Derivations of the subalgebras intermediate the general linear Lie algebra and the diagonal subalgebra over commutative rings

Dengyin Wang and Xian Wang

Department of Mathematics China University of Mining and Technology Xuzhou, 221008, People’s Republic of China
Graduate School of Natural Science and Technology Okayama University Okayama 700-8530, Japan


Abstract: Let $R$ be an arbitrary commutative ring with identity, $\operatorname{gl}(n,R)$ the general linear Lie algebra over $R$, $d(n,R)$ the diagonal subalgebra of $\operatorname{gl}(n,R)$. In case 2 is a unit of $R$, all subalgebras of $\operatorname{gl}(n,R)$ containing $d(n,R)$ are determined and their derivations are given. In case 2 is not a unit partial results are given.

AMSclassification: primary 13C10; secondary 17B40, 17B45.

Keywords: the general linear Lie algebra, derivations of Lie algebras, commutative rings.