Hopf algebras and transcendental numbers

Abstract : In dealing with multiple zeta values, the main diophantine challenge is to prove that known dependence relations among them suffice to deduce all algebraic relations. One tool which should be relevant is the structure of Hopf Algebras, which occurs in several disguises in this context. How to use it is not yet clear, but we point out that it already plays a role in transcendental number theory: Stéphane Fischler deduces interpolation lemmas from zero estimates by using a duality involving bicommutative (commutative and cocommutative) Hopf Algebras. In the first section we state two transcendence results involving values of the exponential function; they are special cases of the linear subgroup Theorem which deals with commutative linear algebraic groups. In the second section, following S.~Fischler, we explain the connection between the data of the linear subgroup Theorem and bicommutative Hopf algebras of finite type. In the third and last section we introduce non-bicommutative Hopf algebras related to multiple zeta values.
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Michel Waldschmidt. Hopf algebras and transcendental numbers. international conference ``Zeta-functions, Topology and Quantum Physics 2003'', Mar 2003, Kinki, Japan. pp.197-219. ⟨hal-00411351⟩

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