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

Let A be an algebra over a field K of characteristic zero, let δ1, . . . , δs ∈ DerK(A) be commuting locally nilpotent K-derivations such that δi(xj) = δij , the Kronecker delta, for some elements x1, . . . , xs ∈ A. A set of algebra generators for the algebra A := ∩i=1ker(δi) is found explicitly and a set of defining relations for the algebra A is described. Similarly, given a set σ1, . . . , σs ∈ AutK(A) of commuting K-automorphisms of the algebra A such that the maps σi − idA are locally nilpotent and σi(xj) = xj + δij , for some elements x1, . . . , xs ∈ A. A set of algebra generators for the algebra A := {a ∈ A |σ1(a) = · · · = σs(a) = a} is found explicitly and a set of defining relations for the algebra A is described. In general, even for a finitely generated noncommutative algebra A the algebras of invariants A and A are not finitely generated, not (left or right) Noetherian and does not satisfy finitely many defining relations (see examples). Though, for a finitely generated commutative algebra A always the opposite is true. The derivations (or automorphisms) just described appear often in may different situations after (possibly) a localization of the algebra A. Mathematics subject classification 2000: 16W22, 13N15, 14R10, 16S15, 16D30.