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 and let δ 1 , …, δ s ∈Der K ( A ) be commuting locally nilpotent K ‐derivations such that δ i ( x j ) equals δ ij , the Kronecker delta, for some elements x 1 , …, x s ∈ A . A set of generators for the algebraA δ : = ⋂ i = 1 s ker ( δ i ) is found explicitly and a set of defining relations for the algebra A δ is described. Similarly, let σ 1 , …, σ s ∈ Aut K ( A ) be commuting K ‐automorphisms of the algebra A is given such that the maps σ i − id A are locally nilpotent and σ i ( x j ) = x j + δ ij , for some elements x 1 , …, x s ∈ A . A set of generators for the algebra A σ : = { a ∈ A | σ 1 ( a ) = … = σ s ( a ) = a } is found explicitly and a set of defining relations for the algebra A σ is described. In general, even for a finitely generated non‐commutative algebra A the algebras of invariants A δ and A σ are not finitely generated, not (left or right) Noetherian and a minimal number of defining relations is infinite. However, for a finitely generated commutative algebra A the opposite is always true. The derivations (or automorphisms) just described appear often in many different situations (possibly) after localization of the algebra A .