A bialgebra theory of Gel'fand-Dorfman algebras with applications to Lie conformal bialgebras
arxiv(2024)
摘要
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.
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