Locally finite-dimensional simple Lie algebras

In this paper the authors introduce a new class of simple infinite-dimensional Lie algebras: locally finite-dimensional Lie algebras. First, general properties of such algebras are studied, and an uncountable family of examples is pointed out. Then the investigation is restricted to simple locally finite-dimensional algebras whose classical finite-dimensional simple factors have totally bounded dimension. The structure of these algebras is determined in the case where the base field is algebraically closed of characteristic $p>7$.