Abstract:
The concepts of homological epimorphism and Arens–Michael envelope were introduced by Joseph Taylor in the early 1970s (under other names). He also began to study the question of under what conditions the Arens–Michael envelope of the universal enveloping algebra of a finite-dimensional Lie algebra is a homological epimorphism. The question turned out to be unexpectedly difficult and new results appeared only in the early 2000s. In particular, A. Yu. Pirkovskii showed that solvability of the Lie algebra is a necessary condition. The final answer is that this condition is also sufficient. On the other hand, for the algebra of analytic functionals of a Lie group, the assumption of solvability is superfluous – if the group is connected, then the indicated envelope is always a homological epimorphism.
