Structure of Polyzetas and Explicit Representation on Transcendence Bases of Shuffle and Stuffle Algebras

Abstract

Polyzetas, indexed by words, satisfy shuffle and quasi-shuffle identities. In this respect, one can explore the multiplicative and algorithmic (locally finite) properties of their generating series. In this paper, we construct pairs of bases in duality on which polyzetas are established in order to compute local coordinates in the infinite dimensional Lie groups where their non-commutative generating series live. We also propose new algorithms leading to the ideal of polynomial relations, homogeneous in weight, among polyzetas (the graded kernel) and their explicit representation (as data structures) in terms of irreducible elements.

About Some Drinfel’d Associators

“…• Invertible solutions of an equation of type S ′ = M 1 S are on the same orbit by multiplication on the right by invertible constant series i.e. let S i , i = 1, 2 be invertible solutions of (DE (1) ), then there exists an unique invertible T ∈ C X such that S 2 = S 1 .T . From this and point (iv) of the theorem, one can parametrize the set of invertible solutions of (DE 2 ).…”

“…. , s r ) as limits, fulfilling identities [14,13,1]. Firstly, they are limits of polylogarithms and secondly, as truncated sums, they are limits of harmonic sums, for z ∈ C, | z |< 1, N ∈ N + :…”

(Pure) transcendence bases in $ϕ$-deformed shuffle bialgebras

“…This series, in the factorized form, encompasses a large part of the combinatorics of Dyson's functional expansions in quantum field theory [18,34]. It is the infinite-dimensional analogue of the theorem of Wei and Norman [2,43,44].…”

On the solutions of the universal differential equation with three regular singularities (On solutions ofKZ3)

“…> . (2.5)4 Here, e denotes the counit defined by e( ) = ⟨ | 1 * ⟩ (for any ∈ ⟨ ⟩) 5. The dual family, i.e.…”

“…Identification allows to obtain homogenous polynomial relations up to weights 12[5] 23. by means of rewriting the system.…”