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Keywords:
Hom-Lie algebra; extension of Hom-Lie algebras and its direct limit
Summary:
We prove that the universal central extension of a direct limit of perfect Hom-Lie algebras $(\mathcal {L}_i, \alpha _{\mathcal {L}_i})$ is (isomorphic to) the direct limit of universal central extensions of $(\mathcal {L}_i, \alpha _{\mathcal {L}_i})$. As an application we provide the universal central extensions of some multiplicative Hom-Lie algebras. More precisely, we consider a family of multiplicative Hom-Lie algebras $\{({\rm sl}_{k}(å), \alpha _k)\}_{k\in I}$ and describe the universal central extension of its direct limit.
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