| Abstract: | ABSTRACT Pairs (A, L) with A a commutative algebra and L a Lie algebra acting on A by derivations, called Lie algops, are studied as algebraic structures over arbitrary fields of arbitrary characteristic. Lie algops possess modules and tensor products—and are considered with respect to a central simple theory. The simplicity problem of determining the faithful unital simple Lie algops ( A, L ) is of interest since the corresponding Lie algebras AL are usually simple (Jordan, 2000
Jordan , D. A. ( 2000 ). On the simplicity of Lie algebras of derivations of commutative algebras . J. Algebra 228 : 580 – 585 . CSA] Crossref], Web of Science ®] , Google Scholar]). For locally finite Lie algops, and up to purely inseparable descent, this problem reduces by way of closures to the closed central simplicity problem of determining those which are closed central simple. The simplicity and representation theories for locally nilpotent separably triangulable unital Lie algops are of particular interest because they relate to the problems of classifying simple Lie algebras of Witt type and their representations. Of these, the simplicity theory reduces to that of Jordan Lie algops. The main Theorems 7.3 and 7.4 reduce the simplicity and representation theories for Jordan Lie algops to the simplicity and representation theories for simple nil and toral Lie algops. |
