Racinet studied the scheme associated with the double shuffle and regularization relations between multiple polylogarithm values at Nth roots of unity and constructed a group scheme attached to the situation. Then, Enriquez and Furusho proved that this group scheme can be essentially identified with the stabilizer of a coproduct element arising in Racinet’s theory with respect to the action of a group of automorphisms of a free Lie algebra. We reformulate Racinet’s construction in terms of crossed products. We then establish a cyclotomic analogue of a stabilizer inclusion arising from Enriquez and Furusho’s work. This yields an explicit group scheme containing Racinet’s double shuffle group scheme, which we also express in the Racinet formalism.
@article{yad1,title={Crossed product interpretation of the double shuffle {L}ie algebra attached to a finite abelian group},author={Yaddaden, Khalef},journal={Publications of the Research Institute for Mathematical Sciences},volume={60},number={3},pages={455–505},year={2024},doi={https://doi.org/10.4171/prims/60-3-2},url={https://ems.press/journals/prims/articles/14298188},}
The cyclotomic double shuffle torsor in terms of Betti and de Rham coproducts
We prove a cyclotomic generalization of Enriquez and Furusho’s result stating that the scheme of double shuffle and regularization relations between multiple zeta values arising from Racinet’s formalism is a torsor of isomorphisms between “Betti” and “de Rham” objects. After reviewing the “de Rham” objects arising from a crossed product interpretation of Racinet’s formalism, we construct the “Betti” objects based on an orbifold fundamental group. We then show that the scheme of double shuffle and regularization relations is a subtorsor of stabilizer schemes relating “Betti” and “de Rham” coproducts.
@article{yad2,title={The cyclotomic double shuffle torsor in terms of Betti and de Rham coproducts},author={Yaddaden, Khalef},journal={Journal of Pure and Applied Algebra},volume={228},number={5},pages={107550},year={2024},issn={0022-4049},doi={https://doi.org/10.1016/j.jpaa.2023.107550},url={https://www.sciencedirect.com/science/article/pii/S0022404923002323},}
Preprints
2024
On a conjecture of Zhao related to standard relations among cyclotomic multiple zeta values
We provide a proof of a conjecture by Zhao concerning the structure of certain relations among cyclotomic multiple zeta values in weight two. We formulate this conjecture in a broader algebraic setting in which we give a natural equivalence between two schemes attached to a finite abelian group G. In particular, when G is the group of roots of unity, these schemes describe the standard relations among cyclotomic multiple zeta values.
@misc{by,title={On a conjecture of Zhao related to standard relations among cyclotomic multiple zeta values},author={Bachmann, Henrik and Yaddaden, Khalef},year={2024},eprint={2411.18952},archiveprefix={arXiv},primaryclass={math.NT},url={https://arxiv.org/abs/2411.18952},}
We reformulate in terms of crossed product Racinet’s construction related to the double shuffle and regularization relations between multiple polylogarithm values at roots of unity. Furthermore, for mutiple zeta values, Enriquez and Furusho showed that a double shuffle scheme is a torsor of isomorphisms relating de Rham and Betti objects. We generalize this result to the cyclotomic case.
Initiation to Mumford’s Geometric Invariant Theory which is caracterized by actions of reductive groups on algebraic varieties. Then give a decomposition of the algebraic variety into a finite number of partitions which allows to a better understanding of the variety. This work is based on this paper of Victoria Hoskins.
We study the set-theoric groupoid structure and equip it with differential structure obtaining what is called a Lie groupoid. Then we consider its action on a smooth manifold. This aims to equip a Lie groupoid with a Riemannian metric with respect to the underlying metrics on the structure. This work is based on this paper of Matias L. del Hoyo, Rui Loja Fernandes.
