Invariants of domains under the actions of restricted Lie algebras
The goal of this paper is to examine the relationship between a domain \(R\) and its subring of invariants \(R^L\), under the action of a finite-dimensional restricted Lie algebra \(L\). In Sections 2 and 3 the author proves a series of technical results involving differential identities, restricted enveloping algebras, smash products, and Frobenius algebras. These technical results all lead up to Section 4, where all of this paper's main results relating \(R\) and \(R^L\) are obtained. It is first shown that there exists a nondegenerate \((R^L, R^L)\)-bimodule ``trace-like'' map \(g : R \to R^L\) and therefore \(I \cap R^L \neq 0\), for every ideal \(I \neq 0\) of \(R\). Next, it is shown that there exists a right \(R^L\)-module embedding \(\phi\) of \(R\) into a finite direct sum of copies of \(R^L\). Using the maps \(g\) and \(\phi\), it is obtained a result which is a combination of several of the main results of this paper.
actions of finite-dimensional restricted Lie algebras
scientific article; zbMATH DE number 848964