values, defined by taking the numerical value of a formal iterated integral. This period map is expected to be injective, hence an isomorphism.