2024/01/24 by Y.S. Hong, Hong, Yangyon, Chengming Bai +3
Mathematics · Physics and Astronomy · #17A30 #17B38 #17B62 #17B69 #17D25 #81T40 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2401.13608
openalex publication_date 2024/01/24 · openalex created_date 2024/01/26 · openalex updated_date 2026/07/28
Gel'fand-Dorfman algebras (GD algebras) give a natural construction of Lie conformal algebras and are in turn characterized by this construction. In this paper, we define the Gel'fand-Dorfman bialgebra (GD bialgebras) and enrich the above construction to a construction of Lie conformal bialgebras by GD bialgebras. As a special case, Novikov bialgebras yield Lie conformal bialgebras. We further introduce the notion of the Gel'fand-Dorfman Yang-Baxter equation (GDYBE), whose skew-symmetric solutions produce GD bialgebras. Moreover, the notions of O-operators on GD algebras and pre-Gel'fand-Dorfman algebras (pre-GD algebras) are introduced to provide skew-symmetric solutions of the GDYBE. The relationships between these notions for GD algebras and the corresponding ones for Lie conformal algebras are given. In particular, there is a natural construction of Lie conformal bialgebras from pre-GD algebras. Finally, GD bialgebras are characterized by certain matched pairs and Manin triples of GD algebras.