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

TL;DR

This paper develops a theory of two-valued algebraic group structures on the complex plane and projective line, introducing new multiplication laws via Buchstaber and Kontsevich polynomials, and presents a novel construction of such a group.

Contribution

It introduces a new two-valued algebraic group structure on the complex plane, distinct from existing coset-based constructions, using polynomial-based multiplication laws.

Findings

Development of a two-valued algebraic group theory on complex lines

Introduction of Buchstaber and Kontsevich polynomials for multiplication laws

Construction of a novel two-valued algebraic group different from known methods

Abstract

The theory of a two-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 our studies is a simple construction of a two-valued algebraic group on $\mathbb C$ different from known coset-construction.