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

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 -bialgebra structures on the power series algebra for certain central simple non-associative algebras . These structures are closely related to a version of the classical Yang-Baxter equation (CYBE) over . If is a Lie algebra, we obtain new proofs for pivotal steps in the known classification of non-degenerate topological Lie bialgebra structures on as well as of non-degenerate solutions of the usual CYBE. If is associative, we achieve the classification of non-triangular topological balanced infinitesimal bialgebra structures on as well as of all non-degenerate solutions of an associative version of the CYBE.
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