Generators and defining relations for ring of invariants of commuting locally nilpotent derivations or automorphisms

Abstract

Let $A$ be an algebra over a field $K$ of characteristic zero, let $\d_1, >..., \d_s\in \Der_K(A)$ be {\em commuting locally nilpotent} $K$-derivations such that $\d_i(x_j)=\d_{ij}$, the Kronecker delta, for some elements $x_1,..., x_s\in A$. A set of algebra generators for the algebra $A^\d:= \cap_{i=1}^s\ker (\d_i)$ is found {\em explicitly} and a set of {\em defining relations} for the algebra $A^\d$ is described. Similarly, given a set $\s_1, ..., \s_s\in \Aut_K(A)$ of {\em commuting} $K$-automorphisms of the algebra $A$ such that the maps $\s_i-{\rm id_A}$ are {\em locally nilpotent} and $\s_i (x_j)=x_j+\d_{ij}$, for some elements $x_1,..., x_s\in A$. A set of algebra generators for the algebra $A^\s:=\{a\in A | \s_1(a)=... =\s_s(a)=a\}$ is found {\em explicitly} and a set of defining relations for the algebra $A^\s$ is described. In general, even for a {\em finitely generated noncommutative} algebra $A$ the algebras of invariants $A^\d $ and $A^\s $ are {\em not} finitely generated, {\em not} (left or right) Noetherian and {\em does not} satisfy finitely many defining relations (see examples). Though, for a {\em finitely generated commutative} algebra $A$ {\em always} the {\em opposite} is true. The derivations (or automorphisms) just described appear often in may different situations after (possibly) a localization of the algebra $A$.