Lie algebras and their universal envelopes.

摘要

The main results of this thesis can be divided into two parts. In the first part, we consider the isomorphism problem for enveloping algebras. Let L be a Lie algebra with universal enveloping algebra U( L). Given another Lie algebra H with the property that U(L) ≅ U( H) as algebras, the isomorphism problem asks what invariants of L are inherited by H. Among other things, we prove that if L is nilpotent then H is nilpotent with the same class as L. We also prove that if L is nilpotent of class at most two then L is isomorphic to H. We also consider the isomorphism problem for restricted enveloping algebras. In contrast to the ordinary Lie algebras, it seems that the restricted case is even more complicated. In passing, we have also given a solution to the Fox problem for free restricted Lie algebras.;In the second part, we study the growth of Lie superalgebras and restricted Lie algebras. We shall see that if L is a finitely generated virtually nilpotent Lie superalgebra then the growth of L is polynomial. In the category of restricted Lie algebras, we extend the seminal work of Passman and Petrogradsky. First we deduce from their results that a finitely generated restricted Lie algebra has polynomial growth precisely when it is virtually nilpotent. Our main result, though, states that the lower central series of every finitely generated restricted Lie algebra with subexponential growth stabilizes. Consequently, there are no residually nilpotent restricted Lie algebras of intermediate growth.