Invariants of Lie superalgebras acting on associative algebras
Let \(R\) be a semiprime algebra over a field \(K\) acted on by a finite-dimensional nilpotent Lie superalgebra \(L\) (such that if \(\text{char }K=p\) then \(L\) is restricted, and if \(\text{char }K=0\) then \(L\) acts on \(R\) as algebraic derivations and algebraic superderivations). The purpose of this paper is to prove a series of ``going up'' results showing what the structure of the subalgebra of invariants \(R^L\) implies with respect to that of \(R\). In addition, these results raise a number of questions, and the authors give counterexamples to many of them. Some of the results in this paper are: If \(R^L\) is right Noetherian (Artinian), then \(R\) is a Noetherian (Artinian) right \(R^L\) module. If \(R^L\) is finite dimensional (has finite Goldie dimension as an \(R^L\)-module), then \(R\) is finite dimensional (has finite Goldie dimension as an \(R\)-module). If \(R\) is prime and \(R^L\) is central, then \(R\) satisfies a polynomial identity. The authors give examples to show that the reverse implications are not always true: for example they give an example of a prime left Noetherian ring \(R\) of arbitrary characteristic with a nilpotent derivation \(\delta\) such that \(R^{(\delta)}\) is not left Noetherian. In another example they show that certain hypotheses on \(R^L\) are needed in order to have \(R\) finitely generated over \(R^L\) even when \(R\) is prime and \(L\) is 1-dimensional.
finite-dimensional nilpotent Lie superalgebras
scientific article; zbMATH DE number 902631