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

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### 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 $$A$$.

### Author and article information

###### Journal
04 April 2006
###### Article
math/0604083