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Derivations of the intermediate Lie algebras between the Lie algebra of diagonal matrices and that of upper triangular matrices over a commutative ring. (English) Zbl 1161.17312

Summary: Let \(R\) be an arbitrary commutative ring with identity, \(\text{gl}(n, R)\) the general linear Lie algebra over \(R\) consisting of all \(n\times n\) matrices over \(R\) and with the bracket operation \([x, y] = xy - yx\), \(\mathbf t\) (resp., \(\mathbf u\)) the Lie subalgebra of \(\text{gl}(n, R)\) consisting of all \(n\times n\) upper triangular (resp., strictly upper triangular) matrices over \(R\) and \(\mathbf d\) the Lie subalgebra of \(\text{gl}(n, R)\) consisting of all \(n\times n\) diagonal matrices over \(R\). The aim of this article is to give an explicit description of the derivation algebras of the intermediate Lie algebras between \(\mathbf d\) and \(\mathbf t\).

MSC:

17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
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