tailieunhanh - 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. | Turk J Math 35 (2011) , 405 – 436. ¨ ITAK ˙ c TUB doi: On generalized Witt algebras in one variable Ki-Bong Nam, Jonathan Pakianathan Abstract 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. Key Words: Infinite dimensional Lie algebra, Virasoro algebra 1. Introduction Throughout this paper, we will work over a field k of characteristic zero. Also note that there will be no finiteness constraints on the dimension of the Lie algebras in this paper — in fact, most of the Lie algebras that we will consider will be infinite dimensional. We now sketch the basic results and ideas of this paper in this introductory section. Precise definitions of the concepts can be found within the paper. Let R be the field of fractions of the power series algebra k[[x]] . Following [6], we define a stable algebra to be a subalgebra of R which is closed under formal differentiation ∂ . Notice that we .