Extending structures for Gel'fand-Dorfman bialgebras

TL;DR

This paper studies how to extend Gel'fand-Dorfman bialgebras by classifying all possible algebraic structures on larger spaces that contain a given bialgebra as a subalgebra, using a new unified product approach.

Contribution

It introduces a novel framework and the object al{GH}^2(V,A) for classifying extending structures of Gel'fand-Dorfman bialgebras, generalizing existing theories.

Findings

Constructed the al{GH}^2(V,A) object for classification

Developed the unified product concept for Gel'fand-Dorfman bialgebras

Analyzed the case when the complement has dimension one

Abstract

Gel'fand-Dorfman bialgebra, which is both a Lie algebra and a Novikov algebra with some compatibility condition, appears in the study of Hamiltonian pairs in completely integrable systems and a class of special Lie conformal algebras called quadratic Lie conformal algebras. In this paper, we investigate the extending structures problem for Gel'fand-Dorfman bialgebras, which is equivalent to some extending structures problem of quadratic Lie conformal algebras. Explicitly, given a Gel'fand-Dorfman bialgebra $(A, \circ, [\cdot,\cdot])$, this problem asks that how to describe and classify all Gel'fand-Dorfman bialgebraic structures on a vector space $E$ $(A\subset E$) such that $(A, \circ, [\cdot,\cdot])$ is a subalgebra of $E$ up to an isomorphism whose restriction on $A$ is the identity map. Motivated by the theories of extending structures for Lie algebras and Novikov algebras, we construct an object $\mathcal{GH}^2(V,A)$ to answer the extending structures problem by introducing a definition of unified product for Gel'fand-Dorfman bialgebras, where $V$ is a complement of $A$ in $E$. In particular, we investigate the special case when $\text{dim}(V)=1$ in detail.