Let $(mathfrak{g}, [cdot,cdot], delta_mathfrak{g})$ be a fixed Lie bialgebra, $E$ be a vector space containing $mathfrak{g}$ as a subspace and $V$ be a complement of $mathfrak{g}$ in $E$. A natural problem is that how to classify all Lie bialgebraic structures on $E$ such that $(mathfrak{g}, [cdot,cdot], delta_mathfrak{g})$ is a Lie sub-bialgebra up to an isomorphism of Lie bialgebras whose restriction on $mathfrak{g}$ is the identity map. This problem is called the extending structures problem. In this paper, we introduce a general co-product on $E$, called the unified co-product of $(mathfrak{g},delta_mathfrak{g})$ by $V$. With this unified co-product and the unified product of $(mathfrak{g}, [cdot,cdot])$ by $V$ developed in cite{AM1}, the unified bi-product of $(mathfrak{g}, [cdot,cdot], delta_mathfrak{g})$ by $V$ is introduced. Moreover, we show that any $E$ in the extending structures problem is isomorphic to a unified bi-product of $(mathfrak{g}, [cdot,cdot], delta_mathfrak{g})$ by $V$. Then an object $mathcal{HBI}_{mathfrak{g}}^2(V,mathfrak{g})$ is constructed to classify all $E$ in the extending structures problem. Moreover, several special unified bi-products are also introduced. In particular, the unified bi-products when $text{dim} V=1$ are investigated in detail.
تحميل البحث