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Journal of Lie Theory 33 (2023), No. 3, 783--798
Copyright Heldermann Verlag 2023

Extending Structures for Lie Bialgebras

Yanyong Hong
School of Mathematics, Hangzhou Normal University, Hangzhou, China


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 bi-product 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 sub-bialgebra up to an isomorphism of Lie bialgebras whose restriction on $\mathfrak{g}$ is the identity map. Moreover, several special unified bi-products are also introduced. In particular, the unified bi-products when $\text{dim} V=1$ are investigated in detail.

Keywords: Lie bialgebra, extending structure.

MSC: 17A30, 17B62, 17B65, 17B69.

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