Enveloping algebras of simple three-dimensional Lie algebras

Abstract

The Killing form of a simple three-dimensional Lie algebra is shown to classify the Lie algebra and its enveloping algebra. The Lie algebra is isomorphic to the two-by-two trace-zero matrices (the so-called split case) if and only if the form is isotropic in the sense of quadratic form theory. In characteristic zero the skew field of quotients of the enveloping algebra is shown to be isomorphic to a Weyl skew field if and only if the Lie algebra is split. Automorphisms of the Lie algebra and its enveloping algebra are induced by isometries of the Killing form in the nonsplit case.