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

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

2006-04-04

arxiv.org/abs/math/0604083v1

Mathematics

Rings and Algebras

21 pages

Scientific paper

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 $A$.

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