
Journal of Lie Theory 33 (2023), No. 3, 783798 Copyright Heldermann Verlag 2023 Extending Structures for Lie Bialgebras Yanyong Hong School of Mathematics, Hangzhou Normal University, Hangzhou, China yyhong@hznu.edu.cn [Abstractpdf] Let $(\mathfrak{g}, [\cdot,\cdot], \delta_\mathfrak{g})$ be a fixed Lie bialgebra and $V$ be a vector space. In this paper, we introduce the notion of a unified biproduct of $(\mathfrak{g}, [\cdot,\cdot], \delta_\mathfrak{g})$ by $V$ and give a theoretical answer to the extending structures problem, i.e. how to classify all Lie bialgebraic structures on $E=\mathfrak{g}\oplus V$ such that $(\mathfrak{g}, [\cdot,\cdot], \delta_\mathfrak{g})$ is a Lie subbialgebra up to an isomorphism of Lie bialgebras whose restriction on $\mathfrak{g}$ is the identity map. Moreover, several special unified biproducts are also introduced. In particular, the unified biproducts when $\text{dim} V=1$ are investigated in detail. Keywords: Lie bialgebra, extending structure. MSC: 17A30, 17B62, 17B65, 17B69. [ Fulltextpdf (138 KB)] for subscribers only. 