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

Dengyin Wang and Xian Wang

Address:
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

E-mail: wdengyin@126.com

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.