On generalized Witt algebras in one variable

We study a class of infinite dimensional Lie algebras called generalized Witt algebras (in one variable). These include the classical Witt algebra and the centerless Virasoro algebra as important examples. We show that any such generalized Witt algebra is a semisimple, indecomposable Lie algebra which does not contain any abelian Lie subalgebras of dimension greater than one. We develop an invariant of these generalized Witt algebras called the spectrum, and use it to show that there exist infinite families of nonisomorphic, simple, generalized Witt algebras and infinite families of nonisomorphic, nonsimple, generalized Witt algebras. We develop a machinery that can be used to study the endomorphisms of a generalized Witt algebra in the case that the spectrum is "discrete". We use this to show, that among other things, every nonzero Lie algebra endomorphism of the classical Witt algebra is an automorphism and every endomorphism of the centerless Virasoro algebra fixes a canonical element up to scalar multiplication. However, not every injective Lie algebra endomorphism of the centerless Virasoro algebra is an automorphism.