Universal central extension of direct limits of Hom-Lie algebras. (English) Zbl 07088784

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 \(\{(\mathrm{sl}_k(\mathcal{A}),\alpha_k)\}_{k\in I}\) and describe the universal central extension of its direct limit.


17A30 Nonassociative algebras satisfying other identities
17B55 Homological methods in Lie (super)algebras
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
17B99 Lie algebras and Lie superalgebras
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