Solutions of the braid equation attached to power series

Abstract

Let K be an algebraically closed characteristic 0 field and let V be the K-vector space with basis $\{x_0,x_1,x_2,\dots \}$. In this paper we use the theory developed in [17] to associate with each power series $f\in K[[X]]$ satisfying $f(0)=1$, an involutive solution $s(f):V\otimes V\to V\otimes V$, of the braid equation, and we begin the study of this correspondence. For $n\ge 2$, let V_n be the subspace of V generated by $\{x_0,\dots ,x_{n-1}\}$. Each one of the solution s(f) induces by restriction a solution $s_n(f)$ on $V_n\otimes V_n$. We also begin the study of these solutions.

KEYWORDS:

• Braid equation
• coalgebras
• non-degenerate solution

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16T25
• 16T15

Additional information

Funding

Jorge A. Guccione and Juan J. Guccione were supported by CONICET PIP 2021-2023 GI,11220200100423CO and CONCYTEC-FONDECYT within the framework of the contest “Proyectos de Investigación Básica 2020-01” [contract number 120-2020-FONDECYT]. Christian Valqui was supported by CONCYTEC-FONDECYT within the framework of the contest “Proyectos de Investigación Básica 2020-01” [contract number 120-2020-FONDECYT].