Classification of $D$-bialgebra structures on power series algebras

TL;DR

This paper classifies $D$-bialgebra structures on power series algebras over certain central simple algebras using algebraic geometry, providing new proofs and extending classifications related to the classical Yang-Baxter equation.

Contribution

It introduces a geometric approach to classify $D$-bialgebra structures on power series algebras for both Lie and associative cases, extending existing results.

Findings

Classified non-degenerate topological Lie bialgebra structures on $A[[z]]$

Classified non-triangular balanced infinitesimal bialgebra structures for associative $A$

Provided new proofs for known classifications in the Lie case

Abstract

In this paper, we use algebro-geometric methods in order to derive classification results for so-called $D$-bialgebra structures on the power series algebra $A[\![z]\!]$ for certain central simple non-associative algebras $A$. These structures are closely related to a version of the classical Yang-Baxter equation (CYBE) over $A$. If $A$ is a Lie algebra, we obtain new proofs for pivotal steps in the known classification of non-degenerate topological Lie bialgebra structures on $A[\![z]\!]$ as well as of non-degenerate solutions of the usual CYBE. If $A$ is associative, we achieve the classification of non-triangular topological balanced infinitesimal bialgebra structures on $A[\![z]\!]$ as well as of all non-degenerate solutions of an associative version of the CYBE.