  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.
