V. M. Buchstaber, I. Yu. Gaiur, V. N. Rubtsov, “Algebraic $2$-valued group structures on $\mathbb P^1$, Kontsevich-type polynomials, and multiplication formulas. I”, Izv. RAN. Ser. Mat., 2026, Volume 90, Issue 1,Pages 37

Algebraic $2$-valued group structures on $\mathbb P^1$, Kontsevich-type polynomials, and multiplication formulas. I

V. M. Buchstaber^a, I. Yu. Gaiur^b, V. N. Rubtsov^c

^a National Research University Higher School of Economics, Moscow
^b Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France
^c Laboratoire Angevin de Recherche en Mathématiques, Université d'Angers, Angers, France

Abstract: The theory of a $2$-valued algebraic group structure on a complex plane and complex projective line is developed. In this theory, depending on the choice of the neutral element, the local multiplication law is given by the Buchstaber polynomial or the generalized Kontsevich polynomial. One of the most exciting results of the first part of our work is a simple construction of a $2$-valued algebraic group structure on $\mathbb C$ different from well known coset-construction.

Keywords: Kontsevich polynomials, $2$-valued groups, Mobius transform.

MSC: Primary 14H45; Secondary 20N99

Received: 04.01.2025
Revised: 18.03.2025

Language: English

DOI: 10.4213/im9691