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

Khalili, Valiollah

Czech. Math. J. 69, No. 1, 275-293 (2019).

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.

MSC:

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

Keywords:

Hom-Lie algebra; extension of Hom-Lie algebras and its direct limit

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