A rigidity theorem for finite group actions on enveloping algebras of semisimple Lie algebras
Let \(G\) be a finite group of automorphisms acting (not necessarily linearly) on the enveloping algebra \(U ({\mathfrak g})\) of a semisimple Lie algebra \({\mathfrak g}\) over an algebraically closed field \(k\) of characteristic zero. The main result of this paper is that if the fixed subalgebra \(U^ G\) is \(k\)-isomorphic to any enveloping algebra, then \(G\) is trivial; the same result is also shown for the \(n\)th Weyl algebra \(A_ n (k)\). In other words, enveloping algebras of semisimple Lie algebras and Weyl algebras do not admit Galois embeddings into themselves; note that symmetric algebras by contrast do admit such embeddings. The proof of the first result uses certain very precise but quite classical facts about primitive ideals in \(U ({\mathfrak g})\); it actually shows more generally that \(U^ G\) cannot even be a quotient of an enveloping algebra if \(G\) acts trivially on the center of \(U ({\mathfrak g})\) (and \(G \neq 1)\). The second result is proved in two ways, one of which involves passing to prime characteristic and the other higher \(K\)-theory. The second proof is much shorter but of course less elementary and unfortunately applies only to linear actions.
scientific article; zbMATH DE number 759446